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

Optimal. Leaf size=91 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{2 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 b \sqrt{b c-a d}} \]

[Out]

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

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Rubi [A]  time = 0.247128, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{d} x^2}{\sqrt{c+d x^4}}\right )}{2 b \sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{2 b \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.208, size = 76, normalized size = 0.84 \[ - \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{x^{2} \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{4}}} \right )}}{2 b \sqrt{a d - b c}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{d} x^{2}}{\sqrt{c + d x^{4}}} \right )}}{2 b \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0879368, size = 90, normalized size = 0.99 \[ \frac{\frac{\log \left (\sqrt{d} \sqrt{c+d x^4}+d x^2\right )}{\sqrt{d}}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{x^2 \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^4}}\right )}{\sqrt{b c-a d}}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 356, normalized size = 3.9 \[{\frac{1}{2\,b}\ln \left ({x}^{2}\sqrt{d}+\sqrt{d{x}^{4}+c} \right ){\frac{1}{\sqrt{d}}}}+{\frac{a}{4\,b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}-{\frac{a}{4\,b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ab}}}{\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(x^5/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.346451, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{6} -{\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 2 \, \log \left (-2 \, \sqrt{d x^{4} + c} d x^{2} -{\left (2 \, d x^{4} + c\right )} \sqrt{d}\right )}{8 \, b \sqrt{d}}, \frac{\sqrt{-d} \sqrt{-\frac{a}{b c - a d}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} - 4 \,{\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{6} -{\left (a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{4} + c} \sqrt{-\frac{a}{b c - a d}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right ) + 4 \, \arctan \left (\frac{\sqrt{-d} x^{2}}{\sqrt{d x^{4} + c}}\right )}{8 \, b \sqrt{-d}}, -\frac{\sqrt{d} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c}{\left (b c - a d\right )} x^{2} \sqrt{\frac{a}{b c - a d}}}\right ) - \log \left (-2 \, \sqrt{d x^{4} + c} d x^{2} -{\left (2 \, d x^{4} + c\right )} \sqrt{d}\right )}{4 \, b \sqrt{d}}, -\frac{\sqrt{-d} \sqrt{\frac{a}{b c - a d}} \arctan \left (\frac{{\left (b c - 2 \, a d\right )} x^{4} - a c}{2 \, \sqrt{d x^{4} + c}{\left (b c - a d\right )} x^{2} \sqrt{\frac{a}{b c - a d}}}\right ) - 2 \, \arctan \left (\frac{\sqrt{-d} x^{2}}{\sqrt{d x^{4} + c}}\right )}{4 \, b \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**5/((a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.224328, size = 107, normalized size = 1.18 \[ \frac{1}{2} \, c{\left (\frac{a \arctan \left (\frac{a \sqrt{d + \frac{c}{x^{4}}}}{\sqrt{a b c - a^{2} d}}\right )}{\sqrt{a b c - a^{2} d} b c} - \frac{\arctan \left (\frac{\sqrt{d + \frac{c}{x^{4}}}}{\sqrt{-d}}\right )}{b c \sqrt{-d}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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