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

Optimal. Leaf size=268 \[ -\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+n+1))-a B (b c (m+1)-a d (m+2 n+1)))}{a^2 b^3 e (m+1) n}-\frac{d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+n+1))-a B (2 b c (m+n+1)-a d (m+2 n+1)))}{a b^3 e (m+1) n}-\frac{d^2 x^{n+1} (e x)^m (A b (m+n+1)-a B (m+2 n+1))}{a b^2 n (m+n+1)}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 1.66299, antiderivative size = 268, 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 \[ -\frac{(e x)^{m+1} (b c-a d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+n+1))-a B (b c (m+1)-a d (m+2 n+1)))}{a^2 b^3 e (m+1) n}-\frac{d (e x)^{m+1} (A b (2 b c (m+1)-a d (m+n+1))-a B (2 b c (m+n+1)-a d (m+2 n+1)))}{a b^3 e (m+1) n}-\frac{d^2 x^{n+1} (e x)^m (A b (m+n+1)-a B (m+2 n+1))}{a b^2 n (m+n+1)}+\frac{(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{a b e n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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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)*(c+d*x**n)**2/(a+b*x**n)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.88662, size = 227, normalized size = 0.85 \[ \frac{x (e x)^m \left (\frac{(a d-b c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m+n+1))+a B (a d (m+2 n+1)-b c (m+1)))}{a^2 (m+1) n}+\frac{b^2 c (-2 a A d-a B c+A b c)}{a n \left (a+b x^n\right )}+b d \left (A d \left (\frac{a}{a n+b n x^n}+\frac{1}{m+1}\right )+B \left (2 c \left (\frac{a}{a n+b n x^n}+\frac{1}{m+1}\right )+\frac{d x^n}{m+n+1}\right )\right )+a B d^2 \left (-\frac{a}{a n+b n x^n}-\frac{2}{m+1}\right )\right )}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 180., size = 0, normalized size = 0. \[ \text{hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*d^2*x^(3*n) + A*c^2 + (2*B*c*d + A*d^2)*x^(2*n) + (B*c^2 + 2*A*c*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(-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)**2/(a+b*x**n)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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