3.720 \(\int \frac{x^7}{\left (a+b x^8\right ) \sqrt{c+d x^8}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \sqrt{b c-a d}} \]

[Out]

-ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]]/(4*Sqrt[b]*Sqrt[b*c - a*d])

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Rubi [A]  time = 0.135698, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

-ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]]/(4*Sqrt[b]*Sqrt[b*c - a*d])

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Rubi in Sympy [A]  time = 16.0112, size = 42, normalized size = 0.82 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{8}}}{\sqrt{a d - b c}} \right )}}{4 \sqrt{b} \sqrt{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

atan(sqrt(b)*sqrt(c + d*x**8)/sqrt(a*d - b*c))/(4*sqrt(b)*sqrt(a*d - b*c))

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Mathematica [A]  time = 0.042007, size = 51, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^8}}{\sqrt{b c-a d}}\right )}{4 \sqrt{b} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((a + b*x^8)*Sqrt[c + d*x^8]),x]

[Out]

-ArcTanh[(Sqrt[b]*Sqrt[c + d*x^8])/Sqrt[b*c - a*d]]/(4*Sqrt[b]*Sqrt[b*c - a*d])

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{{x}^{7}}{b{x}^{8}+a}{\frac{1}{\sqrt{d{x}^{8}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^8+a)/(d*x^8+c)^(1/2),x)

[Out]

int(x^7/(b*x^8+a)/(d*x^8+c)^(1/2),x)

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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^8 + a)*sqrt(d*x^8 + c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224657, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{{\left (b d x^{8} + 2 \, b c - a d\right )} \sqrt{b^{2} c - a b d} - 2 \, \sqrt{d x^{8} + c}{\left (b^{2} c - a b d\right )}}{b x^{8} + a}\right )}{8 \, \sqrt{b^{2} c - a b d}}, -\frac{\arctan \left (-\frac{b c - a d}{\sqrt{d x^{8} + c} \sqrt{-b^{2} c + a b d}}\right )}{4 \, \sqrt{-b^{2} c + a b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\left (a + b x^{8}\right ) \sqrt{c + d x^{8}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**8+a)/(d*x**8+c)**(1/2),x)

[Out]

Integral(x**7/((a + b*x**8)*sqrt(c + d*x**8)), x)

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GIAC/XCAS [A]  time = 0.211107, size = 54, normalized size = 1.06 \[ \frac{\arctan \left (\frac{\sqrt{d x^{8} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{4 \, \sqrt{-b^{2} c + a b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((b*x^8 + a)*sqrt(d*x^8 + c)),x, algorithm="giac")

[Out]

1/4*arctan(sqrt(d*x^8 + c)*b/sqrt(-b^2*c + a*b*d))/sqrt(-b^2*c + a*b*d)