3.5.87 \(\int \frac {1}{x^{7/2} (a+b x^2) (c+d x^2)^3} \, dx\) [487]

Optimal. Leaf size=743 \[ -\frac {32 b^2 c^2-189 a b c d+117 a^2 d^2}{80 a c^3 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac {b^{17/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {b^{17/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3} \]

[Out]

1/80*(-117*a^2*d^2+189*a*b*c*d-32*b^2*c^2)/a/c^3/(-a*d+b*c)^2/x^(5/2)-1/4*d/c/(-a*d+b*c)/x^(5/2)/(d*x^2+c)^2-1
/16*d*(-13*a*d+21*b*c)/c^2/(-a*d+b*c)^2/x^(5/2)/(d*x^2+c)-1/2*b^(17/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4
))/a^(9/4)/(-a*d+b*c)^3*2^(1/2)+1/2*b^(17/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a*d+b*c)^3*2^
(1/2)+1/64*d^(9/4)*(117*a^2*d^2-306*a*b*c*d+221*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(17/4)/(-
a*d+b*c)^3*2^(1/2)-1/64*d^(9/4)*(117*a^2*d^2-306*a*b*c*d+221*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4)
)/c^(17/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(17/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a
*d+b*c)^3*2^(1/2)-1/4*b^(17/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)^3*2^(1
/2)-1/128*d^(9/4)*(117*a^2*d^2-306*a*b*c*d+221*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/
c^(17/4)/(-a*d+b*c)^3*2^(1/2)+1/128*d^(9/4)*(117*a^2*d^2-306*a*b*c*d+221*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)
*d^(1/4)*2^(1/2)*x^(1/2))/c^(17/4)/(-a*d+b*c)^3*2^(1/2)+1/16*(117*a^3*d^3-189*a^2*b*c*d^2+32*a*b^2*c^2*d+32*b^
3*c^3)/a^2/c^4/(-a*d+b*c)^2/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.83, antiderivative size = 743, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {477, 483, 593, 597, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {b^{17/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {117 a^2 d^2-189 a b c d+32 b^2 c^2}{80 a c^3 x^{5/2} (b c-a d)^2}+\frac {117 a^3 d^3-189 a^2 b c d^2+32 a b^2 c^2 d+32 b^3 c^3}{16 a^2 c^4 \sqrt {x} (b c-a d)^2}-\frac {d (21 b c-13 a d)}{16 c^2 x^{5/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac {d}{4 c x^{5/2} \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-1/80*(32*b^2*c^2 - 189*a*b*c*d + 117*a^2*d^2)/(a*c^3*(b*c - a*d)^2*x^(5/2)) + (32*b^3*c^3 + 32*a*b^2*c^2*d -
189*a^2*b*c*d^2 + 117*a^3*d^3)/(16*a^2*c^4*(b*c - a*d)^2*Sqrt[x]) - d/(4*c*(b*c - a*d)*x^(5/2)*(c + d*x^2)^2)
- (d*(21*b*c - 13*a*d))/(16*c^2*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)) - (b^(17/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b*c - a*d)^3) + (b^(17/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sq
rt[2]*a^(9/4)*(b*c - a*d)^3) + (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*
Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^3) - (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*Ar
cTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(17/4)*(b*c - a*d)^3) + (b^(17/4)*Log[Sqrt[a] - Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (b^(17/4)*Log[Sqrt[a] + Sqrt[2]
*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)^3) - (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d
+ 117*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^3
) + (d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]
*x])/(64*Sqrt[2]*c^(17/4)*(b*c - a*d)^3)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 483

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*(e*
x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a*d)
*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n
*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ
[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}+\frac {\text {Subst}\left (\int \frac {8 b c-13 a d-13 b d x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)}\\ &=-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {32 b^2 c^2-189 a b c d+117 a^2 d^2-9 b d (21 b c-13 a d) x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {5 \left (32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3\right )+5 b d \left (32 b^2 c^2-189 a b c d+117 a^2 d^2\right ) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{80 a c^3 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (5 \left (32 b^4 c^4+32 a b^3 c^3 d+32 a^2 b^2 c^2 d^2-189 a^3 b c d^3+117 a^4 d^4\right )+5 b d \left (32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{80 a^2 c^4 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {160 b^5 c^4 x^2}{(b c-a d) \left (a+b x^4\right )}+\frac {5 a^2 d^3 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{80 a^2 c^4 (b c-a d)^2}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {\left (2 b^5\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^3}-\frac {\left (d^3 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^4 (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac {b^{9/2} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^3}+\frac {b^{9/2} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^3}+\frac {\left (d^{5/2} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^4 (b c-a d)^3}-\frac {\left (d^{5/2} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^4 (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^2 (b c-a d)^3}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 a^2 (b c-a d)^3}+\frac {b^{17/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {\left (d^2 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^4 (b c-a d)^3}-\frac {\left (d^2 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^4 (b c-a d)^3}-\frac {\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b^{17/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {b^{17/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}-\frac {\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}\\ &=-\frac {\frac {32 b^2 c}{a}-189 b d+\frac {117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac {32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt {x}}-\frac {d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac {d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac {b^{17/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {b^{17/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {b^{17/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {b^{17/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^3}-\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}+\frac {d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{17/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 1.18, size = 462, normalized size = 0.62 \begin {gather*} \frac {1}{320} \left (\frac {4 \left (160 b^3 c^3 x^2 \left (c+d x^2\right )^2-32 a b^2 c^2 \left (c-5 d x^2\right ) \left (c+d x^2\right )^2+a^2 b c d \left (64 c^3-672 c^2 d x^2-1701 c d^2 x^4-945 d^3 x^6\right )+a^3 d^2 \left (-32 c^3+416 c^2 d x^2+1053 c d^2 x^4+585 d^3 x^6\right )\right )}{a^2 c^4 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac {160 \sqrt {2} b^{17/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4} (-b c+a d)^3}+\frac {5 \sqrt {2} d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{17/4} (b c-a d)^3}+\frac {160 \sqrt {2} b^{17/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4} (-b c+a d)^3}+\frac {5 \sqrt {2} d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{17/4} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((4*(160*b^3*c^3*x^2*(c + d*x^2)^2 - 32*a*b^2*c^2*(c - 5*d*x^2)*(c + d*x^2)^2 + a^2*b*c*d*(64*c^3 - 672*c^2*d*
x^2 - 1701*c*d^2*x^4 - 945*d^3*x^6) + a^3*d^2*(-32*c^3 + 416*c^2*d*x^2 + 1053*c*d^2*x^4 + 585*d^3*x^6)))/(a^2*
c^4*(b*c - a*d)^2*x^(5/2)*(c + d*x^2)^2) + (160*Sqrt[2]*b^(17/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x])])/(a^(9/4)*(-(b*c) + a*d)^3) + (5*Sqrt[2]*d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*
ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(17/4)*(b*c - a*d)^3) + (160*Sqrt[2]*b^(17
/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(9/4)*(-(b*c) + a*d)^3) + (5*Sqrt[2]*
d^(9/4)*(221*b^2*c^2 - 306*a*b*c*d + 117*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]
*x)])/(c^(17/4)*(b*c - a*d)^3))/320

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Maple [A]
time = 0.23, size = 368, normalized size = 0.50 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-1/4*b^4/a^2/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^
(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))+2
*d^3/c^4/(a*d-b*c)^3*((1/32*d*(21*a^2*d^2-50*a*b*c*d+29*b^2*c^2)*x^(7/2)+(25/32*a^2*c*d^2-29/16*a*b*c^2*d+33/3
2*b^2*c^3)*x^(3/2))/(d*x^2+c)^2+1/8*(117/32*a^2*d^2-153/16*a*b*c*d+221/32*b^2*c^2)/d/(c/d)^(1/4)*2^(1/2)*(ln((
x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)
^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))-2/5/a/c^3/x^(5/2)-2*(-3*a*d-b*c)/a^2/c^4/x^(1/2)

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Maxima [A]
time = 0.61, size = 756, normalized size = 1.02 \begin {gather*} \frac {b^{5} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )}} - \frac {{\left (221 \, b^{2} c^{2} d^{3} - 306 \, a b c d^{4} + 117 \, a^{2} d^{5}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, {\left (b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 3 \, a^{2} b c^{5} d^{2} - a^{3} c^{4} d^{3}\right )}} - \frac {32 \, a b^{2} c^{5} - 64 \, a^{2} b c^{4} d + 32 \, a^{3} c^{3} d^{2} - 5 \, {\left (32 \, b^{3} c^{3} d^{2} + 32 \, a b^{2} c^{2} d^{3} - 189 \, a^{2} b c d^{4} + 117 \, a^{3} d^{5}\right )} x^{6} - {\left (320 \, b^{3} c^{4} d + 288 \, a b^{2} c^{3} d^{2} - 1701 \, a^{2} b c^{2} d^{3} + 1053 \, a^{3} c d^{4}\right )} x^{4} - 32 \, {\left (5 \, b^{3} c^{5} + 3 \, a b^{2} c^{4} d - 21 \, a^{2} b c^{3} d^{2} + 13 \, a^{3} c^{2} d^{3}\right )} x^{2}}{80 \, {\left ({\left (a^{2} b^{2} c^{6} d^{2} - 2 \, a^{3} b c^{5} d^{3} + a^{4} c^{4} d^{4}\right )} x^{\frac {13}{2}} + 2 \, {\left (a^{2} b^{2} c^{7} d - 2 \, a^{3} b c^{6} d^{2} + a^{4} c^{5} d^{3}\right )} x^{\frac {9}{2}} + {\left (a^{2} b^{2} c^{8} - 2 \, a^{3} b c^{7} d + a^{4} c^{6} d^{2}\right )} x^{\frac {5}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/4*b^5*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sq
rt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqr
t(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x
+ sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^
(3/4)))/(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) - 1/128*(221*b^2*c^2*d^3 - 306*a*b*c*d^4 + 1
17*a^2*d^5)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))
/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))
/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d
)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4
)*d^(3/4)))/(b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3) - 1/80*(32*a*b^2*c^5 - 64*a^2*b*c^4*d +
32*a^3*c^3*d^2 - 5*(32*b^3*c^3*d^2 + 32*a*b^2*c^2*d^3 - 189*a^2*b*c*d^4 + 117*a^3*d^5)*x^6 - (320*b^3*c^4*d +
288*a*b^2*c^3*d^2 - 1701*a^2*b*c^2*d^3 + 1053*a^3*c*d^4)*x^4 - 32*(5*b^3*c^5 + 3*a*b^2*c^4*d - 21*a^2*b*c^3*d^
2 + 13*a^3*c^2*d^3)*x^2)/((a^2*b^2*c^6*d^2 - 2*a^3*b*c^5*d^3 + a^4*c^4*d^4)*x^(13/2) + 2*(a^2*b^2*c^7*d - 2*a^
3*b*c^6*d^2 + a^4*c^5*d^3)*x^(9/2) + (a^2*b^2*c^8 - 2*a^3*b*c^7*d + a^4*c^6*d^2)*x^(5/2))

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(7/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]
time = 2.92, size = 1000, normalized size = 1.35 \begin {gather*} \frac {\left (a b^{3}\right )^{\frac {3}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} b^{2} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b^{2} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{3} b^{3} c^{3} - 3 \, \sqrt {2} a^{4} b^{2} c^{2} d + 3 \, \sqrt {2} a^{5} b c d^{2} - \sqrt {2} a^{6} d^{3}\right )}} - \frac {{\left (221 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 306 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{32 \, {\left (\sqrt {2} b^{3} c^{8} - 3 \, \sqrt {2} a b^{2} c^{7} d + 3 \, \sqrt {2} a^{2} b c^{6} d^{2} - \sqrt {2} a^{3} c^{5} d^{3}\right )}} - \frac {{\left (221 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 306 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{32 \, {\left (\sqrt {2} b^{3} c^{8} - 3 \, \sqrt {2} a b^{2} c^{7} d + 3 \, \sqrt {2} a^{2} b c^{6} d^{2} - \sqrt {2} a^{3} c^{5} d^{3}\right )}} + \frac {{\left (221 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 306 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{3} c^{8} - 3 \, \sqrt {2} a b^{2} c^{7} d + 3 \, \sqrt {2} a^{2} b c^{6} d^{2} - \sqrt {2} a^{3} c^{5} d^{3}\right )}} - \frac {{\left (221 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 306 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 117 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{3} c^{8} - 3 \, \sqrt {2} a b^{2} c^{7} d + 3 \, \sqrt {2} a^{2} b c^{6} d^{2} - \sqrt {2} a^{3} c^{5} d^{3}\right )}} - \frac {29 \, b c d^{4} x^{\frac {7}{2}} - 21 \, a d^{5} x^{\frac {7}{2}} + 33 \, b c^{2} d^{3} x^{\frac {3}{2}} - 25 \, a c d^{4} x^{\frac {3}{2}}}{16 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} + \frac {2 \, {\left (5 \, b c x^{2} + 15 \, a d x^{2} - a c\right )}}{5 \, a^{2} c^{4} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

(a*b^3)^(3/4)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^3*c^3 - 3*s
qrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) + (a*b^3)^(3/4)*b^2*arctan(-1/2*sqrt(2)*(sqrt(
2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^
2 - sqrt(2)*a^6*d^3) - 1/2*(a*b^3)^(3/4)*b^2*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^3
*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 - sqrt(2)*a^6*d^3) + 1/2*(a*b^3)^(3/4)*b^2*log(-sqrt(2)
*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^3*c^3 - 3*sqrt(2)*a^4*b^2*c^2*d + 3*sqrt(2)*a^5*b*c*d^2 -
sqrt(2)*a^6*d^3) - 1/32*(221*(c*d^3)^(3/4)*b^2*c^2 - 306*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*a
rctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^8 - 3*sqrt(2)*a*b^2*c^7*d + 3*
sqrt(2)*a^2*b*c^6*d^2 - sqrt(2)*a^3*c^5*d^3) - 1/32*(221*(c*d^3)^(3/4)*b^2*c^2 - 306*(c*d^3)^(3/4)*a*b*c*d + 1
17*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^8
- 3*sqrt(2)*a*b^2*c^7*d + 3*sqrt(2)*a^2*b*c^6*d^2 - sqrt(2)*a^3*c^5*d^3) + 1/64*(221*(c*d^3)^(3/4)*b^2*c^2 - 3
06*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2
)*b^3*c^8 - 3*sqrt(2)*a*b^2*c^7*d + 3*sqrt(2)*a^2*b*c^6*d^2 - sqrt(2)*a^3*c^5*d^3) - 1/64*(221*(c*d^3)^(3/4)*b
^2*c^2 - 306*(c*d^3)^(3/4)*a*b*c*d + 117*(c*d^3)^(3/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/
d))/(sqrt(2)*b^3*c^8 - 3*sqrt(2)*a*b^2*c^7*d + 3*sqrt(2)*a^2*b*c^6*d^2 - sqrt(2)*a^3*c^5*d^3) - 1/16*(29*b*c*d
^4*x^(7/2) - 21*a*d^5*x^(7/2) + 33*b*c^2*d^3*x^(3/2) - 25*a*c*d^4*x^(3/2))/((b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d
^2)*(d*x^2 + c)^2) + 2/5*(5*b*c*x^2 + 15*a*d*x^2 - a*c)/(a^2*c^4*x^(5/2))

________________________________________________________________________________________

Mupad [B]
time = 7.03, size = 2500, normalized size = 3.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^3),x)

[Out]

((2*x^2*(13*a*d + 5*b*c))/(5*a^2*c^2) - 2/(5*a*c) + (x^4*(1053*a^3*d^4 + 320*b^3*c^3*d + 288*a*b^2*c^2*d^2 - 1
701*a^2*b*c*d^3))/(80*a^2*c^2*(b^2*c^3 + a^2*c*d^2 - 2*a*b*c^2*d)) + (d^2*x^6*(117*a^3*d^3 + 32*b^3*c^3 + 32*a
*b^2*c^2*d - 189*a^2*b*c*d^2))/(16*a^2*c^3*(b^2*c^3 + a^2*c*d^2 - 2*a*b*c^2*d)))/(c^2*x^(5/2) + d^2*x^(13/2) +
2*c*d*x^(9/2)) - atan((a^11*b^22*c^29*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^10*b^11*c^11*d
+ 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d^5 + 14784*a^1
5*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^1
0 - 192*a^20*b*c*d^11))^(5/4)*33554432i + a^19*b^10*d^17*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192
*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c
^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 105
6*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(1/4)*374777442i + a^33*c^7*d^22*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a
^9*b^12*c^12 - 192*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4
- 12672*a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18
*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(5/4)*448561152i + a^8*b^21*c^11*d^6*x^(1/2)*(-b^1
7/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 +
7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b
^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(1/4)*100026368i - a^9*b^20*
c^10*d^7*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 35
20*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5
*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(1/4)*
276996096i + a^10*b^19*c^9*d^8*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^10*b^11*c^11*d + 1056*a
^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^
6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*
a^20*b*c*d^11))^(1/4)*297676800i + a^11*b^18*c^8*d^9*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^1
0*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d
^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^
19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(1/4)*4624241570i - a^12*b^17*c^7*d^10*x^(1/2)*(-b^17/(16*a^21*d^12 + 16
*a^9*b^12*c^12 - 192*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^
4 - 12672*a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^
18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(1/4)*26395336656i + a^13*b^16*c^6*d^11*x^(1/2)*
(-b^17/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d
^3 + 7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a
^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(1/4)*64982364408i - a^
14*b^15*c^5*d^12*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*
d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*
a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11)
)^(1/4)*92624356656i + a^15*b^14*c^4*d^13*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^10*b^11*c^11
*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d^5 + 14784*
a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*
d^10 - 192*a^20*b*c*d^11))^(1/4)*83665919628i - a^16*b^13*c^3*d^14*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^12*
c^12 - 192*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*a^13*b^8*c^8*d^4 - 12672*
a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^4*d^8 - 3520*a^18*b^3*c^3
*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(1/4)*49036424112i + a^17*b^12*c^2*d^15*x^(1/2)*(-b^17/(16
*a^21*d^12 + 16*a^9*b^12*c^12 - 192*a^10*b^11*c^11*d + 1056*a^11*b^10*c^10*d^2 - 3520*a^12*b^9*c^9*d^3 + 7920*
a^13*b^8*c^8*d^4 - 12672*a^14*b^7*c^7*d^5 + 14784*a^15*b^6*c^6*d^6 - 12672*a^16*b^5*c^5*d^7 + 7920*a^17*b^4*c^
4*d^8 - 3520*a^18*b^3*c^3*d^9 + 1056*a^19*b^2*c^2*d^10 - 192*a^20*b*c*d^11))^(1/4)*18213050232i + a^13*b^20*c^
27*d^2*x^(1/2)*(-b^17/(16*a^21*d^12 + 16*a^9*b^...

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