3.502 \(\int \frac{1}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=805 \[ \frac{(7 b c-19 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{15/4}}{4 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{(7 b c-19 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{15/4}}{4 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{(7 b c-19 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{15/4}}{8 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{(7 b c-19 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{15/4}}{8 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{3/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{d \left (8 b^2 c^2+27 a b d c-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (d x^2+c\right )}-\frac{56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (d x^2+c\right )^2} \]

[Out]

-(56*b^3*c^3 - 96*a*b^2*c^2*d + 189*a^2*b*c*d^2 - 77*a^3*d^3)/(48*a^2*c^3*(b*c -
a*d)^3*x^(3/2)) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)^2)
+ b/(2*a*(b*c - a*d)*x^(3/2)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 27*a*
b*c*d - 11*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*x^(3/2)*(c + d*x^2)) + (b^(15/4)*(7
*b*c - 19*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4
)*(b*c - a*d)^4) - (b^(15/4)*(7*b*c - 19*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) + (d^(11/4)*(285*b^2*c^2 - 266*a
*b*c*d + 77*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*
c^(15/4)*(b*c - a*d)^4) - (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Arc
Tan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^4)
+ (b^(15/4)*(7*b*c - 19*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) - (b^(15/4)*(7*b*c - 19*a*d)*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c -
a*d)^4) + (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[
2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c - a*d)^4) - (
d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*
d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c - a*d)^4)

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Rubi [A]  time = 2.92887, antiderivative size = 805, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ \frac{(7 b c-19 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{15/4}}{4 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{(7 b c-19 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{15/4}}{4 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{(7 b c-19 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{15/4}}{8 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{(7 b c-19 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{15/4}}{8 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{3/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{d \left (8 b^2 c^2+27 a b d c-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (d x^2+c\right )}-\frac{56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (d x^2+c\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

-(56*b^3*c^3 - 96*a*b^2*c^2*d + 189*a^2*b*c*d^2 - 77*a^3*d^3)/(48*a^2*c^3*(b*c -
a*d)^3*x^(3/2)) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)^2)
+ b/(2*a*(b*c - a*d)*x^(3/2)*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 27*a*
b*c*d - 11*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*x^(3/2)*(c + d*x^2)) + (b^(15/4)*(7
*b*c - 19*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4
)*(b*c - a*d)^4) - (b^(15/4)*(7*b*c - 19*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) + (d^(11/4)*(285*b^2*c^2 - 266*a
*b*c*d + 77*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*
c^(15/4)*(b*c - a*d)^4) - (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Arc
Tan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*(b*c - a*d)^4)
+ (b^(15/4)*(7*b*c - 19*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) - (b^(15/4)*(7*b*c - 19*a*d)*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c -
a*d)^4) + (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[
2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c - a*d)^4) - (
d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*
d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c - a*d)^4)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 4.37201, size = 707, normalized size = 0.88 \[ \frac{1}{384} \left (\frac{24 \sqrt{2} b^{15/4} (7 b c-19 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} (b c-a d)^4}+\frac{24 \sqrt{2} b^{15/4} (19 a d-7 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} (b c-a d)^4}+\frac{48 \sqrt{2} b^{15/4} (7 b c-19 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{11/4} (b c-a d)^4}+\frac{48 \sqrt{2} b^{15/4} (19 a d-7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{11/4} (b c-a d)^4}+\frac{192 b^4 \sqrt{x}}{a^2 \left (a+b x^2\right ) (a d-b c)^3}+\frac{3 \sqrt{2} d^{11/4} \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{15/4} (b c-a d)^4}-\frac{3 \sqrt{2} d^{11/4} \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{15/4} (b c-a d)^4}+\frac{6 \sqrt{2} d^{11/4} \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{15/4} (b c-a d)^4}-\frac{6 \sqrt{2} d^{11/4} \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{15/4} (b c-a d)^4}-\frac{256}{a^2 c^3 x^{3/2}}+\frac{24 d^3 \sqrt{x} (15 a d-31 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{96 d^3 \sqrt{x}}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(-256/(a^2*c^3*x^(3/2)) + (192*b^4*Sqrt[x])/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) -
(96*d^3*Sqrt[x])/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (24*d^3*(-31*b*c + 15*a*d)
*Sqrt[x])/(c^3*(b*c - a*d)^3*(c + d*x^2)) + (48*Sqrt[2]*b^(15/4)*(7*b*c - 19*a*d
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*(b*c - a*d)^4) + (48*
Sqrt[2]*b^(15/4)*(-7*b*c + 19*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(a^(11/4)*(b*c - a*d)^4) + (6*Sqrt[2]*d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77
*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(15/4)*(b*c - a*d)^4
) - (6*Sqrt[2]*d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*ArcTan[1 + (Sqr
t[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(15/4)*(b*c - a*d)^4) + (24*Sqrt[2]*b^(15/4)*
(7*b*c - 19*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^
(11/4)*(b*c - a*d)^4) + (24*Sqrt[2]*b^(15/4)*(-7*b*c + 19*a*d)*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*(b*c - a*d)^4) + (3*Sqrt[2]
*d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)
*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(15/4)*(b*c - a*d)^4) - (3*Sqrt[2]*d^(11/4)*(2
85*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqr
t[x] + Sqrt[d]*x])/(c^(15/4)*(b*c - a*d)^4))/384

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Maple [A]  time = 0.049, size = 1143, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

-15/16*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*a^2+23/8*d^5/c^2/(a*d-b*c)^4/(d*x
^2+c)^2*x^(5/2)*a*b-31/16*d^4/c/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*b^2-19/16*d^5/c^
2/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*a^2+27/8*d^4/c/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)
*a*b-35/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*b^2-77/64*d^5/c^4/(a*d-b*c)^4*(c/
d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2+133/32*d^4/c^3/(a*d-b
*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-285/64*d^3/c
^2/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2-77/
64*d^5/c^4/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)
*a^2+133/32*d^4/c^3/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x
^(1/2)-1)*a*b-285/64*d^3/c^2/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d
)^(1/4)*x^(1/2)-1)*b^2-77/128*d^5/c^4/(a*d-b*c)^4*(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))
)*a^2+133/64*d^4/c^3/(a*d-b*c)^4*(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)))*a*b-285/128*d^3
/c^2/(a*d-b*c)^4*(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)))*b^2-2/3/a^2/c^3/x^(3/2)+1/2*b^4
/a/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*d-1/2*b^5/a^2/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*c+1
9/8*b^4/a^2/(a*d-b*c)^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1
)*d-7/8*b^5/a^3/(a*d-b*c)^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/
2)+1)*c+19/8*b^4/a^2/(a*d-b*c)^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*
x^(1/2)-1)*d-7/8*b^5/a^3/(a*d-b*c)^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1
/4)*x^(1/2)-1)*c+19/16*b^4/a^2/(a*d-b*c)^4*(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)))*d-7/1
6*b^5/a^3/(a*d-b*c)^4*(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)))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(5/2)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.534704, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^3*x^(5/2)),x, algorithm="giac")

[Out]

Done