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

Optimal. Leaf size=633 \[ -\frac{\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} (b c-a d)^3}-\frac{\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{b^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{b^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^3}+\frac{b^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{d x^{3/2} (13 b c-5 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d x^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(d*x^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (d*(13*b*c - 5*a*d)*x^(3/2))/(16*
c^2*(b*c - a*d)^2*(c + d*x^2)) - (b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a
^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*
Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (d^(1/4)*(45*b^2*c^2 - 18*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
^(9/4)*(b*c - a*d)^3) - (d^(1/4)*(45*b^2*c^2 - 18*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^(9/4)*(b*c - a*d)^3) + (b^(9
/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/
4)*(b*c - a*d)^3) - (b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x])/(2*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) - (d^(1/4)*(45*b^2*c^2 - 18*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^(9/4)*(b*c - a*d)^3) + (d^(1/4)*(45*b^2*c^2 - 18*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^(9/4)*(b*c
- a*d)^3)

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Rubi [A]  time = 1.77318, antiderivative size = 633, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} (b c-a d)^3}-\frac{\sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{9/4} (b c-a d)^3}+\frac{b^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{b^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^3}+\frac{b^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} (b c-a d)^3}-\frac{d x^{3/2} (13 b c-5 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d x^{3/2}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*x^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)^2) - (d*(13*b*c - 5*a*d)*x^(3/2))/(16*
c^2*(b*c - a*d)^2*(c + d*x^2)) - (b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a
^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*
Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)^3) + (d^(1/4)*(45*b^2*c^2 - 18*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
^(9/4)*(b*c - a*d)^3) - (d^(1/4)*(45*b^2*c^2 - 18*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^(9/4)*(b*c - a*d)^3) + (b^(9
/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/
4)*(b*c - a*d)^3) - (b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x])/(2*Sqrt[2]*a^(1/4)*(b*c - a*d)^3) - (d^(1/4)*(45*b^2*c^2 - 18*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^(9/4)*(b*c - a*d)^3) + (d^(1/4)*(45*b^2*c^2 - 18*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^(9/4)*(b*c
- a*d)^3)

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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(x**(1/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.20567, size = 620, normalized size = 0.98 \[ \frac{1}{128} \left (\frac{\sqrt{2} \sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{9/4} (a d-b c)^3}+\frac{\sqrt{2} \sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{9/4} (b c-a d)^3}+\frac{2 \sqrt{2} \sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{9/4} (b c-a d)^3}-\frac{2 \sqrt{2} \sqrt [4]{d} \left (5 a^2 d^2-18 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{9/4} (b c-a d)^3}+\frac{32 \sqrt{2} b^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a} (b c-a d)^3}+\frac{32 \sqrt{2} b^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a} (a d-b c)^3}+\frac{64 \sqrt{2} b^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} (a d-b c)^3}-\frac{64 \sqrt{2} b^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a} (a d-b c)^3}+\frac{8 d x^{3/2} (5 a d-13 b c)}{c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{32 d x^{3/2}}{c \left (c+d x^2\right )^2 (b c-a d)}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-32*d*x^(3/2))/(c*(b*c - a*d)*(c + d*x^2)^2) + (8*d*(-13*b*c + 5*a*d)*x^(3/2))
/(c^2*(b*c - a*d)^2*(c + d*x^2)) + (64*Sqrt[2]*b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/
4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) - (64*Sqrt[2]*b^(9/4)*ArcTan[1
+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(1/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*d^
(1/4)*(45*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])
/c^(1/4)])/(c^(9/4)*(b*c - a*d)^3) - (2*Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*a*b*c*d
+ 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(9/4)*(b*c - a*d
)^3) + (32*Sqrt[2]*b^(9/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[
b]*x])/(a^(1/4)*(b*c - a*d)^3) + (32*Sqrt[2]*b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(
45*b^2*c^2 - 18*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])/(c^(9/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(1/4)*(45*b^2*c^2 - 18*
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])
/(c^(9/4)*(b*c - a*d)^3))/128

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Maple [A]  time = 0.027, size = 855, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/16*d^4/(a*d-b*c)^3/(d*x^2+c)^2/c^2*x^(7/2)*a^2-9/8*d^3/(a*d-b*c)^3/(d*x^2+c)^2
/c*x^(7/2)*a*b+13/16*d^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(7/2)*b^2+9/16*d^3/(a*d-b*c)^
3/(d*x^2+c)^2/c*x^(3/2)*a^2-13/8*d^2/(a*d-b*c)^3/(d*x^2+c)^2*x^(3/2)*a*b+17/16*d
/(a*d-b*c)^3/(d*x^2+c)^2*c*x^(3/2)*b^2+5/64*d^2/(a*d-b*c)^3/c^2/(c/d)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-9/32*d/(a*d-b*c)^3/c/(c/d)^(1/4)*2
^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b+45/64/(a*d-b*c)^3/(c/d)^(1/4)*2
^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+5/128*d^2/(a*d-b*c)^3/c^2/(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^2-9/64*d/(a*d-b*c)^3/c/(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+45/128/(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)))*b^2+5/64*d^2/(a*d-b*c
)^3/c^2/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-9/32*d/(a*
d-b*c)^3/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b+45/64/(
a*d-b*c)^3/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2-1/4*b^2
/(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*b^2/(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*b^2/(a*d-b*c)^3/(a/b)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 83.491, size = 7324, normalized size = 11.57 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(128*(-b^9/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*
a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^
6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b
^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))^(1/4)*(b^2*c^6 - 2*a*b*c^5*d + a^2*
c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c^5*d - 2*a*b
*c^4*d^2 + a^2*c^3*d^3)*x^2)*arctan(-(a*b^9*c^9 - 9*a^2*b^8*c^8*d + 36*a^3*b^7*c
^7*d^2 - 84*a^4*b^6*c^6*d^3 + 126*a^5*b^5*c^5*d^4 - 126*a^6*b^4*c^4*d^5 + 84*a^7
*b^3*c^3*d^6 - 36*a^8*b^2*c^2*d^7 + 9*a^9*b*c*d^8 - a^10*d^9)*(-b^9/(a*b^12*c^12
- 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8
*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495
*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^1
1 + a^13*d^12))^(3/4)/(b^7*sqrt(x) + sqrt(b^14*x - (a*b^15*c^6 - 6*a^2*b^14*c^5*
d + 15*a^3*b^13*c^4*d^2 - 20*a^4*b^12*c^3*d^3 + 15*a^5*b^11*c^2*d^4 - 6*a^6*b^10
*c*d^5 + a^7*b^9*d^6)*sqrt(-b^9/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*
c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924
*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*
d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))))) - 4*(b^2*c^6 - 2*
a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b
^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*(-(4100625*b^8*c^8*d - 6561000*a*b^
7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*
c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8 + 62
5*a^8*d^9)/(b^12*c^21 - 12*a*b^11*c^20*d + 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*c^
18*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 - 79
2*a^7*b^5*c^14*d^7 + 495*a^8*b^4*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c
^11*d^10 - 12*a^11*b*c^10*d^11 + a^12*c^9*d^12))^(1/4)*arctan(-(b^9*c^16 - 9*a*b
^8*c^15*d + 36*a^2*b^7*c^14*d^2 - 84*a^3*b^6*c^13*d^3 + 126*a^4*b^5*c^12*d^4 - 1
26*a^5*b^4*c^11*d^5 + 84*a^6*b^3*c^10*d^6 - 36*a^7*b^2*c^9*d^7 + 9*a^8*b*c^8*d^8
- a^9*c^7*d^9)*(-(4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c
^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*
d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8 + 625*a^8*d^9)/(b^12*c^21 - 12*a*
b^11*c^20*d + 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*c^18*d^3 + 495*a^4*b^8*c^17*d^4
- 792*a^5*b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 - 792*a^7*b^5*c^14*d^7 + 495*a^8*
b^4*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c^11*d^10 - 12*a^11*b*c^10*d^1
1 + a^12*c^9*d^12))^(3/4)/((91125*b^6*c^6*d - 109350*a*b^5*c^5*d^2 + 74115*a^2*b
^4*c^4*d^3 - 30132*a^3*b^3*c^3*d^4 + 8235*a^4*b^2*c^2*d^5 - 1350*a^5*b*c*d^6 + 1
25*a^6*d^7)*sqrt(x) + sqrt((8303765625*b^12*c^12*d^2 - 19929037500*a*b^11*c^11*d
^3 + 25464881250*a^2*b^10*c^10*d^4 - 21700507500*a^3*b^9*c^9*d^5 + 13583730375*a
^4*b^8*c^8*d^6 - 6513498360*a^5*b^7*c^7*d^7 + 2446637724*a^6*b^6*c^6*d^8 - 72372
2040*a^7*b^5*c^5*d^9 + 167700375*a^8*b^4*c^4*d^10 - 29767500*a^9*b^3*c^3*d^11 +
3881250*a^10*b^2*c^2*d^12 - 337500*a^11*b*c*d^13 + 15625*a^12*d^14)*x - (4100625
*b^14*c^19*d - 31164750*a*b^13*c^18*d^2 + 106634475*a^2*b^12*c^17*d^3 - 21821886
0*a^3*b^11*c^16*d^4 + 299819961*a^4*b^10*c^15*d^5 - 294812946*a^5*b^9*c^14*d^6 +
216070155*a^6*b^8*c^13*d^7 - 121167720*a^7*b^7*c^12*d^8 + 52746475*a^8*b^6*c^11
*d^9 - 17893266*a^9*b^5*c^10*d^10 + 4697241*a^10*b^4*c^9*d^11 - 933740*a^11*b^3*
c^8*d^12 + 134475*a^12*b^2*c^7*d^13 - 12750*a^13*b*c^6*d^14 + 625*a^14*c^5*d^15)
*sqrt(-(4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 32
36760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100
*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8 + 625*a^8*d^9)/(b^12*c^21 - 12*a*b^11*c^20*d
+ 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*c^18*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*
b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 - 792*a^7*b^5*c^14*d^7 + 495*a^8*b^4*c^13*d^
8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c^11*d^10 - 12*a^11*b*c^10*d^11 + a^12*c^
9*d^12))))) - 32*(-b^9/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2
- 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*
c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*
a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))^(1/4)*(b^2*c^6 - 2*a*b*c^5*d
+ a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c^5*d -
2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*log(b^7*sqrt(x) + (a*b^9*c^9 - 9*a^2*b^8*c^8*
d + 36*a^3*b^7*c^7*d^2 - 84*a^4*b^6*c^6*d^3 + 126*a^5*b^5*c^5*d^4 - 126*a^6*b^4*
c^4*d^5 + 84*a^7*b^3*c^3*d^6 - 36*a^8*b^2*c^2*d^7 + 9*a^9*b*c*d^8 - a^10*d^9)*(-
b^9/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d
^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b
^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 -
12*a^12*b*c*d^11 + a^13*d^12))^(3/4)) + 32*(-b^9/(a*b^12*c^12 - 12*a^2*b^11*c^1
1*d + 66*a^3*b^10*c^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6
*b^7*c^7*d^5 + 924*a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 -
220*a^10*b^3*c^3*d^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))^(1
/4)*(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^
2*d^4)*x^4 + 2*(b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*log(b^7*sqrt(x) -
(a*b^9*c^9 - 9*a^2*b^8*c^8*d + 36*a^3*b^7*c^7*d^2 - 84*a^4*b^6*c^6*d^3 + 126*a^5
*b^5*c^5*d^4 - 126*a^6*b^4*c^4*d^5 + 84*a^7*b^3*c^3*d^6 - 36*a^8*b^2*c^2*d^7 + 9
*a^9*b*c*d^8 - a^10*d^9)*(-b^9/(a*b^12*c^12 - 12*a^2*b^11*c^11*d + 66*a^3*b^10*c
^10*d^2 - 220*a^4*b^9*c^9*d^3 + 495*a^5*b^8*c^8*d^4 - 792*a^6*b^7*c^7*d^5 + 924*
a^7*b^6*c^6*d^6 - 792*a^8*b^5*c^5*d^7 + 495*a^9*b^4*c^4*d^8 - 220*a^10*b^3*c^3*d
^9 + 66*a^11*b^2*c^2*d^10 - 12*a^12*b*c*d^11 + a^13*d^12))^(3/4)) + (b^2*c^6 - 2
*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(
b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*(-(4100625*b^8*c^8*d - 6561000*a*b
^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4
*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8 + 6
25*a^8*d^9)/(b^12*c^21 - 12*a*b^11*c^20*d + 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*c
^18*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 - 7
92*a^7*b^5*c^14*d^7 + 495*a^8*b^4*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*
c^11*d^10 - 12*a^11*b*c^10*d^11 + a^12*c^9*d^12))^(1/4)*log((b^9*c^16 - 9*a*b^8*
c^15*d + 36*a^2*b^7*c^14*d^2 - 84*a^3*b^6*c^13*d^3 + 126*a^4*b^5*c^12*d^4 - 126*
a^5*b^4*c^11*d^5 + 84*a^6*b^3*c^10*d^6 - 36*a^7*b^2*c^9*d^7 + 9*a^8*b*c^8*d^8 -
a^9*c^7*d^9)*(-(4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*
d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6
+ 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8 + 625*a^8*d^9)/(b^12*c^21 - 12*a*b^1
1*c^20*d + 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*c^18*d^3 + 495*a^4*b^8*c^17*d^4 -
792*a^5*b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 - 792*a^7*b^5*c^14*d^7 + 495*a^8*b^4
*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c^11*d^10 - 12*a^11*b*c^10*d^11 +
a^12*c^9*d^12))^(3/4) + (91125*b^6*c^6*d - 109350*a*b^5*c^5*d^2 + 74115*a^2*b^4
*c^4*d^3 - 30132*a^3*b^3*c^3*d^4 + 8235*a^4*b^2*c^2*d^5 - 1350*a^5*b*c*d^6 + 125
*a^6*d^7)*sqrt(x)) - (b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b
*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*(
-(4100625*b^8*c^8*d - 6561000*a*b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*
a^3*b^5*c^5*d^4 + 1283526*a^4*b^4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b
^2*c^2*d^7 - 9000*a^7*b*c*d^8 + 625*a^8*d^9)/(b^12*c^21 - 12*a*b^11*c^20*d + 66*
a^2*b^10*c^19*d^2 - 220*a^3*b^9*c^18*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*b^7*c^
16*d^5 + 924*a^6*b^6*c^15*d^6 - 792*a^7*b^5*c^14*d^7 + 495*a^8*b^4*c^13*d^8 - 22
0*a^9*b^3*c^12*d^9 + 66*a^10*b^2*c^11*d^10 - 12*a^11*b*c^10*d^11 + a^12*c^9*d^12
))^(1/4)*log(-(b^9*c^16 - 9*a*b^8*c^15*d + 36*a^2*b^7*c^14*d^2 - 84*a^3*b^6*c^13
*d^3 + 126*a^4*b^5*c^12*d^4 - 126*a^5*b^4*c^11*d^5 + 84*a^6*b^3*c^10*d^6 - 36*a^
7*b^2*c^9*d^7 + 9*a^8*b*c^8*d^8 - a^9*c^7*d^9)*(-(4100625*b^8*c^8*d - 6561000*a*
b^7*c^7*d^2 + 5759100*a^2*b^6*c^6*d^3 - 3236760*a^3*b^5*c^5*d^4 + 1283526*a^4*b^
4*c^4*d^5 - 359640*a^5*b^3*c^3*d^6 + 71100*a^6*b^2*c^2*d^7 - 9000*a^7*b*c*d^8 +
625*a^8*d^9)/(b^12*c^21 - 12*a*b^11*c^20*d + 66*a^2*b^10*c^19*d^2 - 220*a^3*b^9*
c^18*d^3 + 495*a^4*b^8*c^17*d^4 - 792*a^5*b^7*c^16*d^5 + 924*a^6*b^6*c^15*d^6 -
792*a^7*b^5*c^14*d^7 + 495*a^8*b^4*c^13*d^8 - 220*a^9*b^3*c^12*d^9 + 66*a^10*b^2
*c^11*d^10 - 12*a^11*b*c^10*d^11 + a^12*c^9*d^12))^(3/4) + (91125*b^6*c^6*d - 10
9350*a*b^5*c^5*d^2 + 74115*a^2*b^4*c^4*d^3 - 30132*a^3*b^3*c^3*d^4 + 8235*a^4*b^
2*c^2*d^5 - 1350*a^5*b*c*d^6 + 125*a^6*d^7)*sqrt(x)) + 4*((13*b*c*d^2 - 5*a*d^3)
*x^3 + (17*b*c^2*d - 9*a*c*d^2)*x)*sqrt(x))/(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2
+ (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c^5*d - 2*a*b*c^4*d^
2 + a^2*c^3*d^3)*x^2)

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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(x**(1/2)/(b*x**2+a)/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.426414, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done