3.637 \(\int \frac{x^7}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx\)

Optimal. Leaf size=74 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4}}{2 b d} \]

[Out]

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

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Rubi [A]  time = 0.182875, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4}}{2 b d} \]

Antiderivative was successfully verified.

[In]  Int[x^7/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.331, size = 60, normalized size = 0.81 \[ - \frac{a \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{4}}}{\sqrt{a d - b c}} \right )}}{2 b^{\frac{3}{2}} \sqrt{a d - b c}} + \frac{\sqrt{c + d x^{4}}}{2 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0898224, size = 74, normalized size = 1. \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^4}}{\sqrt{b c-a d}}\right )}{2 b^{3/2} \sqrt{b c-a d}}+\frac{\sqrt{c+d x^4}}{2 b d} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((a + b*x^4)*Sqrt[c + d*x^4]),x]

[Out]

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

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Maple [B]  time = 0.012, size = 335, normalized size = 4.5 \[{\frac{1}{2\,bd}\sqrt{d{x}^{4}+c}}+{\frac{a}{4\,{b}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{a}{4\,{b}^{2}}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{\sqrt{-ab}d}{b} \left ({x}^{2}+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ({x}^{2}+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^4+a)/(d*x^4+c)^(1/2),x)

[Out]

1/2*(d*x^4+c)^(1/2)/b/d+1/4*a/b^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^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^7/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250947, size = 1, normalized size = 0.01 \[ \left [\frac{a d \log \left (\frac{{\left (b d x^{4} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} + 2 \, \sqrt{d x^{4} + c}{\left (b^{2} c - a b d\right )}}{b x^{4} + a}\right ) + 2 \, \sqrt{d x^{4} + c} \sqrt{b^{2} c - a b d}}{4 \, \sqrt{b^{2} c - a b d} b d}, \frac{a d \arctan \left (-\frac{b c - a d}{\sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}\right ) + \sqrt{d x^{4} + c} \sqrt{-b^{2} c + a b d}}{2 \, \sqrt{-b^{2} c + a b d} b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**4+a)/(d*x**4+c)**(1/2),x)

[Out]

Integral(x**7/((a + b*x**4)*sqrt(c + d*x**4)), x)

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GIAC/XCAS [A]  time = 0.212602, size = 86, normalized size = 1.16 \[ -\frac{\frac{a d \arctan \left (\frac{\sqrt{d x^{4} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b} - \frac{\sqrt{d x^{4} + c}}{b}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^4 + a)*sqrt(d*x^4 + c)),x, algorithm="giac")

[Out]

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