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

Optimal. Leaf size=127 \[ \frac{(e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]

[Out]

((A*b - a*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)) + ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F
1[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 = 0.378685, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{(e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*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)) + ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F
1[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 = 90.2279, size = 199, normalized size = 1.57 \[ \frac{A d \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c e \left (m + 1\right ) \left (a d - b c\right )} - \frac{A b \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + 1\right ) \left (a d - b c\right )} + \frac{B d x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c e \left (a d - b c\right ) \left (m + n + 1\right )} - \frac{B b x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (a d - b c\right ) \left (m + n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*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)) - A*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)) + B*d*x**n*(e*x)**(-n)*(e*x)**(m + n + 1)*hype
r((1, (m + n + 1)/n), ((m + 2*n + 1)/n,), -d*x**n/c)/(c*e*(a*d - b*c)*(m + n + 1
)) - B*b*x**n*(e*x)**(-n)*(e*x)**(m + n + 1)*hyper((1, (m + n + 1)/n), ((m + 2*n
+ 1)/n,), -b*x**n/a)/(a*e*(a*d - b*c)*(m + n + 1))

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Mathematica [A]  time = 0.179007, size = 102, normalized size = 0.8 \[ \frac{x (e x)^m \left ((a B c-A b c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+a (A d-B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )\right )}{a c (m+1) (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*((-(A*b*c) + a*B*c)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -(
(b*x^n)/a)] + a*(-(B*c) + 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 = 0.097, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^n + A)*(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. \[{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b d x^{2 \, n} + a c +{\left (b c + a d\right )} x^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^n + A)*(e*x)^m/(b*d*x^(2*n) + a*c + (b*c + a*d)*x^n), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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