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

Optimal. Leaf size=228 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (A d (m-n+1) (b c (m+1)-a d (m-2 n+1))+B c (m+1) (a d (m-n+1)-b c (m+n+1)))}{2 c^3 d^2 e (m+1) n^2}-\frac{(e x)^{m+1} (a d (A d (m-2 n+1)-B c (m-n+1))-b c (A d (m+1)-B c (m+n+1)))}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}-\frac{(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{2 c d e n \left (c+d x^n\right )^2} \]

[Out]

-((b*c - a*d)*(e*x)^(1 + m)*(A + B*x^n))/(2*c*d*e*n*(c + d*x^n)^2) - ((a*d*(A*d*
(1 + m - 2*n) - B*c*(1 + m - n)) - b*c*(A*d*(1 + m) - B*c*(1 + m + n)))*(e*x)^(1
+ m))/(2*c^2*d^2*e*n^2*(c + d*x^n)) - ((A*d*(b*c*(1 + m) - a*d*(1 + m - 2*n))*(
1 + m - n) + B*c*(1 + m)*(a*d*(1 + m - n) - b*c*(1 + m + n)))*(e*x)^(1 + m)*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*d^2*e*(1 + m)*n
^2)

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Rubi [A]  time = 0.771661, antiderivative size = 228, 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 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (A d (m-n+1) (b c (m+1)-a d (m-2 n+1))+B c (m+1) (a d (m-n+1)-b c (m+n+1)))}{2 c^3 d^2 e (m+1) n^2}-\frac{(e x)^{m+1} (a d (A d (m-2 n+1)-B c (m-n+1))-b c (A d (m+1)-B c (m+n+1)))}{2 c^2 d^2 e n^2 \left (c+d x^n\right )}-\frac{(e x)^{m+1} (b c-a d) \left (A+B x^n\right )}{2 c d e n \left (c+d x^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*(e*x)^(1 + m)*(A + B*x^n))/(2*c*d*e*n*(c + d*x^n)^2) - ((a*d*(A*d*
(1 + m - 2*n) - B*c*(1 + m - n)) - b*c*(A*d*(1 + m) - B*c*(1 + m + n)))*(e*x)^(1
+ m))/(2*c^2*d^2*e*n^2*(c + d*x^n)) - ((A*d*(b*c*(1 + m) - a*d*(1 + m - 2*n))*(
1 + m - n) + B*c*(1 + m)*(a*d*(1 + m - n) - b*c*(1 + m + n)))*(e*x)^(1 + m)*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*d^2*e*(1 + m)*n
^2)

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Rubi in Sympy [A]  time = 57.3409, size = 199, normalized size = 0.87 \[ \frac{\left (e x\right )^{m + 1} \left (a + b x^{n}\right ) \left (A d - B c\right )}{2 c d e n \left (c + d x^{n}\right )^{2}} - \frac{\left (e x\right )^{m + 1} \left (a d \left (- 2 A d n + \left (m + 1\right ) \left (A d - B c\right )\right ) - b c \left (- 2 A d n + \left (A d - B c\right ) \left (m + n + 1\right )\right )\right )}{2 c^{2} d^{2} e n^{2} \left (c + d x^{n}\right )} + \frac{\left (e x\right )^{m + 1} \left (a d \left (- 2 A d n + \left (m + 1\right ) \left (A d - B c\right )\right ) \left (m - n + 1\right ) - b c \left (m + 1\right ) \left (- 2 A d n + \left (A d - B c\right ) \left (m + n + 1\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c^{3} d^{2} e n^{2} \left (m + 1\right )} \]

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)**3,x)

[Out]

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

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Mathematica [B]  time = 1.00353, size = 1153, normalized size = 5.06 \[ \frac{x (e x)^m \left (b B (m+1) n c^4-A b d (m+1) n c^3-a B d (m+1) n c^3-b B (m+1) \left (d x^n+c\right ) c^3-b B m (m+1) \left (d x^n+c\right ) c^3-2 b B (m+1) n \left (d x^n+c\right ) c^3+a A d^2 (m+1) n c^2+A b d (m+1) \left (d x^n+c\right ) c^2+a B d (m+1) \left (d x^n+c\right ) c^2+A b d m (m+1) \left (d x^n+c\right ) c^2+a B d m (m+1) \left (d x^n+c\right ) c^2+b B m^2 \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c^2+b B \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c^2+2 b B m \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c^2+b B n \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c^2+b B m n \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c^2-a A d^2 (m+1) \left (d x^n+c\right ) c-a A d^2 m (m+1) \left (d x^n+c\right ) c+2 a A d^2 (m+1) n \left (d x^n+c\right ) c-A b d m^2 \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c-a B d m^2 \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c-A b d \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c-a B d \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c-2 A b d m \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c-2 a B d m \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c+A b d n \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c+a B d n \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c+A b d m n \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c+a B d m n \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) c+a A d^2 \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )+a A d^2 m^2 \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )+2 a A d^2 n^2 \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )+2 a A d^2 m \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )-3 a A d^2 n \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )-3 a A d^2 m n \left (d x^n+c\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )\right )}{2 c^3 d^2 (m+1) n^2 \left (d x^n+c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*(b*B*c^4*(1 + m)*n - A*b*c^3*d*(1 + m)*n - a*B*c^3*d*(1 + m)*n + a*A*
c^2*d^2*(1 + m)*n - b*B*c^3*(1 + m)*(c + d*x^n) + A*b*c^2*d*(1 + m)*(c + d*x^n)
+ a*B*c^2*d*(1 + m)*(c + d*x^n) - a*A*c*d^2*(1 + m)*(c + d*x^n) - b*B*c^3*m*(1 +
m)*(c + d*x^n) + A*b*c^2*d*m*(1 + m)*(c + d*x^n) + a*B*c^2*d*m*(1 + m)*(c + d*x
^n) - a*A*c*d^2*m*(1 + m)*(c + d*x^n) - 2*b*B*c^3*(1 + m)*n*(c + d*x^n) + 2*a*A*
c*d^2*(1 + m)*n*(c + d*x^n) + b*B*c^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)
/n, (1 + m + n)/n, -((d*x^n)/c)] - A*b*c*d*(c + d*x^n)^2*Hypergeometric2F1[1, (1
+ m)/n, (1 + m + n)/n, -((d*x^n)/c)] - a*B*c*d*(c + d*x^n)^2*Hypergeometric2F1[
1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + a*A*d^2*(c + d*x^n)^2*Hypergeometri
c2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*b*B*c^2*m*(c + d*x^n)^2*Hype
rgeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*A*b*c*d*m*(c + d*x^
n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 2*a*B*c*d*m*
(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + 2*a
*A*d^2*m*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/
c)] + b*B*c^2*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -
((d*x^n)/c)] - A*b*c*d*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m
+ n)/n, -((d*x^n)/c)] - a*B*c*d*m^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n
, (1 + m + n)/n, -((d*x^n)/c)] + a*A*d^2*m^2*(c + d*x^n)^2*Hypergeometric2F1[1,
(1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + b*B*c^2*n*(c + d*x^n)^2*Hypergeometric
2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + A*b*c*d*n*(c + d*x^n)^2*Hyperge
ometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + a*B*c*d*n*(c + d*x^n)^2*
Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] - 3*a*A*d^2*n*(c +
d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] + b*B*c^2*
m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)] +
A*b*c*d*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x
^n)/c)] + a*B*c*d*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/
n, -((d*x^n)/c)] - 3*a*A*d^2*m*n*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (
1 + m + n)/n, -((d*x^n)/c)] + 2*a*A*d^2*n^2*(c + d*x^n)^2*Hypergeometric2F1[1, (
1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(2*c^3*d^2*(1 + m)*n^2*(c + d*x^n)^2)

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) \left ( A+B{x}^{n} \right ) }{ \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \]

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)^3,x)

[Out]

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

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

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)^3,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*b*x^(2*n) + A*a + (B*a + A*b)*x^n)*(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)*(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. \[ \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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