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

Optimal. Leaf size=177 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (b c (a B (m+1)-A b (m-n+1))+a d (A b (m+1)-a B (m+n+1)))}{a^2 b^2 e (m+1) n}-\frac{d (e x)^{m+1} (A b (m+1)-a B (m+n+1))}{a b^2 e (m+1) n}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{a b e n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.694834, antiderivative size = 177, 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 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (b c (a B (m+1)-A b (m-n+1))+a d (A b (m+1)-a B (m+n+1)))}{a^2 b^2 e (m+1) n}-\frac{d (e x)^{m+1} (A b (m+1)-a B (m+n+1))}{a b^2 e (m+1) n}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )}{a b e n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.5419, size = 148, normalized size = 0.84 \[ \frac{\left (e x\right )^{m + 1} \left (c + d x^{n}\right ) \left (A b - B a\right )}{a b e n \left (a + b x^{n}\right )} - \frac{d \left (e x\right )^{m + 1} \left (- A b n + \left (A b - B a\right ) \left (m + n + 1\right )\right )}{a b^{2} e n \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (a d \left (- A b n + \left (A b - B a\right ) \left (m + n + 1\right )\right ) - b c \left (- A b n + \left (m + 1\right ) \left (A b - B a\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} b^{2} e n \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.39017, size = 195, normalized size = 1.1 \[ \frac{x (e x)^m \left (a^2 (-B) d-\frac{\, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+1))+a B (a d (m+n+1)-b c (m+1)))}{m+1}+\frac{A b (b c (m-n+1)-a d (m+1))+a B (a d (m+n+1)-b c (m+1))}{m+1}+\frac{a (a B-A b) (a d-b c)}{a+b x^n}+a b (A d+B c)+\frac{A b^2 c (-m+n-1)}{m+1}\right )}{a^2 b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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