3.44 \(\int \frac{(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx\)

Optimal. Leaf size=304 \[ -\frac{(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-n (1-p)+1)-B c (m+n p+1))) F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c^2 d e (m+1) n (b c-a d)}-\frac{b (e x)^{m+1} (m+n p+1) (B c-A d) \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c d e (m+1) n (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^{p+1}}{c e n (b c-a d) \left (c+d x^n\right )} \]

[Out]

((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^(1 + p))/(c*(b*c - a*d)*e*n*(c + d*x^n))
- ((a*d*(B*c*(1 + m) - A*d*(1 + m - n)) + b*c*(A*d*(1 + m - n*(1 - p)) - B*c*(1
+ m + n*p)))*(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^2*d*(b*c - a*d)*e*(1 + m)*n*(1 + (b*x^n)/a)^p
) - (b*(B*c - A*d)*(1 + m + n*p)*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(
1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(c*d*(b*c - a*d)*e*(1 + m)*n*(1 + (b
*x^n)/a)^p)

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Rubi [A]  time = 1.50539, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ -\frac{(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-n (1-p)+1)-B c (m+n p+1))) F_1\left (\frac{m+1}{n};-p,1;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c^2 d e (m+1) n (b c-a d)}-\frac{b (e x)^{m+1} (m+n p+1) (B c-A d) \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c d e (m+1) n (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^{p+1}}{c e n (b c-a d) \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^m*(a + b*x^n)^p*(A + B*x^n))/(c + d*x^n)^2,x]

[Out]

((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^(1 + p))/(c*(b*c - a*d)*e*n*(c + d*x^n))
- ((a*d*(B*c*(1 + m) - A*d*(1 + m - n)) + b*c*(A*d*(1 + m - n*(1 - p)) - B*c*(1
+ m + n*p)))*(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^2*d*(b*c - a*d)*e*(1 + m)*n*(1 + (b*x^n)/a)^p
) - (b*(B*c - A*d)*(1 + m + n*p)*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(
1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(c*d*(b*c - a*d)*e*(1 + m)*n*(1 + (b
*x^n)/a)^p)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.69733, size = 438, normalized size = 1.44 \[ \frac{a c x (e x)^m \left (a+b x^n\right )^p \left (\frac{A (m+n+1)^2 F_1\left (\frac{m+1}{n};-p,2;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{(m+1) \left (n x^n \left (b c p F_1\left (\frac{m+n+1}{n};1-p,2;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-2 a d F_1\left (\frac{m+n+1}{n};-p,3;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )+a c (m+n+1) F_1\left (\frac{m+1}{n};-p,2;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )}+\frac{B (m+2 n+1) x^n F_1\left (\frac{m+n+1}{n};-p,2;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{n x^n \left (b c p F_1\left (\frac{m+2 n+1}{n};1-p,2;\frac{m+3 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-2 a d F_1\left (\frac{m+2 n+1}{n};-p,3;\frac{m+3 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )+a c (m+2 n+1) F_1\left (\frac{m+n+1}{n};-p,2;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}\right )}{(m+n+1) \left (c+d x^n\right )^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((e*x)^m*(a + b*x^n)^p*(A + B*x^n))/(c + d*x^n)^2,x]

[Out]

(a*c*x*(e*x)^m*(a + b*x^n)^p*((A*(1 + m + n)^2*AppellF1[(1 + m)/n, -p, 2, (1 + m
+ n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m)*(a*c*(1 + m + n)*AppellF1[(1 + m)
/n, -p, 2, (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, 2, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)] - 2*a*d*Appe
llF1[(1 + m + n)/n, -p, 3, (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, 2, (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, 2, (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, 2,
(1 + m + 3*n)/n, -((b*x^n)/a), -((d*x^n)/c)] - 2*a*d*AppellF1[(1 + m + 2*n)/n,
-p, 3, (1 + m + 3*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))))/((1 + m + n)*(c + d*x^n)
^2)

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Maple [F]  time = 0.13, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( A+B{x}^{n} \right ) }{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]

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)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \]

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)^2,x, algorithm="maxima")

[Out]

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

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

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)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \]

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)^2,x, algorithm="giac")

[Out]

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