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

Optimal. Leaf size=805 \[ \frac{(5 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{(5 b c-17 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{40 b^3 c^3-96 a b^2 d c^2+125 a^2 b d^2 c-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+25 a b d c-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (d x^2+c\right )^2} \]

[Out]

-(40*b^3*c^3 - 96*a*b^2*c^2*d + 125*a^2*b*c*d^2 - 45*a^3*d^3)/(16*a^2*c^3*(b*c -
a*d)^3*Sqrt[x]) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)^2)
+ b/(2*a*(b*c - a*d)*Sqrt[x]*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 25*a*
b*c*d - 9*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*Sqrt[x]*(c + d*x^2)) + (b^(13/4)*(5*
b*c - 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*
(b*c - a*d)^4) - (b^(13/4)*(5*b*c - 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])
/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) + (d^(9/4)*(221*b^2*c^2 - 170*a*b*c
*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(1
3/4)*(b*c - a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[1
+ (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^4) - (b^
(13/4)*(5*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^(9/4)*(b*c - a*d)^4) + (b^(13/4)*(5*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^(9/4)*(b*c - a*d)^4)
- (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/
4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(b*c - a*d)^4) + (d^(9/4)*
(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*S
qrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(b*c - a*d)^4)

_______________________________________________________________________________________

Rubi [A]  time = 3.05594, antiderivative size = 805, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ \frac{(5 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{13/4}}{4 \sqrt{2} a^{9/4} (b c-a d)^4}-\frac{(5 b c-17 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{(5 b c-17 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{13/4}}{8 \sqrt{2} a^{9/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) \sqrt{x} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}+\frac{d^{9/4} \left (221 b^2 c^2-170 a b d c+45 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{13/4} (b c-a d)^4}-\frac{40 b^3 c^3-96 a b^2 d c^2+125 a^2 b d^2 c-45 a^3 d^3}{16 a^2 c^3 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+25 a b d c-9 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 \sqrt{x} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 \sqrt{x} \left (d x^2+c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(40*b^3*c^3 - 96*a*b^2*c^2*d + 125*a^2*b*c*d^2 - 45*a^3*d^3)/(16*a^2*c^3*(b*c -
a*d)^3*Sqrt[x]) + (d*(2*b*c + a*d))/(4*a*c*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2)^2)
+ b/(2*a*(b*c - a*d)*Sqrt[x]*(a + b*x^2)*(c + d*x^2)^2) + (d*(8*b^2*c^2 + 25*a*
b*c*d - 9*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*Sqrt[x]*(c + d*x^2)) + (b^(13/4)*(5*
b*c - 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*
(b*c - a*d)^4) - (b^(13/4)*(5*b*c - 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])
/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*(b*c - a*d)^4) + (d^(9/4)*(221*b^2*c^2 - 170*a*b*c
*d + 45*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(1
3/4)*(b*c - a*d)^4) - (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[1
+ (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(13/4)*(b*c - a*d)^4) - (b^
(13/4)*(5*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^(9/4)*(b*c - a*d)^4) + (b^(13/4)*(5*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^(9/4)*(b*c - a*d)^4)
- (d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/
4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(b*c - a*d)^4) + (d^(9/4)*
(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*S
qrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(13/4)*(b*c - a*d)^4)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 6.14473, size = 706, normalized size = 0.88 \[ \frac{1}{128} \left (\frac{8 \sqrt{2} b^{13/4} (17 a d-5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^4}+\frac{8 \sqrt{2} b^{13/4} (5 b c-17 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^4}+\frac{16 \sqrt{2} b^{13/4} (5 b c-17 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{9/4} (b c-a d)^4}+\frac{16 \sqrt{2} b^{13/4} (17 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{9/4} (b c-a d)^4}+\frac{64 b^4 x^{3/2}}{a^2 \left (a+b x^2\right ) (a d-b c)^3}-\frac{\sqrt{2} d^{9/4} \left (45 a^2 d^2-170 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^4}+\frac{\sqrt{2} d^{9/4} \left (45 a^2 d^2-170 a b c d+221 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{13/4} (b c-a d)^4}+\frac{2 \sqrt{2} d^{9/4} \left (45 a^2 d^2-170 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{13/4} (b c-a d)^4}-\frac{2 \sqrt{2} d^{9/4} \left (45 a^2 d^2-170 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{13/4} (b c-a d)^4}-\frac{256}{a^2 c^3 \sqrt{x}}+\frac{8 d^3 x^{3/2} (13 a d-29 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{32 d^3 x^{3/2}}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-256/(a^2*c^3*Sqrt[x]) + (64*b^4*x^(3/2))/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) -
(32*d^3*x^(3/2))/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (8*d^3*(-29*b*c + 13*a*d)*x
^(3/2))/(c^3*(b*c - a*d)^3*(c + d*x^2)) + (16*Sqrt[2]*b^(13/4)*(5*b*c - 17*a*d)*
ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*(b*c - a*d)^4) + (16*Sqr
t[2]*b^(13/4)*(-5*b*c + 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
a^(9/4)*(b*c - a*d)^4) + (2*Sqrt[2]*d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*
d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(13/4)*(b*c - a*d)^4) - (
2*Sqrt[2]*d^(9/4)*(221*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d
^(1/4)*Sqrt[x])/c^(1/4)])/(c^(13/4)*(b*c - a*d)^4) + (8*Sqrt[2]*b^(13/4)*(-5*b*c
+ 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*
(b*c - a*d)^4) + (8*Sqrt[2]*b^(13/4)*(5*b*c - 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*(b*c - a*d)^4) - (Sqrt[2]*d^(9/4)*(22
1*b^2*c^2 - 170*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt
[x] + Sqrt[d]*x])/(c^(13/4)*(b*c - a*d)^4) + (Sqrt[2]*d^(9/4)*(221*b^2*c^2 - 170
*a*b*c*d + 45*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x
])/(c^(13/4)*(b*c - a*d)^4))/128

_______________________________________________________________________________________

Maple [A]  time = 0.043, 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^(3/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

-13/16*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*a^2+21/8*d^5/c^2/(a*d-b*c)^4/(d*x
^2+c)^2*x^(7/2)*a*b-29/16*d^4/c/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*b^2-17/16*d^5/c^
2/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*a^2+25/8*d^4/c/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)
*a*b-33/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*b^2-45/128*d^4/c^3/(a*d-b*c)^4/(c
/d)^(1/4)*2^(1/2)*a^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)))-45/64*d^4/c^3/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*
a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-45/64*d^4/c^3/(a*d-b*c)^4/(c/d)^(1/4)*
2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+85/64*d^3/c^2/(a*d-b*c)^4/(c/d
)^(1/4)*2^(1/2)*a*b*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)))+85/32*d^3/c^2/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*a*
b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+85/32*d^3/c^2/(a*d-b*c)^4/(c/d)^(1/4)*2^
(1/2)*a*b*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-221/128*d^2/c/(a*d-b*c)^4/(c/d)^
(1/4)*2^(1/2)*b^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)))-221/64*d^2/c/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2)*b^2*a
rctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-221/64*d^2/c/(a*d-b*c)^4/(c/d)^(1/4)*2^(1/2
)*b^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/a^2/c^3/x^(1/2)+1/2*b^4/a/(a*d-b*c
)^4*x^(3/2)/(b*x^2+a)*d-1/2*b^5/a^2/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*c+17/16*b^3/a/
(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*d*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)))+17/8*b^3/a/(a*d-b*c)^4/(a/b)^(1/4)
*2^(1/2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+17/8*b^3/a/(a*d-b*c)^4/(a/b)^(1
/4)*2^(1/2)*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-5/16*b^4/a^2/(a*d-b*c)^4/(a/
b)^(1/4)*2^(1/2)*c*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)))-5/8*b^4/a^2/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*c*arc
tan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-5/8*b^4/a^2/(a*d-b*c)^4/(a/b)^(1/4)*2^(1/2)*c
*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

_______________________________________________________________________________________

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^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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^(3/2)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.664914, 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^(3/2)),x, algorithm="giac")

[Out]

Done