3.480 \(\int \frac{x^{7/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=631 \[ -\frac{a^{5/4} b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{5/4} b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}-\frac{a^{5/4} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} (b c-a d)^3}+\frac{a^{5/4} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} (b c-a d)^3}-\frac{\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac{\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\sqrt{x} (b c-9 a d)}{16 d \left (c+d x^2\right ) (b c-a d)^2}-\frac{c \sqrt{x}}{4 d \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(c*Sqrt[x])/(4*d*(b*c - a*d)*(c + d*x^2)^2) + ((b*c - 9*a*d)*Sqrt[x])/(16*d*(b*
c - a*d)^2*(c + d*x^2)) - (a^(5/4)*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/
a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) + (a^(5/4)*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d
^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(5/4)*(
b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4
)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3) - (a^(5/4)*b^(3/
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^(5/4)*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt
[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) - ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*Log[Sq
rt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(5/4
)*(b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c
^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3)

_______________________________________________________________________________________

Rubi [A]  time = 1.67614, antiderivative size = 631, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{a^{5/4} b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{a^{5/4} b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}-\frac{a^{5/4} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} (b c-a d)^3}+\frac{a^{5/4} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} (b c-a d)^3}-\frac{\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}-\frac{\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{3/4} d^{5/4} (b c-a d)^3}+\frac{\sqrt{x} (b c-9 a d)}{16 d \left (c+d x^2\right ) (b c-a d)^2}-\frac{c \sqrt{x}}{4 d \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(c*Sqrt[x])/(4*d*(b*c - a*d)*(c + d*x^2)^2) + ((b*c - 9*a*d)*Sqrt[x])/(16*d*(b*
c - a*d)^2*(c + d*x^2)) - (a^(5/4)*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/
a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) + (a^(5/4)*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(Sqrt[2]*(b*c - a*d)^3) - ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d
^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(5/4)*(
b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4
)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3) - (a^(5/4)*b^(3/
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^(5/4)*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt
[b]*x])/(2*Sqrt[2]*(b*c - a*d)^3) - ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*Log[Sq
rt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(5/4
)*(b*c - a*d)^3) + ((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c
^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(3/4)*d^(5/4)*(b*c - a*d)^3)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 1.05653, size = 640, normalized size = 1.01 \[ \frac{-32 \sqrt{2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left (c+d x^2\right )^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+32 \sqrt{2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left (c+d x^2\right )^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-64 \sqrt{2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left (c+d x^2\right )^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+64 \sqrt{2} a^{5/4} b^{3/4} c^{3/4} d^{5/4} \left (c+d x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\sqrt{2} \left (c+d x^2\right )^2 \left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+\sqrt{2} \left (c+d x^2\right )^2 \left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-2 \sqrt{2} \left (c+d x^2\right )^2 \left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+2 \sqrt{2} \left (c+d x^2\right )^2 \left (-5 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )+8 c^{3/4} \sqrt [4]{d} \sqrt{x} \left (c+d x^2\right ) (b c-9 a d) (b c-a d)-32 c^{7/4} \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{128 c^{3/4} d^{5/4} \left (c+d x^2\right )^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

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

[Out]

(-32*c^(7/4)*d^(1/4)*(b*c - a*d)^2*Sqrt[x] + 8*c^(3/4)*d^(1/4)*(b*c - 9*a*d)*(b*
c - a*d)*Sqrt[x]*(c + d*x^2) - 64*Sqrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d
*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 64*Sqrt[2]*a^(5/4)*b^(3/
4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] -
2*Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*ArcTan[1 - (Sqrt[2
]*d^(1/4)*Sqrt[x])/c^(1/4)] + 2*Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c
+ d*x^2)^2*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 32*Sqrt[2]*a^(5/4)*b^
(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x] + 32*Sqrt[2]*a^(5/4)*b^(3/4)*c^(3/4)*d^(5/4)*(c + d*x^2)^2*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - Sqrt[2]*(3*b^2*c^2 - 30*a
*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
+ Sqrt[d]*x] + Sqrt[2]*(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*(c + d*x^2)^2*Log[Sq
rt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(128*c^(3/4)*d^(5/4)*(b*c
- a*d)^3*(c + d*x^2)^2)

_______________________________________________________________________________________

Maple [A]  time = 0.029, size = 839, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/16/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/2)*a^2*d^2+5/8/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/2
)*c*a*b*d-1/16/(a*d-b*c)^3/(d*x^2+c)^2*x^(5/2)*b^2*c^2-5/16/(a*d-b*c)^3/(d*x^2+c
)^2*c*d*x^(1/2)*a^2+1/8/(a*d-b*c)^3/(d*x^2+c)^2*c^2*x^(1/2)*a*b+3/16/(a*d-b*c)^3
/(d*x^2+c)^2*c^3/d*x^(1/2)*b^2+5/64/(a*d-b*c)^3*d*(c/d)^(1/4)/c*2^(1/2)*arctan(2
^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2+15/32/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-3/64/(a*d-b*c)^3/d*(c/d)^(1/4)*c*2^(1/2)*arcta
n(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+5/64/(a*d-b*c)^3*d*(c/d)^(1/4)/c*2^(1/2)*ar
ctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2+15/32/(a*d-b*c)^3*(c/d)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-3/64/(a*d-b*c)^3/d*(c/d)^(1/4)*c*2^(1/2)
*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+5/128/(a*d-b*c)^3*d*(c/d)^(1/4)/c*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+15/64/(a*d-b*c)^3*(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-3/128
/(a*d-b*c)^3/d*(c/d)^(1/4)*c*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^2-1/4*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)))-1/2*a*b/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/b)^(1/4)*x^(1/2)+1)-1/2*a*b/(a*d-b*c)^3*(a/b)^(1/4)*2^(1/2)*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(x^(7/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 101.111, size = 6067, normalized size = 9.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(128*(-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a
^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6
- 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2
*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^2*c^4*d - 2*a*b*c^3*d^2 + a^
2*c^2*d^3 + (b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2*d^5)*x^4 + 2*(b^2*c^3*d^2 - 2*a*b*c
^2*d^3 + a^2*c*d^4)*x^2)*arctan(-(-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^
2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^
5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^
3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^3*c^3
- 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(a*b*sqrt(x) + sqrt(a^2*b^2*x + (b^6
*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*
d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*sqrt(-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*
a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*
d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*
b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))))) - 4*(b^2*
c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2*d^5)*x^4
+ 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2)*(-(81*b^8*c^8 - 3240*a*b^7*c^
7*d + 48060*a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 +
513000*a^5*b^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*d^7 + 625*a^8*d^
8)/(b^12*c^15*d^5 - 12*a*b^11*c^14*d^6 + 66*a^2*b^10*c^13*d^7 - 220*a^3*b^9*c^12
*d^8 + 495*a^4*b^8*c^11*d^9 - 792*a^5*b^7*c^10*d^10 + 924*a^6*b^6*c^9*d^11 - 792
*a^7*b^5*c^8*d^12 + 495*a^8*b^4*c^7*d^13 - 220*a^9*b^3*c^6*d^14 + 66*a^10*b^2*c^
5*d^15 - 12*a^11*b*c^4*d^16 + a^12*c^3*d^17))^(1/4)*arctan((b^3*c^4*d - 3*a*b^2*
c^3*d^2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*(-(81*b^8*c^8 - 3240*a*b^7*c^7*d + 48060*
a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 + 513000*a^5*b
^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*d^7 + 625*a^8*d^8)/(b^12*c^1
5*d^5 - 12*a*b^11*c^14*d^6 + 66*a^2*b^10*c^13*d^7 - 220*a^3*b^9*c^12*d^8 + 495*a
^4*b^8*c^11*d^9 - 792*a^5*b^7*c^10*d^10 + 924*a^6*b^6*c^9*d^11 - 792*a^7*b^5*c^8
*d^12 + 495*a^8*b^4*c^7*d^13 - 220*a^9*b^3*c^6*d^14 + 66*a^10*b^2*c^5*d^15 - 12*
a^11*b*c^4*d^16 + a^12*c^3*d^17))^(1/4)/((3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*sq
rt(x) - sqrt((9*b^4*c^4 - 180*a*b^3*c^3*d + 870*a^2*b^2*c^2*d^2 + 300*a^3*b*c*d^
3 + 25*a^4*d^4)*x + (b^6*c^8*d^2 - 6*a*b^5*c^7*d^3 + 15*a^2*b^4*c^6*d^4 - 20*a^3
*b^3*c^5*d^5 + 15*a^4*b^2*c^4*d^6 - 6*a^5*b*c^3*d^7 + a^6*c^2*d^8)*sqrt(-(81*b^8
*c^8 - 3240*a*b^7*c^7*d + 48060*a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 64935
0*a^4*b^4*c^4*d^4 + 513000*a^5*b^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^6 + 15000*a^7*
b*c*d^7 + 625*a^8*d^8)/(b^12*c^15*d^5 - 12*a*b^11*c^14*d^6 + 66*a^2*b^10*c^13*d^
7 - 220*a^3*b^9*c^12*d^8 + 495*a^4*b^8*c^11*d^9 - 792*a^5*b^7*c^10*d^10 + 924*a^
6*b^6*c^9*d^11 - 792*a^7*b^5*c^8*d^12 + 495*a^8*b^4*c^7*d^13 - 220*a^9*b^3*c^6*d
^14 + 66*a^10*b^2*c^5*d^15 - 12*a^11*b*c^4*d^16 + a^12*c^3*d^17))))) + 32*(-a^5*
b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 +
495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c
^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a
^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2
*c^2*d^3 - 2*a*b*c*d^4 + a^2*d^5)*x^4 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d
^4)*x^2)*log(a*b*sqrt(x) + (-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10
*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 92
4*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*
d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^3*c^3 - 3*a
*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)) - 32*(-a^5*b^3/(b^12*c^12 - 12*a*b^11*c^1
1*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5
*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 -
220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12))^(1/
4)*(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3 - 2*a*b*c*d^4 + a^2*d
^5)*x^4 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2)*log(a*b*sqrt(x) - (-a
^5*b^3/(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^
3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^
5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 1
2*a^11*b*c*d^11 + a^12*d^12))^(1/4)*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a
^3*d^3)) - (b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3 - 2*a*b*c*d^4
+ a^2*d^5)*x^4 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2)*(-(81*b^8*c^8
- 3240*a*b^7*c^7*d + 48060*a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 649350*a^
4*b^4*c^4*d^4 + 513000*a^5*b^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*
d^7 + 625*a^8*d^8)/(b^12*c^15*d^5 - 12*a*b^11*c^14*d^6 + 66*a^2*b^10*c^13*d^7 -
220*a^3*b^9*c^12*d^8 + 495*a^4*b^8*c^11*d^9 - 792*a^5*b^7*c^10*d^10 + 924*a^6*b^
6*c^9*d^11 - 792*a^7*b^5*c^8*d^12 + 495*a^8*b^4*c^7*d^13 - 220*a^9*b^3*c^6*d^14
+ 66*a^10*b^2*c^5*d^15 - 12*a^11*b*c^4*d^16 + a^12*c^3*d^17))^(1/4)*log(-(3*b^2*
c^2 - 30*a*b*c*d - 5*a^2*d^2)*sqrt(x) + (b^3*c^4*d - 3*a*b^2*c^3*d^2 + 3*a^2*b*c
^2*d^3 - a^3*c*d^4)*(-(81*b^8*c^8 - 3240*a*b^7*c^7*d + 48060*a^2*b^6*c^6*d^2 - 3
07800*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 + 513000*a^5*b^3*c^3*d^5 + 133500
*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*d^7 + 625*a^8*d^8)/(b^12*c^15*d^5 - 12*a*b^11*c
^14*d^6 + 66*a^2*b^10*c^13*d^7 - 220*a^3*b^9*c^12*d^8 + 495*a^4*b^8*c^11*d^9 - 7
92*a^5*b^7*c^10*d^10 + 924*a^6*b^6*c^9*d^11 - 792*a^7*b^5*c^8*d^12 + 495*a^8*b^4
*c^7*d^13 - 220*a^9*b^3*c^6*d^14 + 66*a^10*b^2*c^5*d^15 - 12*a^11*b*c^4*d^16 + a
^12*c^3*d^17))^(1/4)) + (b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3
- 2*a*b*c*d^4 + a^2*d^5)*x^4 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2)*
(-(81*b^8*c^8 - 3240*a*b^7*c^7*d + 48060*a^2*b^6*c^6*d^2 - 307800*a^3*b^5*c^5*d^
3 + 649350*a^4*b^4*c^4*d^4 + 513000*a^5*b^3*c^3*d^5 + 133500*a^6*b^2*c^2*d^6 + 1
5000*a^7*b*c*d^7 + 625*a^8*d^8)/(b^12*c^15*d^5 - 12*a*b^11*c^14*d^6 + 66*a^2*b^1
0*c^13*d^7 - 220*a^3*b^9*c^12*d^8 + 495*a^4*b^8*c^11*d^9 - 792*a^5*b^7*c^10*d^10
+ 924*a^6*b^6*c^9*d^11 - 792*a^7*b^5*c^8*d^12 + 495*a^8*b^4*c^7*d^13 - 220*a^9*
b^3*c^6*d^14 + 66*a^10*b^2*c^5*d^15 - 12*a^11*b*c^4*d^16 + a^12*c^3*d^17))^(1/4)
*log(-(3*b^2*c^2 - 30*a*b*c*d - 5*a^2*d^2)*sqrt(x) - (b^3*c^4*d - 3*a*b^2*c^3*d^
2 + 3*a^2*b*c^2*d^3 - a^3*c*d^4)*(-(81*b^8*c^8 - 3240*a*b^7*c^7*d + 48060*a^2*b^
6*c^6*d^2 - 307800*a^3*b^5*c^5*d^3 + 649350*a^4*b^4*c^4*d^4 + 513000*a^5*b^3*c^3
*d^5 + 133500*a^6*b^2*c^2*d^6 + 15000*a^7*b*c*d^7 + 625*a^8*d^8)/(b^12*c^15*d^5
- 12*a*b^11*c^14*d^6 + 66*a^2*b^10*c^13*d^7 - 220*a^3*b^9*c^12*d^8 + 495*a^4*b^8
*c^11*d^9 - 792*a^5*b^7*c^10*d^10 + 924*a^6*b^6*c^9*d^11 - 792*a^7*b^5*c^8*d^12
+ 495*a^8*b^4*c^7*d^13 - 220*a^9*b^3*c^6*d^14 + 66*a^10*b^2*c^5*d^15 - 12*a^11*b
*c^4*d^16 + a^12*c^3*d^17))^(1/4)) - 4*(3*b*c^2 + 5*a*c*d - (b*c*d - 9*a*d^2)*x^
2)*sqrt(x))/(b^2*c^4*d - 2*a*b*c^3*d^2 + a^2*c^2*d^3 + (b^2*c^2*d^3 - 2*a*b*c*d^
4 + a^2*d^5)*x^4 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*x^2)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.394611, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done