3.862 \(\int \frac{1}{x^3 \left (a+b x^n\right ) \left (c+d x^n\right )} \, dx\)

Optimal. Leaf size=95 \[ \frac{d \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c x^2 (b c-a d)}-\frac{b \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2 (b c-a d)} \]

[Out]

-(b*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((b*x^n)/a)])/(2*a*(b*c - a*d)*x^2
) + (d*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((d*x^n)/c)])/(2*c*(b*c - a*d)*
x^2)

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Rubi [A]  time = 0.134278, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{d \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{d x^n}{c}\right )}{2 c x^2 (b c-a d)}-\frac{b \, _2F_1\left (1,-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(b*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((b*x^n)/a)])/(2*a*(b*c - a*d)*x^2
) + (d*Hypergeometric2F1[1, -2/n, -((2 - n)/n), -((d*x^n)/c)])/(2*c*(b*c - a*d)*
x^2)

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Rubi in Sympy [A]  time = 18.9581, size = 63, normalized size = 0.66 \[ - \frac{d{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c x^{2} \left (a d - b c\right )} + \frac{b{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a x^{2} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*hyper((1, -2/n), ((n - 2)/n,), -d*x**n/c)/(2*c*x**2*(a*d - b*c)) + b*hyper((1
, -2/n), ((n - 2)/n,), -b*x**n/a)/(2*a*x**2*(a*d - b*c))

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Mathematica [A]  time = 0.0835613, size = 77, normalized size = 0.81 \[ \frac{b c \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )-a d \, _2F_1\left (1,-\frac{2}{n};\frac{n-2}{n};-\frac{d x^n}{c}\right )}{2 a c x^2 (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

(b*c*Hypergeometric2F1[1, -2/n, (-2 + n)/n, -((b*x^n)/a)] - a*d*Hypergeometric2F
1[1, -2/n, (-2 + n)/n, -((d*x^n)/c)])/(2*a*c*(-(b*c) + a*d)*x^2)

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3} \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x^3/(a+b*x^n)/(c+d*x^n),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)*(d*x^n + c)*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(b*d*x^3*x^(2*n) + (b*c + a*d)*x^3*x^n + a*c*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a+b*x**n)/(c+d*x**n),x)

[Out]

Integral(1/(x**3*(a + b*x**n)*(c + d*x**n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)*(d*x^n + c)*x^3), x)