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

Optimal. Leaf size=624 \[ \frac{b^{5/4} (b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d x^{3/2} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2}+\frac{b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 1.83762, antiderivative size = 624, 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 \[ \frac{b^{5/4} (b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d x^{3/2} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2}+\frac{b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 3.30798, size = 589, normalized size = 0.94 \[ \frac{1}{16} \left (\frac{\sqrt{2} b^{5/4} (9 a d-b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (a d-b c)^3}+\frac{\sqrt{2} b^{5/4} (b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{5/4} (9 a d-b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4} (a d-b c)^3}+\frac{8 b^2 x^{3/2}}{a \left (a+b x^2\right ) (b c-a d)^2}+\frac{\sqrt{2} d^{5/4} (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4} (b c-a d)^3}+\frac{\sqrt{2} d^{5/4} (a d-9 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{5/4} (a d-9 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4} (b c-a d)^3}+\frac{8 d^2 x^{3/2}}{c \left (c+d x^2\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((8*b^2*x^(3/2))/(a*(b*c - a*d)^2*(a + b*x^2)) + (8*d^2*x^(3/2))/(c*(b*c - a*d)^
2*(c + d*x^2)) + (2*Sqrt[2]*b^(5/4)*(b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sq
rt[x])/a^(1/4)])/(a^(5/4)*(-(b*c) + a*d)^3) + (2*Sqrt[2]*b^(5/4)*(-(b*c) + 9*a*d
)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(5/4)*(-(b*c) + a*d)^3) + (2
*Sqrt[2]*d^(5/4)*(-9*b*c + a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(
c^(5/4)*(b*c - a*d)^3) + (2*Sqrt[2]*d^(5/4)*(9*b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^
(1/4)*Sqrt[x])/c^(1/4)])/(c^(5/4)*(b*c - a*d)^3) + (Sqrt[2]*b^(5/4)*(-(b*c) + 9*
a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(-(b*c
) + a*d)^3) + (Sqrt[2]*b^(5/4)*(b*c - 9*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/
4)*Sqrt[x] + Sqrt[b]*x])/(a^(5/4)*(-(b*c) + a*d)^3) + (Sqrt[2]*d^(5/4)*(9*b*c -
a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(5/4)*(b*c -
a*d)^3) + (Sqrt[2]*d^(5/4)*(-9*b*c + a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)
*Sqrt[x] + Sqrt[d]*x])/(c^(5/4)*(b*c - a*d)^3))/16

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Maple [A]  time = 0.03, size = 778, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^3/(a*d-b*c)^3/c*x^(3/2)/(d*x^2+c)*a-1/2*d^2/(a*d-b*c)^3*x^(3/2)/(d*x^2+c)*
b+1/8*d^2/(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-9/8*d/(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+1/16*d^2/(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-9/16*d/(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+1/8*d^2/(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-9/8*d/(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+1/2*b^2/(a*d-b*c)^3*x^(3/2)/(b*x^2+a)*d-1/2*b^3/
(a*d-b*c)^3/a*x^(3/2)/(b*x^2+a)*c+9/8*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)*d-1/8*b^2/(a*d-b*c)^3/a/(a/b)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c+9/16*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)))*d-1/16*b^2/(a*d-b*c)^3/a/(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+9/8*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)*d-1/8*b^2/(a*d-b*c)^
3/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*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(sqrt(x)/((b*x^2 + a)^2*(d*x^2 + c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 73.8499, size = 7488, normalized size = 12. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(4*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2
*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3
)*x^2)*(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d^3 +
6561*a^4*b^5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 2
20*a^8*b^9*c^9*d^3 + 495*a^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c
^6*d^6 - 792*a^12*b^5*c^5*d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c^3*d^9 + 66
*a^15*b^2*c^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12))^(1/4)*arctan((a^4*b^9*c^9 -
9*a^5*b^8*c^8*d + 36*a^6*b^7*c^7*d^2 - 84*a^7*b^6*c^6*d^3 + 126*a^8*b^5*c^5*d^4
- 126*a^9*b^4*c^4*d^5 + 84*a^10*b^3*c^3*d^6 - 36*a^11*b^2*c^2*d^7 + 9*a^12*b*c*d
^8 - a^13*d^9)*(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*
c*d^3 + 6561*a^4*b^5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10
*d^2 - 220*a^8*b^9*c^9*d^3 + 495*a^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^
11*b^6*c^6*d^6 - 792*a^12*b^5*c^5*d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c^3*
d^9 + 66*a^15*b^2*c^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12))^(3/4)/((b^7*c^3 - 27
*a*b^6*c^2*d + 243*a^2*b^5*c*d^2 - 729*a^3*b^4*d^3)*sqrt(x) - sqrt((b^14*c^6 - 5
4*a*b^13*c^5*d + 1215*a^2*b^12*c^4*d^2 - 14580*a^3*b^11*c^3*d^3 + 98415*a^4*b^10
*c^2*d^4 - 354294*a^5*b^9*c*d^5 + 531441*a^6*b^8*d^6)*x - (a^3*b^15*c^10 - 42*a^
4*b^14*c^9*d + 717*a^5*b^13*c^8*d^2 - 6392*a^6*b^12*c^7*d^3 + 32082*a^7*b^11*c^6
*d^4 - 93372*a^8*b^10*c^5*d^5 + 164242*a^9*b^9*c^4*d^6 - 177912*a^10*b^8*c^3*d^7
+ 116397*a^11*b^7*c^2*d^8 - 42282*a^12*b^6*c*d^9 + 6561*a^13*b^5*d^10)*sqrt(-(b
^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d^3 + 6561*a^4*b^
5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 220*a^8*b^9*
c^9*d^3 + 495*a^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c^6*d^6 - 79
2*a^12*b^5*c^5*d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c^3*d^9 + 66*a^15*b^2*c
^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12))))) + 4*(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a
^4*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 -
a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^2)*(-(6561*b^4*c^4*d^5 - 2916*a*b^3
*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8 + a^4*d^9)/(b^12*c^17 - 12*a*b^1
1*c^16*d + 66*a^2*b^10*c^15*d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 -
792*a^5*b^7*c^12*d^5 + 924*a^6*b^6*c^11*d^6 - 792*a^7*b^5*c^10*d^7 + 495*a^8*b^4
*c^9*d^8 - 220*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^11 + a^1
2*c^5*d^12))^(1/4)*arctan((b^9*c^13 - 9*a*b^8*c^12*d + 36*a^2*b^7*c^11*d^2 - 84*
a^3*b^6*c^10*d^3 + 126*a^4*b^5*c^9*d^4 - 126*a^5*b^4*c^8*d^5 + 84*a^6*b^3*c^7*d^
6 - 36*a^7*b^2*c^6*d^7 + 9*a^8*b*c^5*d^8 - a^9*c^4*d^9)*(-(6561*b^4*c^4*d^5 - 29
16*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8 + a^4*d^9)/(b^12*c^17 -
12*a*b^11*c^16*d + 66*a^2*b^10*c^15*d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^1
3*d^4 - 792*a^5*b^7*c^12*d^5 + 924*a^6*b^6*c^11*d^6 - 792*a^7*b^5*c^10*d^7 + 495
*a^8*b^4*c^9*d^8 - 220*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^
11 + a^12*c^5*d^12))^(3/4)/((729*b^3*c^3*d^4 - 243*a*b^2*c^2*d^5 + 27*a^2*b*c*d^
6 - a^3*d^7)*sqrt(x) - sqrt((531441*b^6*c^6*d^8 - 354294*a*b^5*c^5*d^9 + 98415*a
^2*b^4*c^4*d^10 - 14580*a^3*b^3*c^3*d^11 + 1215*a^4*b^2*c^2*d^12 - 54*a^5*b*c*d^
13 + a^6*d^14)*x - (6561*b^10*c^13*d^5 - 42282*a*b^9*c^12*d^6 + 116397*a^2*b^8*c
^11*d^7 - 177912*a^3*b^7*c^10*d^8 + 164242*a^4*b^6*c^9*d^9 - 93372*a^5*b^5*c^8*d
^10 + 32082*a^6*b^4*c^7*d^11 - 6392*a^7*b^3*c^6*d^12 + 717*a^8*b^2*c^5*d^13 - 42
*a^9*b*c^4*d^14 + a^10*c^3*d^15)*sqrt(-(6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 +
486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8 + a^4*d^9)/(b^12*c^17 - 12*a*b^11*c^16*d +
66*a^2*b^10*c^15*d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5*b^7
*c^12*d^5 + 924*a^6*b^6*c^11*d^6 - 792*a^7*b^5*c^10*d^7 + 495*a^8*b^4*c^9*d^8 -
220*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^11 + a^12*c^5*d^12)
)))) - (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2
*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3
)*x^2)*(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d^3 +
6561*a^4*b^5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 2
20*a^8*b^9*c^9*d^3 + 495*a^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c
^6*d^6 - 792*a^12*b^5*c^5*d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c^3*d^9 + 66
*a^15*b^2*c^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12))^(1/4)*log((a^4*b^9*c^9 - 9*a
^5*b^8*c^8*d + 36*a^6*b^7*c^7*d^2 - 84*a^7*b^6*c^6*d^3 + 126*a^8*b^5*c^5*d^4 - 1
26*a^9*b^4*c^4*d^5 + 84*a^10*b^3*c^3*d^6 - 36*a^11*b^2*c^2*d^7 + 9*a^12*b*c*d^8
- a^13*d^9)*(-(b^9*c^4 - 36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d
^3 + 6561*a^4*b^5*d^4)/(a^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^
2 - 220*a^8*b^9*c^9*d^3 + 495*a^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*
b^6*c^6*d^6 - 792*a^12*b^5*c^5*d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c^3*d^9
+ 66*a^15*b^2*c^2*d^10 - 12*a^16*b*c*d^11 + a^17*d^12))^(3/4) - (b^7*c^3 - 27*a
*b^6*c^2*d + 243*a^2*b^5*c*d^2 - 729*a^3*b^4*d^3)*sqrt(x)) + (a^2*b^2*c^4 - 2*a^
3*b*c^3*d + a^4*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^4 +
(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^2)*(-(b^9*c^4 - 36*a*b
^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d^3 + 6561*a^4*b^5*d^4)/(a^5*b^1
2*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 220*a^8*b^9*c^9*d^3 + 495*a
^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c^6*d^6 - 792*a^12*b^5*c^5*
d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c^3*d^9 + 66*a^15*b^2*c^2*d^10 - 12*a^
16*b*c*d^11 + a^17*d^12))^(1/4)*log(-(a^4*b^9*c^9 - 9*a^5*b^8*c^8*d + 36*a^6*b^7
*c^7*d^2 - 84*a^7*b^6*c^6*d^3 + 126*a^8*b^5*c^5*d^4 - 126*a^9*b^4*c^4*d^5 + 84*a
^10*b^3*c^3*d^6 - 36*a^11*b^2*c^2*d^7 + 9*a^12*b*c*d^8 - a^13*d^9)*(-(b^9*c^4 -
36*a*b^8*c^3*d + 486*a^2*b^7*c^2*d^2 - 2916*a^3*b^6*c*d^3 + 6561*a^4*b^5*d^4)/(a
^5*b^12*c^12 - 12*a^6*b^11*c^11*d + 66*a^7*b^10*c^10*d^2 - 220*a^8*b^9*c^9*d^3 +
495*a^9*b^8*c^8*d^4 - 792*a^10*b^7*c^7*d^5 + 924*a^11*b^6*c^6*d^6 - 792*a^12*b^
5*c^5*d^7 + 495*a^13*b^4*c^4*d^8 - 220*a^14*b^3*c^3*d^9 + 66*a^15*b^2*c^2*d^10 -
12*a^16*b*c*d^11 + a^17*d^12))^(3/4) - (b^7*c^3 - 27*a*b^6*c^2*d + 243*a^2*b^5*
c*d^2 - 729*a^3*b^4*d^3)*sqrt(x)) - (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 +
(a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^2*b^2*c^3*
d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^2)*(-(6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 4
86*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8 + a^4*d^9)/(b^12*c^17 - 12*a*b^11*c^16*d + 6
6*a^2*b^10*c^15*d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5*b^7*
c^12*d^5 + 924*a^6*b^6*c^11*d^6 - 792*a^7*b^5*c^10*d^7 + 495*a^8*b^4*c^9*d^8 - 2
20*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^11 + a^12*c^5*d^12))
^(1/4)*log((b^9*c^13 - 9*a*b^8*c^12*d + 36*a^2*b^7*c^11*d^2 - 84*a^3*b^6*c^10*d^
3 + 126*a^4*b^5*c^9*d^4 - 126*a^5*b^4*c^8*d^5 + 84*a^6*b^3*c^7*d^6 - 36*a^7*b^2*
c^6*d^7 + 9*a^8*b*c^5*d^8 - a^9*c^4*d^9)*(-(6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^
6 + 486*a^2*b^2*c^2*d^7 - 36*a^3*b*c*d^8 + a^4*d^9)/(b^12*c^17 - 12*a*b^11*c^16*
d + 66*a^2*b^10*c^15*d^2 - 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5
*b^7*c^12*d^5 + 924*a^6*b^6*c^11*d^6 - 792*a^7*b^5*c^10*d^7 + 495*a^8*b^4*c^9*d^
8 - 220*a^9*b^3*c^8*d^9 + 66*a^10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^11 + a^12*c^5*d
^12))^(3/4) - (729*b^3*c^3*d^4 - 243*a*b^2*c^2*d^5 + 27*a^2*b*c*d^6 - a^3*d^7)*s
qrt(x)) + (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*
c^2*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*
d^3)*x^2)*(-(6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 - 36*a^
3*b*c*d^8 + a^4*d^9)/(b^12*c^17 - 12*a*b^11*c^16*d + 66*a^2*b^10*c^15*d^2 - 220*
a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5*b^7*c^12*d^5 + 924*a^6*b^6*c^1
1*d^6 - 792*a^7*b^5*c^10*d^7 + 495*a^8*b^4*c^9*d^8 - 220*a^9*b^3*c^8*d^9 + 66*a^
10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^11 + a^12*c^5*d^12))^(1/4)*log(-(b^9*c^13 - 9*
a*b^8*c^12*d + 36*a^2*b^7*c^11*d^2 - 84*a^3*b^6*c^10*d^3 + 126*a^4*b^5*c^9*d^4 -
126*a^5*b^4*c^8*d^5 + 84*a^6*b^3*c^7*d^6 - 36*a^7*b^2*c^6*d^7 + 9*a^8*b*c^5*d^8
- a^9*c^4*d^9)*(-(6561*b^4*c^4*d^5 - 2916*a*b^3*c^3*d^6 + 486*a^2*b^2*c^2*d^7 -
36*a^3*b*c*d^8 + a^4*d^9)/(b^12*c^17 - 12*a*b^11*c^16*d + 66*a^2*b^10*c^15*d^2
- 220*a^3*b^9*c^14*d^3 + 495*a^4*b^8*c^13*d^4 - 792*a^5*b^7*c^12*d^5 + 924*a^6*b
^6*c^11*d^6 - 792*a^7*b^5*c^10*d^7 + 495*a^8*b^4*c^9*d^8 - 220*a^9*b^3*c^8*d^9 +
66*a^10*b^2*c^7*d^10 - 12*a^11*b*c^6*d^11 + a^12*c^5*d^12))^(3/4) - (729*b^3*c^
3*d^4 - 243*a*b^2*c^2*d^5 + 27*a^2*b*c*d^6 - a^3*d^7)*sqrt(x)) + 4*((b^2*c*d + a
*b*d^2)*x^3 + (b^2*c^2 + a^2*d^2)*x)*sqrt(x))/(a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4
*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x^4 + (a*b^3*c^4 - a^
2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*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)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x}}{{\left (b x^{2} + a\right )}^{2}{\left (d x^{2} + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x)/((b*x^2 + a)^2*(d*x^2 + c)^2), x)