3.406 \(\int \frac{1}{c^7 x^7 \left (\frac{a}{x^4}+b x^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ \frac{2}{a c^7 (n+4) x^2 \sqrt{\frac{a}{x^4}+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x^2 \sqrt{\frac{a}{x^4}+b x^n}}\right )}{a^{3/2} c^7 (n+4)} \]

[Out]

2/(a*c^7*(4 + n)*x^2*Sqrt[a/x^4 + b*x^n]) - (2*ArcTanh[Sqrt[a]/(x^2*Sqrt[a/x^4 +
b*x^n])])/(a^(3/2)*c^7*(4 + n))

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Rubi [A]  time = 0.253784, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2}{a c^7 (n+4) x^2 \sqrt{\frac{a}{x^4}+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a}}{x^2 \sqrt{\frac{a}{x^4}+b x^n}}\right )}{a^{3/2} c^7 (n+4)} \]

Antiderivative was successfully verified.

[In]  Int[1/(c^7*x^7*(a/x^4 + b*x^n)^(3/2)),x]

[Out]

2/(a*c^7*(4 + n)*x^2*Sqrt[a/x^4 + b*x^n]) - (2*ArcTanh[Sqrt[a]/(x^2*Sqrt[a/x^4 +
b*x^n])])/(a^(3/2)*c^7*(4 + n))

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Rubi in Sympy [A]  time = 19.7097, size = 63, normalized size = 0.88 \[ \frac{2}{a c^{7} x^{2} \left (n + 4\right ) \sqrt{\frac{a}{x^{4}} + b x^{n}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a}}{x^{2} \sqrt{\frac{a}{x^{4}} + b x^{n}}} \right )}}{a^{\frac{3}{2}} c^{7} \left (n + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/c**7/x**7/(a/x**4+b*x**n)**(3/2),x)

[Out]

2/(a*c**7*x**2*(n + 4)*sqrt(a/x**4 + b*x**n)) - 2*atanh(sqrt(a)/(x**2*sqrt(a/x**
4 + b*x**n)))/(a**(3/2)*c**7*(n + 4))

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Mathematica [A]  time = 0.191948, size = 100, normalized size = 1.39 \[ \frac{2 \left (-\sqrt{a+b x^{n+4}} \log \left (\sqrt{a} \sqrt{a+b x^{n+4}}+a\right )+\log \left (x^{\frac{n+4}{2}}\right ) \sqrt{a+b x^{n+4}}+\sqrt{a}\right )}{a^{3/2} c^7 (n+4) x^2 \sqrt{\frac{a}{x^4}+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(c^7*x^7*(a/x^4 + b*x^n)^(3/2)),x]

[Out]

(2*(Sqrt[a] + Sqrt[a + b*x^(4 + n)]*Log[x^((4 + n)/2)] - Sqrt[a + b*x^(4 + n)]*L
og[a + Sqrt[a]*Sqrt[a + b*x^(4 + n)]]))/(a^(3/2)*c^7*(4 + n)*x^2*Sqrt[a/x^4 + b*
x^n])

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{1}{{c}^{7}{x}^{7}} \left ({\frac{a}{{x}^{4}}}+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/c^7/x^7/(a/x^4+b*x^n)^(3/2),x)

[Out]

int(1/c^7/x^7/(a/x^4+b*x^n)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{{\left (b x^{n} + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} x^{7}}\,{d x}}{c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a/x^4)^(3/2)*c^7*x^7),x, algorithm="maxima")

[Out]

integrate(1/((b*x^n + a/x^4)^(3/2)*x^7), x)/c^7

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a/x^4)^(3/2)*c^7*x^7),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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(1/c**7/x**7/(a/x**4+b*x**n)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c^{7} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a/x^4)^(3/2)*c^7*x^7),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a/x^4)^(3/2)*c^7*x^7), x)