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

Optimal. Leaf size=114 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.177392, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)}-\frac{d (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.8722, size = 80, normalized size = 0.7 \[ \frac{d \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c e \left (m + 1\right ) \left (a d - b c\right )} - \frac{b \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.109462, size = 88, normalized size = 0.77 \[ \frac{x (e x)^m \left (a d \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )-b c \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )\right )}{a c (m+1) (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.093, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m}}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x)^m/(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. \[ \int \frac{\left (e x\right )^{m}}{\left (a + b x^{n}\right ) \left (c + d x^{n}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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