3.661 \(\int \frac{x^5}{\left (a+b x^4\right )^2 \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=93 \[ \frac{c \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^2 \sqrt{c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)} \]

[Out]

-(x^2*Sqrt[c + d*x^4])/(4*(b*c - a*d)*(a + b*x^4)) + (c*ArcTan[(Sqrt[b*c - a*d]*
x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(4*Sqrt[a]*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.252799, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{c \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{4 \sqrt{a} (b c-a d)^{3/2}}-\frac{x^2 \sqrt{c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

-(x^2*Sqrt[c + d*x^4])/(4*(b*c - a*d)*(a + b*x^4)) + (c*ArcTan[(Sqrt[b*c - a*d]*
x^2)/(Sqrt[a]*Sqrt[c + d*x^4])])/(4*Sqrt[a]*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 29.4446, size = 76, normalized size = 0.82 \[ \frac{x^{2} \sqrt{c + d x^{4}}}{4 \left (a + b x^{4}\right ) \left (a d - b c\right )} - \frac{c \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{4 \sqrt{a} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

x**2*sqrt(c + d*x**4)/(4*(a + b*x**4)*(a*d - b*c)) - c*atanh(x**2*sqrt(a*d - b*c
)/(sqrt(a)*sqrt(c + d*x**4)))/(4*sqrt(a)*(a*d - b*c)**(3/2))

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Mathematica [A]  time = 0.163808, size = 90, normalized size = 0.97 \[ \frac{\frac{x^2 \sqrt{c+d x^4}}{a+b x^4}-\frac{c \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{\sqrt{a} \sqrt{b c-a d}}}{4 a d-4 b c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]

[Out]

((x^2*Sqrt[c + d*x^4])/(a + b*x^4) - (c*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sq
rt[c + d*x^4])])/(Sqrt[a]*Sqrt[b*c - a*d]))/(-4*b*c + 4*a*d)

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Maple [B]  time = 0.017, size = 861, normalized size = 9.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)

[Out]

-1/8/b/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(
x^2-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(-a
*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2-1/b*(-a*b)^(1/2)))+1
/8/b/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x^
2+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b
)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2+1/b*(-a*b)^(1/2)))+1/8
/b/(a*d-b*c)/(x^2-1/b*(-a*b)^(1/2))*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)
/b*(x^2-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/8/b^2*d*(-a*b)^(1/2)/(a*d-b*c)/(-
(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^(1/2))+
2*(-(a*d-b*c)/b)^(1/2)*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(
-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2-1/b*(-a*b)^(1/2)))+1/8/b/(a*d-b*c)/(x^2+1/
b*(-a*b)^(1/2))*((x^2+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(
1/2))-(a*d-b*c)/b)^(1/2)+1/8/b^2*d*(-a*b)^(1/2)/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*l
n((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/
2)*((x^2+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*
c)/b)^(1/2))/(x^2+1/b*(-a*b)^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{{\left (b x^{4} + a\right )}^{2} \sqrt{d x^{4} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

integrate(x^5/((b*x^4 + a)^2*sqrt(d*x^4 + c)), x)

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Fricas [A]  time = 0.344302, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{d x^{4} + c} \sqrt{-a b c + a^{2} d} x^{2} +{\left (b c x^{4} + a c\right )} \log \left (-\frac{4 \,{\left ({\left (a b^{2} c^{2} - 3 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} x^{6} -{\left (a^{2} b c^{2} - a^{3} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} -{\left ({\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2}\right )} \sqrt{-a b c + a^{2} d}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{16 \,{\left ({\left (b^{2} c - a b d\right )} x^{4} + a b c - a^{2} d\right )} \sqrt{-a b c + a^{2} d}}, -\frac{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2} -{\left (b c x^{4} + a c\right )} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c} \sqrt{a b c - a^{2} d} x^{2}}\right )}{8 \,{\left ({\left (b^{2} c - a b d\right )} x^{4} + a b c - a^{2} d\right )} \sqrt{a b c - a^{2} d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

[-1/16*(4*sqrt(d*x^4 + c)*sqrt(-a*b*c + a^2*d)*x^2 + (b*c*x^4 + a*c)*log(-(4*((a
*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*x^6 - (a^2*b*c^2 - a^3*c*d)*x^2)*sqrt(d*x^4
+ c) - ((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 +
a^2*c^2)*sqrt(-a*b*c + a^2*d))/(b^2*x^8 + 2*a*b*x^4 + a^2)))/(((b^2*c - a*b*d)*x
^4 + a*b*c - a^2*d)*sqrt(-a*b*c + a^2*d)), -1/8*(2*sqrt(d*x^4 + c)*sqrt(a*b*c -
a^2*d)*x^2 - (b*c*x^4 + a*c)*arctan(1/2*((b*c - 2*a*d)*x^4 - a*c)/(sqrt(d*x^4 +
c)*sqrt(a*b*c - a^2*d)*x^2)))/(((b^2*c - a*b*d)*x^4 + a*b*c - a^2*d)*sqrt(a*b*c
- a^2*d))]

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.224907, size = 124, normalized size = 1.33 \[ -\frac{1}{4} \, c{\left (\frac{\arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d}{\left (b c - a d\right )}} + \frac{\sqrt{d + \frac{c}{x^{4}}}}{{\left (b c + a{\left (d + \frac{c}{x^{4}}\right )} - a d\right )}{\left (b c - a d\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/((b*x^4 + a)^2*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

-1/4*c*(arctan(a*sqrt(d + c/x^4)/sqrt(a*b*c - a^2*d))/(sqrt(a*b*c - a^2*d)*(b*c
- a*d)) + sqrt(d + c/x^4)/((b*c + a*(d + c/x^4) - a*d)*(b*c - a*d)))