3.5.81 \(\int \frac {x^{5/2}}{(a+b x^2) (c+d x^2)^3} \, dx\) [481]

Optimal. Leaf size=628 \[ \frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{3/4} (b c-a d)^3} \]

[Out]

1/4*x^(3/2)/(-a*d+b*c)/(d*x^2+c)^2+1/16*(3*a*d+5*b*c)*x^(3/2)/c/(-a*d+b*c)^2/(d*x^2+c)+1/2*a^(3/4)*b^(5/4)*arc
tan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/(-a*d+b*c)^3*2^(1/2)-1/2*a^(3/4)*b^(5/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/
2)/a^(1/4))/(-a*d+b*c)^3*2^(1/2)-1/64*(-3*a^2*d^2+30*a*b*c*d+5*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/
4))/c^(5/4)/d^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/64*(-3*a^2*d^2+30*a*b*c*d+5*b^2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/
2)/c^(1/4))/c^(5/4)/d^(3/4)/(-a*d+b*c)^3*2^(1/2)-1/4*a^(3/4)*b^(5/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1
/2)*x^(1/2))/(-a*d+b*c)^3*2^(1/2)+1/4*a^(3/4)*b^(5/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/(-
a*d+b*c)^3*2^(1/2)+1/128*(-3*a^2*d^2+30*a*b*c*d+5*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^(5/4)/d^(3/4)/(-a*d+b*c)^3*2^(1/2)-1/128*(-3*a^2*d^2+30*a*b*c*d+5*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^(5/4)/d^(3/4)/(-a*d+b*c)^3*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.52, antiderivative size = 628, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 482, 593, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {a^{3/4} b^{5/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+30 a b c d+5 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^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+30 a b c d+5 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^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (-3 a^2 d^2+30 a b c d+5 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (-3 a^2 d^2+30 a b c d+5 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^{5/4} d^{3/4} (b c-a d)^3}+\frac {x^{3/2} (3 a d+5 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac {x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

x^(3/2)/(4*(b*c - a*d)*(c + d*x^2)^2) + ((5*b*c + 3*a*d)*x^(3/2))/(16*c*(b*c - a*d)^2*(c + d*x^2)) + (a^(3/4)*
b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - (a^(3/4)*b^(5/4)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(3/4)*(b*c - a*d)^3) + ((5*b^2*c^2 + 30*a*b*c*d - 3*
a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(3/4)*(b*c - a*d)^3) - (a^(3/4)*
b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) + (a^(3/4)*b^(5/
4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) + ((5*b^2*c^2 + 30*a*
b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(3/4)*(b*
c - a*d)^3) - ((5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x]
)/(64*Sqrt[2]*c^(5/4)*d^(3/4)*(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 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && 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 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 {x^{5/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {x^6}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 a-5 b x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 a (9 b c-a d)-b (5 b c+3 a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^2}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \left (\frac {32 a b^2 c x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^2}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {\left (2 a b^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c (b c-a d)^3}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (a b^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}-\frac {\left (a b^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c \sqrt {d} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c \sqrt {d} (b c-a d)^3}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {(a b) \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 (b c-a d)^3}-\frac {(a b) \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 (b c-a d)^3}-\frac {\left (a^{3/4} b^{5/4}\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 )}{2 \sqrt {2} (b c-a d)^3}-\frac {\left (a^{3/4} b^{5/4}\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 )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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 d (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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 d (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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^{5/4} d^{3/4} (b c-a d)^3}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}-\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (a^{3/4} b^{5/4}\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 )}{\sqrt {2} (b c-a d)^3}+\frac {\left (a^{3/4} b^{5/4}\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 )}{\sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 a^2 d^2\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^{5/4} d^{3/4} (b c-a d)^3}\\ &=\frac {x^{3/2}}{4 (b c-a d) \left (c+d x^2\right )^2}+\frac {(5 b c+3 a d) x^{3/2}}{16 c (b c-a d)^2 \left (c+d x^2\right )}+\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 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^{5/4} d^{3/4} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {a^{3/4} b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} (b c-a d)^3}+\frac {\left (5 b^2 c^2+30 a b c d-3 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^{5/4} d^{3/4} (b c-a d)^3}-\frac {\left (5 b^2 c^2+30 a b c d-3 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^{5/4} d^{3/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 1.17, size = 329, normalized size = 0.52 \begin {gather*} \frac {\frac {4 (b c-a d) x^{3/2} \left (a d \left (-c+3 d x^2\right )+b c \left (9 c+5 d x^2\right )\right )}{c \left (c+d x^2\right )^2}+32 \sqrt {2} a^{3/4} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\frac {\sqrt {2} \left (5 b^2 c^2+30 a b c d-3 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^{5/4} d^{3/4}}+32 \sqrt {2} a^{3/4} b^{5/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )-\frac {\sqrt {2} \left (5 b^2 c^2+30 a b c d-3 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^{5/4} d^{3/4}}}{64 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((4*(b*c - a*d)*x^(3/2)*(a*d*(-c + 3*d*x^2) + b*c*(9*c + 5*d*x^2)))/(c*(c + d*x^2)^2) + 32*Sqrt[2]*a^(3/4)*b^(
5/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - (Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d - 3*a^
2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(5/4)*d^(3/4)) + 32*Sqrt[2]*a^(3/4)
*b^(5/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] - (Sqrt[2]*(5*b^2*c^2 + 30*a*b*c*d -
3*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(5/4)*d^(3/4)))/(64*(b*c - a*
d)^3)

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Maple [A]
time = 0.09, size = 330, normalized size = 0.53

method	result	size
derivativedivides	\(\frac {a b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\frac {2 \left (\frac {d \left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c}+\left (-\frac {1}{32} a^{2} d^{2}+\frac {5}{16} a b c d -\frac {9}{32} b^{2} c^{2}\right ) x^{\frac {3}{2}}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-30 a b c d -5 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{\left (a d -b c \right )^{3}}\)	\(330\)
default	\(\frac {a b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\frac {2 \left (\frac {d \left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c}+\left (-\frac {1}{32} a^{2} d^{2}+\frac {5}{16} a b c d -\frac {9}{32} b^{2} c^{2}\right ) x^{\frac {3}{2}}\right )}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (3 a^{2} d^{2}-30 a b c d -5 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{\left (a d -b c \right )^{3}}\)	\(330\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/4*a*b/(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/(a*d
-b*c)^3*((1/32*d*(3*a^2*d^2+2*a*b*c*d-5*b^2*c^2)/c*x^(7/2)+(-1/32*a^2*d^2+5/16*a*b*c*d-9/32*b^2*c^2)*x^(3/2))/
(d*x^2+c)^2+1/256*(3*a^2*d^2-30*a*b*c*d-5*b^2*c^2)/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)))

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Maxima [A]
time = 0.52, size = 583, normalized size = 0.93 \begin {gather*} -\frac {a b^{2} {\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 (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {{\left (5 \, b^{2} c^{2} + 30 \, a b c d - 3 \, a^{2} d^{2}\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^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}} + \frac {{\left (5 \, b c d + 3 \, a d^{2}\right )} x^{\frac {7}{2}} + {\left (9 \, b c^{2} - a c d\right )} x^{\frac {3}{2}}}{16 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2} + {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/4*a*b^2*(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)) + 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)))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) + 1/128*(5*b^2*c^2 + 30*a*b*c*d - 3*a^2*d^2)*(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^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3) + 1/16*((5*b*c*d + 3*a*d^2)*x^(7/2) + (9*b*c^2 - a*c*d)*
x^(3/2))/(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2 + (b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^4 + 2*(b^2*c^4*d -
2*a*b*c^3*d^2 + a^2*c^2*d^3)*x^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(x^(5/2)/(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(x**(5/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 963 vs. \(2 (481) = 962\).
time = 1.36, size = 963, normalized size = 1.53 \begin {gather*} \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 30 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \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^{5} d^{3} - 3 \, \sqrt {2} a b^{2} c^{4} d^{4} + 3 \, \sqrt {2} a^{2} b c^{3} d^{5} - \sqrt {2} a^{3} c^{2} d^{6}\right )}} + \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 30 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \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^{5} d^{3} - 3 \, \sqrt {2} a b^{2} c^{4} d^{4} + 3 \, \sqrt {2} a^{2} b c^{3} d^{5} - \sqrt {2} a^{3} c^{2} d^{6}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 30 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \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^{5} d^{3} - 3 \, \sqrt {2} a b^{2} c^{4} d^{4} + 3 \, \sqrt {2} a^{2} b c^{3} d^{5} - \sqrt {2} a^{3} c^{2} d^{6}\right )}} + \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} + 30 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \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^{5} d^{3} - 3 \, \sqrt {2} a b^{2} c^{4} d^{4} + 3 \, \sqrt {2} a^{2} b c^{3} d^{5} - \sqrt {2} a^{3} c^{2} d^{6}\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \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} b^{4} c^{3} - 3 \, \sqrt {2} a b^{3} c^{2} d + 3 \, \sqrt {2} a^{2} b^{2} c d^{2} - \sqrt {2} a^{3} b d^{3}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \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} b^{4} c^{3} - 3 \, \sqrt {2} a b^{3} c^{2} d + 3 \, \sqrt {2} a^{2} b^{2} c d^{2} - \sqrt {2} a^{3} b d^{3}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{4} c^{3} - 3 \, \sqrt {2} a b^{3} c^{2} d + 3 \, \sqrt {2} a^{2} b^{2} c d^{2} - \sqrt {2} a^{3} b d^{3}\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{4} c^{3} - 3 \, \sqrt {2} a b^{3} c^{2} d + 3 \, \sqrt {2} a^{2} b^{2} c d^{2} - \sqrt {2} a^{3} b d^{3}\right )}} + \frac {5 \, b c d x^{\frac {7}{2}} + 3 \, a d^{2} x^{\frac {7}{2}} + 9 \, b c^{2} x^{\frac {3}{2}} - a c d x^{\frac {3}{2}}}{16 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)/(b*x^2+a)/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/32*(5*(c*d^3)^(3/4)*b^2*c^2 + 30*(c*d^3)^(3/4)*a*b*c*d - 3*(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^5*d^3 - 3*sqrt(2)*a*b^2*c^4*d^4 + 3*sqrt(2)*a^2*b*c^3*d
^5 - sqrt(2)*a^3*c^2*d^6) + 1/32*(5*(c*d^3)^(3/4)*b^2*c^2 + 30*(c*d^3)^(3/4)*a*b*c*d - 3*(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^5*d^3 - 3*sqrt(2)*a*b^2*c^
4*d^4 + 3*sqrt(2)*a^2*b*c^3*d^5 - sqrt(2)*a^3*c^2*d^6) - 1/64*(5*(c*d^3)^(3/4)*b^2*c^2 + 30*(c*d^3)^(3/4)*a*b*
c*d - 3*(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^5*d^3 - 3*sqrt(
2)*a*b^2*c^4*d^4 + 3*sqrt(2)*a^2*b*c^3*d^5 - sqrt(2)*a^3*c^2*d^6) + 1/64*(5*(c*d^3)^(3/4)*b^2*c^2 + 30*(c*d^3)
^(3/4)*a*b*c*d - 3*(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^5*d
^3 - 3*sqrt(2)*a*b^2*c^4*d^4 + 3*sqrt(2)*a^2*b*c^3*d^5 - sqrt(2)*a^3*c^2*d^6) - (a*b^3)^(3/4)*arctan(1/2*sqrt(
2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^4*c^3 - 3*sqrt(2)*a*b^3*c^2*d + 3*sqrt(2)*a^2*b^2
*c*d^2 - sqrt(2)*a^3*b*d^3) - (a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))
/(sqrt(2)*b^4*c^3 - 3*sqrt(2)*a*b^3*c^2*d + 3*sqrt(2)*a^2*b^2*c*d^2 - sqrt(2)*a^3*b*d^3) + 1/2*(a*b^3)^(3/4)*l
og(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^4*c^3 - 3*sqrt(2)*a*b^3*c^2*d + 3*sqrt(2)*a^2*b^2*c
*d^2 - sqrt(2)*a^3*b*d^3) - 1/2*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^4*c
^3 - 3*sqrt(2)*a*b^3*c^2*d + 3*sqrt(2)*a^2*b^2*c*d^2 - sqrt(2)*a^3*b*d^3) + 1/16*(5*b*c*d*x^(7/2) + 3*a*d^2*x^
(7/2) + 9*b*c^2*x^(3/2) - a*c*d*x^(3/2))/((b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*(d*x^2 + c)^2)

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Mupad [B]
time = 2.30, size = 2500, normalized size = 3.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/((a + b*x^2)*(c + d*x^2)^3),x)

[Out]

2*atan(((((((27*a^20*b^4*d^20)/16 - (1107*a^19*b^5*c*d^19)/16 + (125*a^3*b^21*c^17*d^3)/16 - (31893*a^4*b^20*c
^16*d^4)/16 + (44481*a^5*b^19*c^15*d^5)/2 - (227605*a^6*b^18*c^14*d^6)/2 + (1382895*a^7*b^17*c^13*d^7)/4 - (27
23535*a^8*b^16*c^12*d^8)/4 + (1760163*a^9*b^15*c^11*d^9)/2 - (1361943*a^10*b^14*c^10*d^10)/2 + (1117215*a^11*b
^13*c^9*d^11)/8 + (2877545*a^12*b^12*c^8*d^12)/8 - (1026465*a^13*b^11*c^7*d^13)/2 + (744837*a^14*b^10*c^6*d^14
)/2 - (688489*a^15*b^9*c^5*d^15)/4 + (208665*a^16*b^8*c^4*d^16)/4 - (20115*a^17*b^7*c^3*d^17)/2 + (2295*a^18*b
^6*c^2*d^18)/2)*1i)/(b^14*c^16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 + 91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13
*d^3 + 1001*a^4*b^10*c^12*d^4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^
8*b^6*c^8*d^8 - 2002*a^9*b^5*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 -
14*a*b^13*c^15*d) - (x^(1/2)*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9
*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920
*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/
4)*(147456*a^19*b^4*c*d^20 + 17186816*a^3*b^20*c^17*d^4 - 201326592*a^4*b^19*c^16*d^5 + 1089601536*a^5*b^18*c^
15*d^6 - 3630694400*a^6*b^17*c^14*d^7 + 8402436096*a^7*b^16*c^13*d^8 - 14511243264*a^8*b^15*c^12*d^9 + 1970208
7680*a^9*b^14*c^11*d^10 - 21851799552*a^10*b^13*c^10*d^11 + 20194099200*a^11*b^12*c^9*d^12 - 15479078912*a^12*
b^11*c^8*d^13 + 9580707840*a^13*b^10*c^7*d^14 - 4594335744*a^14*b^9*c^6*d^15 + 1620770816*a^15*b^8*c^5*d^16 -
393216000*a^16*b^7*c^4*d^17 + 59375616*a^17*b^6*c^3*d^18 - 4718592*a^18*b^5*c^2*d^19))/(4096*(b^12*c^14 + a^12
*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*
b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*
b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^
3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 +
7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 192*a*b^11*c^11*d - 192*a^11*b*c*d^11)
)^(3/4) - (x^(1/2)*(81*a^11*b^8*d^9 + 625*a^3*b^16*c^8*d + 5976*a^10*b^9*c*d^8 + 15000*a^4*b^15*c^7*d^2 + 1335
00*a^5*b^14*c^6*d^3 + 538600*a^6*b^13*c^5*d^4 + 956550*a^7*b^12*c^4*d^5 + 583080*a^8*b^11*c^3*d^6 - 136260*a^9
*b^10*c^2*d^7))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c^1
1*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^9*d^5 + 924*a^6*b^6*c^8*d^6 - 792*a^7*b^5*c^7*d^7 + 495*a^8*b^4*c
^6*d^8 - 220*a^9*b^3*c^5*d^9 + 66*a^10*b^2*c^4*d^10 - 12*a*b^11*c^13*d)))*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*
c^12 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^
6*b^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 -
192*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4) - (((((27*a^20*b^4*d^20)/16 - (1107*a^19*b^5*c*d^19)/16 + (125*
a^3*b^21*c^17*d^3)/16 - (31893*a^4*b^20*c^16*d^4)/16 + (44481*a^5*b^19*c^15*d^5)/2 - (227605*a^6*b^18*c^14*d^6
)/2 + (1382895*a^7*b^17*c^13*d^7)/4 - (2723535*a^8*b^16*c^12*d^8)/4 + (1760163*a^9*b^15*c^11*d^9)/2 - (1361943
*a^10*b^14*c^10*d^10)/2 + (1117215*a^11*b^13*c^9*d^11)/8 + (2877545*a^12*b^12*c^8*d^12)/8 - (1026465*a^13*b^11
*c^7*d^13)/2 + (744837*a^14*b^10*c^6*d^14)/2 - (688489*a^15*b^9*c^5*d^15)/4 + (208665*a^16*b^8*c^4*d^16)/4 - (
20115*a^17*b^7*c^3*d^17)/2 + (2295*a^18*b^6*c^2*d^18)/2)*1i)/(b^14*c^16 + a^14*c^2*d^14 - 14*a^13*b*c^3*d^13 +
91*a^2*b^12*c^14*d^2 - 364*a^3*b^11*c^13*d^3 + 1001*a^4*b^10*c^12*d^4 - 2002*a^5*b^9*c^11*d^5 + 3003*a^6*b^8*
c^10*d^6 - 3432*a^7*b^7*c^9*d^7 + 3003*a^8*b^6*c^8*d^8 - 2002*a^9*b^5*c^7*d^9 + 1001*a^10*b^4*c^6*d^10 - 364*a
^11*b^3*c^5*d^11 + 91*a^12*b^2*c^4*d^12 - 14*a*b^13*c^15*d) + (x^(1/2)*(-(a^3*b^5)/(16*a^12*d^12 + 16*b^12*c^1
2 + 1056*a^2*b^10*c^10*d^2 - 3520*a^3*b^9*c^9*d^3 + 7920*a^4*b^8*c^8*d^4 - 12672*a^5*b^7*c^7*d^5 + 14784*a^6*b
^6*c^6*d^6 - 12672*a^7*b^5*c^5*d^7 + 7920*a^8*b^4*c^4*d^8 - 3520*a^9*b^3*c^3*d^9 + 1056*a^10*b^2*c^2*d^10 - 19
2*a*b^11*c^11*d - 192*a^11*b*c*d^11))^(1/4)*(147456*a^19*b^4*c*d^20 + 17186816*a^3*b^20*c^17*d^4 - 201326592*a
^4*b^19*c^16*d^5 + 1089601536*a^5*b^18*c^15*d^6 - 3630694400*a^6*b^17*c^14*d^7 + 8402436096*a^7*b^16*c^13*d^8
- 14511243264*a^8*b^15*c^12*d^9 + 19702087680*a^9*b^14*c^11*d^10 - 21851799552*a^10*b^13*c^10*d^11 + 201940992
00*a^11*b^12*c^9*d^12 - 15479078912*a^12*b^11*c^8*d^13 + 9580707840*a^13*b^10*c^7*d^14 - 4594335744*a^14*b^9*c
^6*d^15 + 1620770816*a^15*b^8*c^5*d^16 - 393216000*a^16*b^7*c^4*d^17 + 59375616*a^17*b^6*c^3*d^18 - 4718592*a^
18*b^5*c^2*d^19))/(4096*(b^12*c^14 + a^12*c^2*d^12 - 12*a^11*b*c^3*d^11 + 66*a^2*b^10*c^12*d^2 - 220*a^3*b^9*c
^11*d^3 + 495*a^4*b^8*c^10*d^4 - 792*a^5*b^7*c^...

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