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

Optimal. Leaf size=164 \[ \frac{B (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c d e (m+1)} \]

[Out]

-(((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^p*AppellF1[(1 + m)/n, -p, 1, (1 + m + n
)/n, -((b*x^n)/a), -((d*x^n)/c)])/(c*d*e*(1 + m)*(1 + (b*x^n)/a)^p)) + (B*(e*x)^
(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/
a)])/(d*e*(1 + m)*(1 + (b*x^n)/a)^p)

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Rubi [A]  time = 0.474507, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{B (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c d e (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

-(((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^p*AppellF1[(1 + m)/n, -p, 1, (1 + m + n
)/n, -((b*x^n)/a), -((d*x^n)/c)])/(c*d*e*(1 + m)*(1 + (b*x^n)/a)^p)) + (B*(e*x)^
(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/
a)])/(d*e*(1 + m)*(1 + (b*x^n)/a)^p)

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Rubi in Sympy [A]  time = 72.6844, size = 133, normalized size = 0.81 \[ \frac{A \left (e x\right )^{m + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + 1}{n},1,- p,\frac{m + n + 1}{n},- \frac{d x^{n}}{c},- \frac{b x^{n}}{a} \right )}}{c e \left (m + 1\right )} + \frac{B x^{- m} x^{m + n + 1} \left (e x\right )^{m} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m + n + 1}{n},1,- p,\frac{m + 2 n + 1}{n},- \frac{d x^{n}}{c},- \frac{b x^{n}}{a} \right )}}{c \left (m + n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(e*x)**(m + 1)*(1 + b*x**n/a)**(-p)*(a + b*x**n)**p*appellf1((m + 1)/n, 1, -p,
(m + n + 1)/n, -d*x**n/c, -b*x**n/a)/(c*e*(m + 1)) + B*x**(-m)*x**(m + n + 1)*(
e*x)**m*(1 + b*x**n/a)**(-p)*(a + b*x**n)**p*appellf1((m + n + 1)/n, 1, -p, (m +
2*n + 1)/n, -d*x**n/c, -b*x**n/a)/(c*(m + n + 1))

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Mathematica [B]  time = 1.4939, size = 438, normalized size = 2.67 \[ \frac{a c x (e x)^m \left (a+b x^n\right )^p \left (\frac{A (m+n+1)^2 F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{(m+1) \left (n x^n \left (b c p F_1\left (\frac{m+n+1}{n};1-p,1;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-a d F_1\left (\frac{m+n+1}{n};-p,2;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )+a c (m+n+1) F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )}+\frac{B (m+2 n+1) x^n F_1\left (\frac{m+n+1}{n};-p,1;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{n x^n \left (b c p F_1\left (\frac{m+2 n+1}{n};1-p,1;\frac{m+3 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-a d F_1\left (\frac{m+2 n+1}{n};-p,2;\frac{m+3 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )+a c (m+2 n+1) F_1\left (\frac{m+n+1}{n};-p,1;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}\right )}{(m+n+1) \left (c+d x^n\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*c*x*(e*x)^m*(a + b*x^n)^p*((A*(1 + m + n)^2*AppellF1[(1 + m)/n, -p, 1, (1 + m
+ n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m)*(a*c*(1 + m + n)*AppellF1[(1 + m)
/n, -p, 1, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + n*x^n*(b*c*p*AppellF1[(1
+ m + n)/n, 1 - p, 1, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)] - a*d*Appell
F1[(1 + m + n)/n, -p, 2, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))) + (B*(1
+ m + 2*n)*x^n*AppellF1[(1 + m + n)/n, -p, 1, (1 + m + 2*n)/n, -((b*x^n)/a), -(
(d*x^n)/c)])/(a*c*(1 + m + 2*n)*AppellF1[(1 + m + n)/n, -p, 1, (1 + m + 2*n)/n,
-((b*x^n)/a), -((d*x^n)/c)] + n*x^n*(b*c*p*AppellF1[(1 + m + 2*n)/n, 1 - p, 1, (
1 + m + 3*n)/n, -((b*x^n)/a), -((d*x^n)/c)] - a*d*AppellF1[(1 + m + 2*n)/n, -p,
2, (1 + m + 3*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))))/((1 + m + n)*(c + d*x^n))

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( A+B{x}^{n} \right ) }{c+d{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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