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

Optimal. Leaf size=89 \[ \frac{b x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a (b c-a d)}-\frac{d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.111108, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a (b c-a d)}-\frac{d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.6535, size = 60, normalized size = 0.67 \[ \frac{d x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c \left (a d - b c\right )} - \frac{b x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0802597, size = 78, normalized size = 0.88 \[ \frac{b c x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )-a d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 a b c^2-2 a^2 c d} \]

Antiderivative was successfully verified.

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

[Out]

(b*c*x^2*Hypergeometric2F1[1, 2/n, (2 + n)/n, -((b*x^n)/a)] - a*d*x^2*Hypergeome
tric2F1[1, 2/n, (2 + n)/n, -((d*x^n)/c)])/(2*a*b*c^2 - 2*a^2*c*d)

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Maple [F]  time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{x}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^n)/(c+d*x^n),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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