3.5.95 \(\int \frac {1}{x^{7/2} (a+b x^2)^2 (c+d x^2)^2} \, dx\) [495]

Optimal. Leaf size=731 \[ \frac {-9 b^2 c^2+8 a b c d-9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {(b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{13/4} (9 b c-17 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {b^{13/4} (9 b c-17 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} (b c-a d)^3}-\frac {d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {b^{13/4} (9 b c-17 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}-\frac {b^{13/4} (9 b c-17 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {d^{13/4} (17 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {d^{13/4} (17 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^3} \]

[Out]

1/10*(-9*a^2*d^2+8*a*b*c*d-9*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/x^(5/2)+1/2*d*(a*d+b*c)/a/c/(-a*d+b*c)^2/x^(5/2)/(d
*x^2+c)+1/2*b/a/(-a*d+b*c)/x^(5/2)/(b*x^2+a)/(d*x^2+c)-1/8*b^(13/4)*(-17*a*d+9*b*c)*arctan(1-b^(1/4)*2^(1/2)*x
^(1/2)/a^(1/4))/a^(13/4)/(-a*d+b*c)^3*2^(1/2)+1/8*b^(13/4)*(-17*a*d+9*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^
(1/4))/a^(13/4)/(-a*d+b*c)^3*2^(1/2)-1/8*d^(13/4)*(-9*a*d+17*b*c)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^
(13/4)/(-a*d+b*c)^3*2^(1/2)+1/8*d^(13/4)*(-9*a*d+17*b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-
a*d+b*c)^3*2^(1/2)+1/16*b^(13/4)*(-17*a*d+9*b*c)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(13/4
)/(-a*d+b*c)^3*2^(1/2)-1/16*b^(13/4)*(-17*a*d+9*b*c)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(
13/4)/(-a*d+b*c)^3*2^(1/2)+1/16*d^(13/4)*(-9*a*d+17*b*c)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))
/c^(13/4)/(-a*d+b*c)^3*2^(1/2)-1/16*d^(13/4)*(-9*a*d+17*b*c)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1
/2))/c^(13/4)/(-a*d+b*c)^3*2^(1/2)+1/2*(a*d+b*c)*(9*a^2*d^2-17*a*b*c*d+9*b^2*c^2)/a^3/c^3/(-a*d+b*c)^2/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.86, antiderivative size = 731, 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^{13/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (9 b c-17 a d)}{4 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {b^{13/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (9 b c-17 a d)}{4 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {b^{13/4} (9 b c-17 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}-\frac {b^{13/4} (9 b c-17 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}-\frac {9 a^2 d^2-8 a b c d+9 b^2 c^2}{10 a^2 c^2 x^{5/2} (b c-a d)^2}+\frac {(a d+b c) \left (9 a^2 d^2-17 a b c d+9 b^2 c^2\right )}{2 a^3 c^3 \sqrt {x} (b c-a d)^2}-\frac {d^{13/4} (17 b c-9 a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {d^{13/4} (17 b c-9 a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {d^{13/4} (17 b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {d^{13/4} (17 b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {b}{2 a x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d (a d+b c)}{2 a c x^{5/2} \left (c+d x^2\right ) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/10*(9*b^2*c^2 - 8*a*b*c*d + 9*a^2*d^2)/(a^2*c^2*(b*c - a*d)^2*x^(5/2)) + ((b*c + a*d)*(9*b^2*c^2 - 17*a*b*c
*d + 9*a^2*d^2))/(2*a^3*c^3*(b*c - a*d)^2*Sqrt[x]) + (d*(b*c + a*d))/(2*a*c*(b*c - a*d)^2*x^(5/2)*(c + d*x^2))
+ b/(2*a*(b*c - a*d)*x^(5/2)*(a + b*x^2)*(c + d*x^2)) - (b^(13/4)*(9*b*c - 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) + (b^(13/4)*(9*b*c - 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/
4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) - (d^(13/4)*(17*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1
/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (d^(13/4)*(17*b*c - 9*a*d)*ArcTan[1 + (Sqrt[2]*d^(
1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) + (b^(13/4)*(9*b*c - 17*a*d)*Log[Sqrt[a] - Sqrt[2]*
a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) - (b^(13/4)*(9*b*c - 17*a*d)*Log[Sqrt
[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*(b*c - a*d)^3) + (d^(13/4)*(17*b*c - 9
*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(b*c - a*d)^3) - (d^(13/
4)*(17*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13/4)*(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 )^2 \left (c+d x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^6 \left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-9 b c+4 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 )}{2 a (b c-a d)}\\ &=\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-4 \left (9 b^2 c^2-8 a b c d+9 a^2 d^2\right )-36 b d (b c+a d) x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 a c (b c-a d)^2}\\ &=-\frac {9 b^2 c^2-8 a b c d+9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {-20 (b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )-20 b d \left (9 b^2 c^2-8 a b c d+9 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 )}{40 a^2 c^2 (b c-a d)^2}\\ &=-\frac {9 b^2 c^2-8 a b c d+9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {(b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (-20 \left (9 b^4 c^4-8 a b^3 c^3 d-8 a^2 b^2 c^2 d^2-8 a^3 b c d^3+9 a^4 d^4\right )-20 b d (b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{40 a^3 c^3 (b c-a d)^2}\\ &=-\frac {9 b^2 c^2-8 a b c d+9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {(b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \left (-\frac {20 b^4 c^3 (9 b c-17 a d) x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {20 a^3 d^4 (-17 b c+9 a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{40 a^3 c^3 (b c-a d)^2}\\ &=-\frac {9 b^2 c^2-8 a b c d+9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {(b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^4 (9 b c-17 a d)\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^3 (b c-a d)^3}+\frac {\left (d^4 (17 b c-9 a d)\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^3 (b c-a d)^3}\\ &=-\frac {9 b^2 c^2-8 a b c d+9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {(b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (b^{7/2} (9 b c-17 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^3 (b c-a d)^3}+\frac {\left (b^{7/2} (9 b c-17 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^3 (b c-a d)^3}-\frac {\left (d^{7/2} (17 b c-9 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^3 (b c-a d)^3}+\frac {\left (d^{7/2} (17 b c-9 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^3 (b c-a d)^3}\\ &=-\frac {9 b^2 c^2-8 a b c d+9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {(b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^3 (9 b c-17 a d)\right ) \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 )}{8 a^3 (b c-a d)^3}+\frac {\left (b^3 (9 b c-17 a d)\right ) \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 )}{8 a^3 (b c-a d)^3}+\frac {\left (b^{13/4} (9 b c-17 a d)\right ) \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 )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {\left (b^{13/4} (9 b c-17 a d)\right ) \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 )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {\left (d^3 (17 b c-9 a d)\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 )}{8 c^3 (b c-a d)^3}+\frac {\left (d^3 (17 b c-9 a d)\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 )}{8 c^3 (b c-a d)^3}+\frac {\left (d^{13/4} (17 b c-9 a d)\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 )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {\left (d^{13/4} (17 b c-9 a d)\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 )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}\\ &=-\frac {9 b^2 c^2-8 a b c d+9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {(b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{13/4} (9 b c-17 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}-\frac {b^{13/4} (9 b c-17 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {d^{13/4} (17 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {d^{13/4} (17 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {\left (b^{13/4} (9 b c-17 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} (b c-a d)^3}-\frac {\left (b^{13/4} (9 b c-17 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {\left (d^{13/4} (17 b c-9 a d)\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 )}{4 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {\left (d^{13/4} (17 b c-9 a d)\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 )}{4 \sqrt {2} c^{13/4} (b c-a d)^3}\\ &=-\frac {9 b^2 c^2-8 a b c d+9 a^2 d^2}{10 a^2 c^2 (b c-a d)^2 x^{5/2}}+\frac {(b c+a d) \left (9 b^2 c^2-17 a b c d+9 a^2 d^2\right )}{2 a^3 c^3 (b c-a d)^2 \sqrt {x}}+\frac {d (b c+a d)}{2 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac {b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{13/4} (9 b c-17 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {b^{13/4} (9 b c-17 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4} (b c-a d)^3}-\frac {d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {d^{13/4} (17 b c-9 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^3}+\frac {b^{13/4} (9 b c-17 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}-\frac {b^{13/4} (9 b c-17 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4} (b c-a d)^3}+\frac {d^{13/4} (17 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}-\frac {d^{13/4} (17 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 1.33, size = 460, normalized size = 0.63 \begin {gather*} \frac {\frac {4 (b c-a d) \left (45 b^4 c^3 x^4 \left (c+d x^2\right )-4 a b^3 c^2 x^2 \left (-9 c^2+c d x^2+10 d^2 x^4\right )+a^4 d^2 \left (-4 c^2+36 c d x^2+45 d^2 x^4\right )-4 a^2 b^2 c \left (c^3+9 c^2 d x^2+18 c d^2 x^4+10 d^3 x^6\right )+a^3 b d \left (8 c^3-36 c^2 d x^2-4 c d^2 x^4+45 d^3 x^6\right )\right )}{a^3 c^3 x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {5 \sqrt {2} b^{13/4} (-9 b c+17 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{13/4}}+\frac {5 \sqrt {2} d^{13/4} (-17 b c+9 a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{13/4}}+\frac {5 \sqrt {2} b^{13/4} (-9 b c+17 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{13/4}}+\frac {5 \sqrt {2} d^{13/4} (-17 b c+9 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{13/4}}}{40 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(b*c - a*d)*(45*b^4*c^3*x^4*(c + d*x^2) - 4*a*b^3*c^2*x^2*(-9*c^2 + c*d*x^2 + 10*d^2*x^4) + a^4*d^2*(-4*c^
2 + 36*c*d*x^2 + 45*d^2*x^4) - 4*a^2*b^2*c*(c^3 + 9*c^2*d*x^2 + 18*c*d^2*x^4 + 10*d^3*x^6) + a^3*b*d*(8*c^3 -
36*c^2*d*x^2 - 4*c*d^2*x^4 + 45*d^3*x^6)))/(a^3*c^3*x^(5/2)*(a + b*x^2)*(c + d*x^2)) + (5*Sqrt[2]*b^(13/4)*(-9
*b*c + 17*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(13/4) + (5*Sqrt[2]*d^(13/4)
*(-17*b*c + 9*a*d)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/c^(13/4) + (5*Sqrt[2]*b^(1
3/4)*(-9*b*c + 17*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(13/4) + (5*Sqrt[2]
*d^(13/4)*(-17*b*c + 9*a*d)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/c^(13/4))/(40*(b
*c - a*d)^3)

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Maple [A]
time = 0.20, size = 343, normalized size = 0.47 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^4/a^3/(a*d-b*c)^3*((1/4*a*d-1/4*b*c)*x^(3/2)/(b*x^2+a)+1/8*(17/4*a*d-9/4*b*c)/b/(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^4/c^3/(a*d-b*c)^3*((1/4*a*d-1/4*b*c)*x^(3/2)/(d
*x^2+c)+1/8*(9/4*a*d-17/4*b*c)/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^2/c^2/x^(5/2)-2*(-2*a*d-2*b*c)/a^3/c^3/x^(1/2)

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Maxima [A]
time = 0.58, size = 774, normalized size = 1.06 \begin {gather*} \frac {{\left (9 \, b^{5} c - 17 \, a b^{4} d\right )} {\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 )}}{16 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )}} + \frac {{\left (17 \, b c d^{4} - 9 \, a 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 )}}{16 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )}} - \frac {4 \, a^{2} b^{2} c^{4} - 8 \, a^{3} b c^{3} d + 4 \, a^{4} c^{2} d^{2} - 5 \, {\left (9 \, b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} - 8 \, a^{2} b^{2} c d^{3} + 9 \, a^{3} b d^{4}\right )} x^{6} - {\left (45 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 72 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 45 \, a^{4} d^{4}\right )} x^{4} - 36 \, {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}}{10 \, {\left ({\left (a^{3} b^{3} c^{5} d - 2 \, a^{4} b^{2} c^{4} d^{2} + a^{5} b c^{3} d^{3}\right )} x^{\frac {13}{2}} + {\left (a^{3} b^{3} c^{6} - a^{4} b^{2} c^{5} d - a^{5} b c^{4} d^{2} + a^{6} c^{3} d^{3}\right )} x^{\frac {9}{2}} + {\left (a^{4} b^{2} c^{6} - 2 \, a^{5} b c^{5} d + a^{6} c^{4} 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)^2/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

1/16*(9*b^5*c - 17*a*b^4*d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(s
qrt(a)*sqrt(b)))/(sqrt(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))/sqrt(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^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3) + 1/16*(17*b*c*d^4 - 9*a
*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^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3) - 1/10*(4*a^2*b^2*c^4 - 8*a^3*b*c^3*d + 4*a^4*
c^2*d^2 - 5*(9*b^4*c^3*d - 8*a*b^3*c^2*d^2 - 8*a^2*b^2*c*d^3 + 9*a^3*b*d^4)*x^6 - (45*b^4*c^4 - 4*a*b^3*c^3*d
- 72*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 45*a^4*d^4)*x^4 - 36*(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c
*d^3)*x^2)/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^(13/2) + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*
b*c^4*d^2 + a^6*c^3*d^3)*x^(9/2) + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*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)^2/(d*x^2+c)^2,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)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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Giac [A]
time = 2.16, size = 1015, normalized size = 1.39 \begin {gather*} \frac {{\left (9 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{2} c - 17 \, \left (a b^{3}\right )^{\frac {3}{4}} a b d\right )} \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 )}{4 \, {\left (\sqrt {2} a^{4} b^{3} c^{3} - 3 \, \sqrt {2} a^{5} b^{2} c^{2} d + 3 \, \sqrt {2} a^{6} b c d^{2} - \sqrt {2} a^{7} d^{3}\right )}} + \frac {{\left (9 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{2} c - 17 \, \left (a b^{3}\right )^{\frac {3}{4}} a b d\right )} \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 )}{4 \, {\left (\sqrt {2} a^{4} b^{3} c^{3} - 3 \, \sqrt {2} a^{5} b^{2} c^{2} d + 3 \, \sqrt {2} a^{6} b c d^{2} - \sqrt {2} a^{7} d^{3}\right )}} + \frac {{\left (17 \, \left (c d^{3}\right )^{\frac {3}{4}} b c d - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a 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 )}{4 \, {\left (\sqrt {2} b^{3} c^{7} - 3 \, \sqrt {2} a b^{2} c^{6} d + 3 \, \sqrt {2} a^{2} b c^{5} d^{2} - \sqrt {2} a^{3} c^{4} d^{3}\right )}} + \frac {{\left (17 \, \left (c d^{3}\right )^{\frac {3}{4}} b c d - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a 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 )}{4 \, {\left (\sqrt {2} b^{3} c^{7} - 3 \, \sqrt {2} a b^{2} c^{6} d + 3 \, \sqrt {2} a^{2} b c^{5} d^{2} - \sqrt {2} a^{3} c^{4} d^{3}\right )}} - \frac {{\left (9 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{2} c - 17 \, \left (a b^{3}\right )^{\frac {3}{4}} a b d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a^{4} b^{3} c^{3} - 3 \, \sqrt {2} a^{5} b^{2} c^{2} d + 3 \, \sqrt {2} a^{6} b c d^{2} - \sqrt {2} a^{7} d^{3}\right )}} + \frac {{\left (9 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{2} c - 17 \, \left (a b^{3}\right )^{\frac {3}{4}} a b d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a^{4} b^{3} c^{3} - 3 \, \sqrt {2} a^{5} b^{2} c^{2} d + 3 \, \sqrt {2} a^{6} b c d^{2} - \sqrt {2} a^{7} d^{3}\right )}} - \frac {{\left (17 \, \left (c d^{3}\right )^{\frac {3}{4}} b c d - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{3} c^{7} - 3 \, \sqrt {2} a b^{2} c^{6} d + 3 \, \sqrt {2} a^{2} b c^{5} d^{2} - \sqrt {2} a^{3} c^{4} d^{3}\right )}} + \frac {{\left (17 \, \left (c d^{3}\right )^{\frac {3}{4}} b c d - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{3} c^{7} - 3 \, \sqrt {2} a b^{2} c^{6} d + 3 \, \sqrt {2} a^{2} b c^{5} d^{2} - \sqrt {2} a^{3} c^{4} d^{3}\right )}} + \frac {b^{4} c^{3} d x^{\frac {7}{2}} + a^{3} b d^{4} x^{\frac {7}{2}} + b^{4} c^{4} x^{\frac {3}{2}} + a^{4} d^{4} x^{\frac {3}{2}}}{2 \, {\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2}\right )} {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )}} + \frac {2 \, {\left (10 \, b c x^{2} + 10 \, a d x^{2} - a c\right )}}{5 \, a^{3} c^{3} 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)^2/(d*x^2+c)^2,x, algorithm="giac")

[Out]

1/4*(9*(a*b^3)^(3/4)*b^2*c - 17*(a*b^3)^(3/4)*a*b*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b
)^(1/4))/(sqrt(2)*a^4*b^3*c^3 - 3*sqrt(2)*a^5*b^2*c^2*d + 3*sqrt(2)*a^6*b*c*d^2 - sqrt(2)*a^7*d^3) + 1/4*(9*(a
*b^3)^(3/4)*b^2*c - 17*(a*b^3)^(3/4)*a*b*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))
/(sqrt(2)*a^4*b^3*c^3 - 3*sqrt(2)*a^5*b^2*c^2*d + 3*sqrt(2)*a^6*b*c*d^2 - sqrt(2)*a^7*d^3) + 1/4*(17*(c*d^3)^(
3/4)*b*c*d - 9*(c*d^3)^(3/4)*a*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^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1/4*(17*(c*d^3)^(3/4)*b*c*
d - 9*(c*d^3)^(3/4)*a*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^7
- 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) - 1/8*(9*(a*b^3)^(3/4)*b^2*c - 17*(a
*b^3)^(3/4)*a*b*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^4*b^3*c^3 - 3*sqrt(2)*a^5*b^2*c
^2*d + 3*sqrt(2)*a^6*b*c*d^2 - sqrt(2)*a^7*d^3) + 1/8*(9*(a*b^3)^(3/4)*b^2*c - 17*(a*b^3)^(3/4)*a*b*d)*log(-sq
rt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^4*b^3*c^3 - 3*sqrt(2)*a^5*b^2*c^2*d + 3*sqrt(2)*a^6*b*c*
d^2 - sqrt(2)*a^7*d^3) - 1/8*(17*(c*d^3)^(3/4)*b*c*d - 9*(c*d^3)^(3/4)*a*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4)
+ x + sqrt(c/d))/(sqrt(2)*b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1
/8*(17*(c*d^3)^(3/4)*b*c*d - 9*(c*d^3)^(3/4)*a*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)
*b^3*c^7 - 3*sqrt(2)*a*b^2*c^6*d + 3*sqrt(2)*a^2*b*c^5*d^2 - sqrt(2)*a^3*c^4*d^3) + 1/2*(b^4*c^3*d*x^(7/2) + a
^3*b*d^4*x^(7/2) + b^4*c^4*x^(3/2) + a^4*d^4*x^(3/2))/((a^3*b^2*c^5 - 2*a^4*b*c^4*d + a^5*c^3*d^2)*(b*d*x^4 +
b*c*x^2 + a*d*x^2 + a*c)) + 2/5*(10*b*c*x^2 + 10*a*d*x^2 - a*c)/(a^3*c^3*x^(5/2))

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Mupad [B]
time = 6.62, size = 2500, normalized size = 3.42 \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)^2*(c + d*x^2)^2),x)

[Out]

2*atan((2654208*a^16*b^22*c^27*x^(1/2)*(-(6561*b^17*c^4 + 83521*a^4*b^13*d^4 - 176868*a^3*b^14*c*d^3 + 140454*
a^2*b^15*c^2*d^2 - 49572*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a^13*b^12*c^12 - 49152*a^14*b^11*c^11*d + 270336
*a^15*b^10*c^10*d^2 - 901120*a^16*b^9*c^9*d^3 + 2027520*a^17*b^8*c^8*d^4 - 3244032*a^18*b^7*c^7*d^5 + 3784704*
a^19*b^6*c^6*d^6 - 3244032*a^20*b^5*c^5*d^7 + 2027520*a^21*b^4*c^4*d^8 - 901120*a^22*b^3*c^3*d^9 + 270336*a^23
*b^2*c^2*d^10 - 49152*a^24*b*c*d^11))^(5/4) + 15169032*a^22*b^8*d^19*x^(1/2)*(-(6561*b^17*c^4 + 83521*a^4*b^13
*d^4 - 176868*a^3*b^14*c*d^3 + 140454*a^2*b^15*c^2*d^2 - 49572*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a^13*b^12*
c^12 - 49152*a^14*b^11*c^11*d + 270336*a^15*b^10*c^10*d^2 - 901120*a^16*b^9*c^9*d^3 + 2027520*a^17*b^8*c^8*d^4
- 3244032*a^18*b^7*c^7*d^5 + 3784704*a^19*b^6*c^6*d^6 - 3244032*a^20*b^5*c^5*d^7 + 2027520*a^21*b^4*c^4*d^8 -
901120*a^22*b^3*c^3*d^9 + 270336*a^23*b^2*c^2*d^10 - 49152*a^24*b*c*d^11))^(1/4) + 2654208*a^38*c^5*d^22*x^(1
/2)*(-(6561*b^17*c^4 + 83521*a^4*b^13*d^4 - 176868*a^3*b^14*c*d^3 + 140454*a^2*b^15*c^2*d^2 - 49572*a*b^16*c^3
*d)/(4096*a^25*d^12 + 4096*a^13*b^12*c^12 - 49152*a^14*b^11*c^11*d + 270336*a^15*b^10*c^10*d^2 - 901120*a^16*b
^9*c^9*d^3 + 2027520*a^17*b^8*c^8*d^4 - 3244032*a^18*b^7*c^7*d^5 + 3784704*a^19*b^6*c^6*d^6 - 3244032*a^20*b^5
*c^5*d^7 + 2027520*a^21*b^4*c^4*d^8 - 901120*a^22*b^3*c^3*d^9 + 270336*a^23*b^2*c^2*d^10 - 49152*a^24*b*c*d^11
))^(5/4) - 130671792*a^21*b^9*c*d^18*x^(1/2)*(-(6561*b^17*c^4 + 83521*a^4*b^13*d^4 - 176868*a^3*b^14*c*d^3 + 1
40454*a^2*b^15*c^2*d^2 - 49572*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a^13*b^12*c^12 - 49152*a^14*b^11*c^11*d +
270336*a^15*b^10*c^10*d^2 - 901120*a^16*b^9*c^9*d^3 + 2027520*a^17*b^8*c^8*d^4 - 3244032*a^18*b^7*c^7*d^5 + 37
84704*a^19*b^6*c^6*d^6 - 3244032*a^20*b^5*c^5*d^7 + 2027520*a^21*b^4*c^4*d^8 - 901120*a^22*b^3*c^3*d^9 + 27033
6*a^23*b^2*c^2*d^10 - 49152*a^24*b*c*d^11))^(1/4) - 41877504*a^17*b^21*c^26*d*x^(1/2)*(-(6561*b^17*c^4 + 83521
*a^4*b^13*d^4 - 176868*a^3*b^14*c*d^3 + 140454*a^2*b^15*c^2*d^2 - 49572*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a
^13*b^12*c^12 - 49152*a^14*b^11*c^11*d + 270336*a^15*b^10*c^10*d^2 - 901120*a^16*b^9*c^9*d^3 + 2027520*a^17*b^
8*c^8*d^4 - 3244032*a^18*b^7*c^7*d^5 + 3784704*a^19*b^6*c^6*d^6 - 3244032*a^20*b^5*c^5*d^7 + 2027520*a^21*b^4*
c^4*d^8 - 901120*a^22*b^3*c^3*d^9 + 270336*a^23*b^2*c^2*d^10 - 49152*a^24*b*c*d^11))^(5/4) - 41877504*a^37*b*c
^6*d^21*x^(1/2)*(-(6561*b^17*c^4 + 83521*a^4*b^13*d^4 - 176868*a^3*b^14*c*d^3 + 140454*a^2*b^15*c^2*d^2 - 4957
2*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a^13*b^12*c^12 - 49152*a^14*b^11*c^11*d + 270336*a^15*b^10*c^10*d^2 - 9
01120*a^16*b^9*c^9*d^3 + 2027520*a^17*b^8*c^8*d^4 - 3244032*a^18*b^7*c^7*d^5 + 3784704*a^19*b^6*c^6*d^6 - 3244
032*a^20*b^5*c^5*d^7 + 2027520*a^21*b^4*c^4*d^8 - 901120*a^22*b^3*c^3*d^9 + 270336*a^23*b^2*c^2*d^10 - 49152*a
^24*b*c*d^11))^(5/4) + 15169032*a^11*b^19*c^11*d^8*x^(1/2)*(-(6561*b^17*c^4 + 83521*a^4*b^13*d^4 - 176868*a^3*
b^14*c*d^3 + 140454*a^2*b^15*c^2*d^2 - 49572*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a^13*b^12*c^12 - 49152*a^14*
b^11*c^11*d + 270336*a^15*b^10*c^10*d^2 - 901120*a^16*b^9*c^9*d^3 + 2027520*a^17*b^8*c^8*d^4 - 3244032*a^18*b^
7*c^7*d^5 + 3784704*a^19*b^6*c^6*d^6 - 3244032*a^20*b^5*c^5*d^7 + 2027520*a^21*b^4*c^4*d^8 - 901120*a^22*b^3*c
^3*d^9 + 270336*a^23*b^2*c^2*d^10 - 49152*a^24*b*c*d^11))^(1/4) - 130671792*a^12*b^18*c^10*d^9*x^(1/2)*(-(6561
*b^17*c^4 + 83521*a^4*b^13*d^4 - 176868*a^3*b^14*c*d^3 + 140454*a^2*b^15*c^2*d^2 - 49572*a*b^16*c^3*d)/(4096*a
^25*d^12 + 4096*a^13*b^12*c^12 - 49152*a^14*b^11*c^11*d + 270336*a^15*b^10*c^10*d^2 - 901120*a^16*b^9*c^9*d^3
+ 2027520*a^17*b^8*c^8*d^4 - 3244032*a^18*b^7*c^7*d^5 + 3784704*a^19*b^6*c^6*d^6 - 3244032*a^20*b^5*c^5*d^7 +
2027520*a^21*b^4*c^4*d^8 - 901120*a^22*b^3*c^3*d^9 + 270336*a^23*b^2*c^2*d^10 - 49152*a^24*b*c*d^11))^(1/4) +
450333432*a^13*b^17*c^9*d^10*x^(1/2)*(-(6561*b^17*c^4 + 83521*a^4*b^13*d^4 - 176868*a^3*b^14*c*d^3 + 140454*a^
2*b^15*c^2*d^2 - 49572*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a^13*b^12*c^12 - 49152*a^14*b^11*c^11*d + 270336*a
^15*b^10*c^10*d^2 - 901120*a^16*b^9*c^9*d^3 + 2027520*a^17*b^8*c^8*d^4 - 3244032*a^18*b^7*c^7*d^5 + 3784704*a^
19*b^6*c^6*d^6 - 3244032*a^20*b^5*c^5*d^7 + 2027520*a^21*b^4*c^4*d^8 - 901120*a^22*b^3*c^3*d^9 + 270336*a^23*b
^2*c^2*d^10 - 49152*a^24*b*c*d^11))^(1/4) - 784872864*a^14*b^16*c^8*d^11*x^(1/2)*(-(6561*b^17*c^4 + 83521*a^4*
b^13*d^4 - 176868*a^3*b^14*c*d^3 + 140454*a^2*b^15*c^2*d^2 - 49572*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a^13*b
^12*c^12 - 49152*a^14*b^11*c^11*d + 270336*a^15*b^10*c^10*d^2 - 901120*a^16*b^9*c^9*d^3 + 2027520*a^17*b^8*c^8
*d^4 - 3244032*a^18*b^7*c^7*d^5 + 3784704*a^19*b^6*c^6*d^6 - 3244032*a^20*b^5*c^5*d^7 + 2027520*a^21*b^4*c^4*d
^8 - 901120*a^22*b^3*c^3*d^9 + 270336*a^23*b^2*c^2*d^10 - 49152*a^24*b*c*d^11))^(1/4) + 717087608*a^15*b^15*c^
7*d^12*x^(1/2)*(-(6561*b^17*c^4 + 83521*a^4*b^13*d^4 - 176868*a^3*b^14*c*d^3 + 140454*a^2*b^15*c^2*d^2 - 49572
*a*b^16*c^3*d)/(4096*a^25*d^12 + 4096*a^13*b^12...

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