3.38 \(\int \frac{(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx\)

Optimal. Leaf size=112 \[ \frac{(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac{(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]

[Out]

-((B*c - A*d)*(e*x)^(1 + m))/(2*c*d*e*n*(c + d*x^n)^2) + ((B*c*(1 + m) - A*d*(1
+ m - 2*n))*(e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^
n)/c)])/(2*c^3*d*e*(1 + m)*n)

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Rubi [A]  time = 0.174511, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac{(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^m*(A + B*x^n))/(c + d*x^n)^3,x]

[Out]

-((B*c - A*d)*(e*x)^(1 + m))/(2*c*d*e*n*(c + d*x^n)^2) + ((B*c*(1 + m) - A*d*(1
+ m - 2*n))*(e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/n, (1 + m + n)/n, -((d*x^
n)/c)])/(2*c^3*d*e*(1 + m)*n)

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Rubi in Sympy [A]  time = 16.4091, size = 85, normalized size = 0.76 \[ \frac{\left (e x\right )^{m + 1} \left (A d - B c\right )}{2 c d e n \left (c + d x^{n}\right )^{2}} - \frac{\left (e x\right )^{m + 1} \left (A d \left (m - 2 n + 1\right ) - B c \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c^{3} d e n \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(A+B*x**n)/(c+d*x**n)**3,x)

[Out]

(e*x)**(m + 1)*(A*d - B*c)/(2*c*d*e*n*(c + d*x**n)**2) - (e*x)**(m + 1)*(A*d*(m
- 2*n + 1) - B*c*(m + 1))*hyper((2, (m + 1)/n), ((m + n + 1)/n,), -d*x**n/c)/(2*
c**3*d*e*n*(m + 1))

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Mathematica [A]  time = 0.271832, size = 127, normalized size = 1.13 \[ \frac{x (e x)^m \left (-\frac{c^2 n (B c-A d)}{\left (c+d x^n\right )^2}-\frac{(m-n+1) (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{m+1}+\frac{c (B c (m+1)-A d (m-2 n+1))}{c+d x^n}\right )}{2 c^3 d n^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n)^3,x]

[Out]

(x*(e*x)^m*(-((c^2*(B*c - A*d)*n)/(c + d*x^n)^2) + (c*(B*c*(1 + m) - A*d*(1 + m
- 2*n)))/(c + d*x^n) - ((B*c*(1 + m) - A*d*(1 + m - 2*n))*(1 + m - n)*Hypergeome
tric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(1 + m)))/(2*c^3*d*n^2)

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x)

[Out]

int((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -{\left ({\left (m^{2} - m{\left (n - 2\right )} - n + 1\right )} B c e^{m} -{\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} A d e^{m}\right )} \int \frac{x^{m}}{2 \,{\left (c^{2} d^{2} n^{2} x^{n} + c^{3} d n^{2}\right )}}\,{d x} + \frac{{\left (B c^{2} e^{m}{\left (m - n + 1\right )} - A c d e^{m}{\left (m - 3 \, n + 1\right )}\right )} x x^{m} -{\left (A d^{2} e^{m}{\left (m - 2 \, n + 1\right )} - B c d e^{m}{\left (m + 1\right )}\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{2 \,{\left (c^{2} d^{3} n^{2} x^{2 \, n} + 2 \, c^{3} d^{2} n^{2} x^{n} + c^{4} d n^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^3,x, algorithm="maxima")

[Out]

-((m^2 - m*(n - 2) - n + 1)*B*c*e^m - (m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*A*d*
e^m)*integrate(1/2*x^m/(c^2*d^2*n^2*x^n + c^3*d*n^2), x) + 1/2*((B*c^2*e^m*(m -
n + 1) - A*c*d*e^m*(m - 3*n + 1))*x*x^m - (A*d^2*e^m*(m - 2*n + 1) - B*c*d*e^m*(
m + 1))*x*e^(m*log(x) + n*log(x)))/(c^2*d^3*n^2*x^(2*n) + 2*c^3*d^2*n^2*x^n + c^
4*d*n^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^3,x, algorithm="fricas")

[Out]

integral((B*x^n + A)*(e*x)^m/(d^3*x^(3*n) + 3*c*d^2*x^(2*n) + 3*c^2*d*x^n + c^3)
, 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)/(c+d*x**n)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^3,x, algorithm="giac")

[Out]

integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^3, x)