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

Optimal. Leaf size=178 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (A d (b c (m+1)-a d (m-n+1))+B c (a d (m+1)-b c (m+n+1)))}{c^2 d^2 e (m+1) n}-\frac{(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}-\frac{B (e x)^{m+1} (a d (m+1)-b c (m+n+1))}{c d^2 e (m+1) n} \]

[Out]

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

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Rubi [A]  time = 0.698718, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (A d (b c (m+1)-a d (m-n+1))+B c (a d (m+1)-b c (m+n+1)))}{c^2 d^2 e (m+1) n}-\frac{(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{c d e n \left (c+d x^n\right )}-\frac{B (e x)^{m+1} (a d (m+1)-b c (m+n+1))}{c d^2 e (m+1) n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.9914, size = 148, normalized size = 0.83 \[ - \frac{b \left (e x\right )^{m + 1} \left (- A d n + \left (A d - B c\right ) \left (m + n + 1\right )\right )}{c d^{2} e n \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (a + b x^{n}\right ) \left (A d - B c\right )}{c d e n \left (c + d x^{n}\right )} - \frac{\left (e x\right )^{m + 1} \left (a d \left (- A d n + \left (m + 1\right ) \left (A d - B c\right )\right ) - b c \left (- A d n + \left (A d - B c\right ) \left (m + n + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c^{2} d^{2} e n \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.408161, size = 193, normalized size = 1.08 \[ \frac{x (e x)^m \left (-\frac{\, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (A d (m-n+1)-B c (m+1))+b c (B c (m+n+1)-A d (m+1)))}{m+1}+\frac{a d (A d (m-n+1)-B c (m+1))+b c (B c (m+n+1)-A d (m+1))}{m+1}+\frac{c (b c-a d) (B c-A d)}{c+d x^n}+a d \left (\frac{A d (-m+n-1)}{m+1}+B c\right )+b c (A d-B c)\right )}{c^2 d^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) \left ( A+B{x}^{n} \right ) }{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]

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)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)*(e*x)^m/(d*x^n + c)^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)*(e*x)^m/(d*x^n + c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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)**2,x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)*(e*x)^m/(d*x^n + c)^2,x, algorithm="giac")

[Out]

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