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

Optimal. Leaf size=366 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) \left (-a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left (B c (m+1) (m-2 n+1)-A d \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )\right )+b^2 c^2 (m-2 n+1) (A d (m-3 n+1)-B c (m-n+1))\right )}{2 c^3 e (m+1) n^2 (b c-a d)^3}+\frac{b^2 (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (a d (B c (m+1)-A d (m-2 n+1))+b c (A d (m-4 n+1)-B c (m-2 n+1)))}{2 c^2 e n^2 (b c-a d)^2 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (B c-A d)}{2 c e n (b c-a d) \left (c+d x^n\right )^2} \]

[Out]

((B*c - A*d)*(e*x)^(1 + m))/(2*c*(b*c - a*d)*e*n*(c + d*x^n)^2) + ((b*c*(A*d*(1
+ m - 4*n) - B*c*(1 + m - 2*n)) + a*d*(B*c*(1 + m) - A*d*(1 + m - 2*n)))*(e*x)^(
1 + m))/(2*c^2*(b*c - a*d)^2*e*n^2*(c + d*x^n)) + (b^2*(A*b - a*B)*(e*x)^(1 + m)
*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(b*c - a*d)^3*
e*(1 + m)) - ((b^2*c^2*(A*d*(1 + m - 3*n) - B*c*(1 + m - n))*(1 + m - 2*n) - a^2
*d^2*(B*c*(1 + m) - A*d*(1 + m - 2*n))*(1 + m - n) + 2*a*b*c*d*(B*c*(1 + m)*(1 +
m - 2*n) - A*d*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2)))*(e*x)^(1 + m)*Hypergeome
tric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*(b*c - a*d)^3*e*(1 +
m)*n^2)

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Rubi [A]  time = 3.21029, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) \left (-a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left (B c (m+1) (m-2 n+1)-A d \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )\right )+b^2 c^2 (m-2 n+1) (A d (m-3 n+1)-B c (m-n+1))\right )}{2 c^3 e (m+1) n^2 (b c-a d)^3}+\frac{b^2 (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (a d (B c (m+1)-A d (m-2 n+1))+b c (A d (m-4 n+1)-B c (m-2 n+1)))}{2 c^2 e n^2 (b c-a d)^2 \left (c+d x^n\right )}+\frac{(e x)^{m+1} (B c-A d)}{2 c e n (b c-a d) \left (c+d x^n\right )^2} \]

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))/(2*c*(b*c - a*d)*e*n*(c + d*x^n)^2) + ((b*c*(A*d*(1
+ m - 4*n) - B*c*(1 + m - 2*n)) + a*d*(B*c*(1 + m) - A*d*(1 + m - 2*n)))*(e*x)^(
1 + m))/(2*c^2*(b*c - a*d)^2*e*n^2*(c + d*x^n)) + (b^2*(A*b - a*B)*(e*x)^(1 + m)
*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(b*c - a*d)^3*
e*(1 + m)) - ((b^2*c^2*(A*d*(1 + m - 3*n) - B*c*(1 + m - n))*(1 + m - 2*n) - a^2
*d^2*(B*c*(1 + m) - A*d*(1 + m - 2*n))*(1 + m - n) + 2*a*b*c*d*(B*c*(1 + m)*(1 +
m - 2*n) - A*d*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2)))*(e*x)^(1 + m)*Hypergeome
tric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(2*c^3*(b*c - a*d)^3*e*(1 +
m)*n^2)

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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)/(a+b*x**n)/(c+d*x**n)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 3.01253, size = 357, normalized size = 0.98 \[ -\frac{x (e x)^m \left (a \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) \left (a^2 d^2 (m-n+1) (B c (m+1)-A d (m-2 n+1))+2 a b c d \left (A d \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )-B c (m+1) (m-2 n+1)\right )+b^2 c^2 (m-2 n+1) (B c (m-n+1)-A d (m-3 n+1))\right )+2 b^2 c^3 n^2 (A b-a B) \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )-a c (m+1) (b c-a d) \left (a d \left (A d \left (c (m-3 n+1)+d (m-2 n+1) x^n\right )-B c \left (c (m-n+1)+d (m+1) x^n\right )\right )+b c \left (B c \left (c (m-3 n+1)+d (m-2 n+1) x^n\right )-A d \left (c (m-5 n+1)+d (m-4 n+1) x^n\right )\right )\right )\right )}{2 a c^3 (m+1) n^2 (a d-b c)^3 \left (c+d x^n\right )^2} \]

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*(-(a*c*(b*c - a*d)*(1 + m)*(b*c*(-(A*d*(c*(1 + m - 5*n) + d*(1 + m -
4*n)*x^n)) + B*c*(c*(1 + m - 3*n) + d*(1 + m - 2*n)*x^n)) + a*d*(-(B*c*(c*(1 +
m - n) + d*(1 + m)*x^n)) + A*d*(c*(1 + m - 3*n) + d*(1 + m - 2*n)*x^n)))) + 2*b^
2*(A*b - a*B)*c^3*n^2*(c + d*x^n)^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/
n, -((b*x^n)/a)] + a*(b^2*c^2*(-(A*d*(1 + m - 3*n)) + B*c*(1 + m - n))*(1 + m -
2*n) + a^2*d^2*(B*c*(1 + m) - A*d*(1 + m - 2*n))*(1 + m - n) + 2*a*b*c*d*(-(B*c*
(1 + m)*(1 + m - 2*n)) + A*d*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2)))*(c + d*x^n)
^2*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)]))/(2*a*c^3*(-(b*
c) + a*d)^3*(1 + m)*n^2*(c + d*x^n)^2)

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Maple [F]  time = 0.129, 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 (5 \, n - 2\right )} + 6 \, n^{2} - 5 \, n + 1\right )} b^{2} c^{2} d e^{m} - 2 \,{\left (m^{2} - 2 \, m{\left (2 \, n - 1\right )} + 3 \, n^{2} - 4 \, n + 1\right )} a b c d^{2} e^{m} +{\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} a^{2} d^{3} e^{m}\right )} A -{\left ({\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} b^{2} c^{3} e^{m} - 2 \,{\left (m^{2} - 2 \, m{\left (n - 1\right )} - 2 \, n + 1\right )} a b c^{2} d e^{m} +{\left (m^{2} - m{\left (n - 2\right )} - n + 1\right )} a^{2} c d^{2} e^{m}\right )} B\right )} \int -\frac{x^{m}}{2 \,{\left (b^{3} c^{6} n^{2} - 3 \, a b^{2} c^{5} d n^{2} + 3 \, a^{2} b c^{4} d^{2} n^{2} - a^{3} c^{3} d^{3} n^{2} +{\left (b^{3} c^{5} d n^{2} - 3 \, a b^{2} c^{4} d^{2} n^{2} + 3 \, a^{2} b c^{3} d^{3} n^{2} - a^{3} c^{2} d^{4} n^{2}\right )} x^{n}\right )}}\,{d x} +{\left (B a b^{2} e^{m} - A b^{3} e^{m}\right )} \int -\frac{x^{m}}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{n}}\,{d x} - \frac{{\left ({\left (a c d^{2} e^{m}{\left (m - 3 \, n + 1\right )} - b c^{2} d e^{m}{\left (m - 5 \, n + 1\right )}\right )} A -{\left (a c^{2} d e^{m}{\left (m - n + 1\right )} - b c^{3} e^{m}{\left (m - 3 \, 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 - 4 \, n + 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 (b^{2} c^{6} n^{2} - 2 \, a b c^{5} d n^{2} + a^{2} c^{4} d^{2} n^{2} +{\left (b^{2} c^{4} d^{2} n^{2} - 2 \, a b c^{3} d^{3} n^{2} + a^{2} c^{2} d^{4} n^{2}\right )} x^{2 \, n} + 2 \,{\left (b^{2} c^{5} d n^{2} - 2 \, a b c^{4} d^{2} n^{2} + a^{2} c^{3} d^{3} n^{2}\right )} x^{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(((m^2 - m*(5*n - 2) + 6*n^2 - 5*n + 1)*b^2*c^2*d*e^m - 2*(m^2 - 2*m*(2*n - 1) +
3*n^2 - 4*n + 1)*a*b*c*d^2*e^m + (m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*a^2*d^3*
e^m)*A - ((m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*b^2*c^3*e^m - 2*(m^2 - 2*m*(n -
1) - 2*n + 1)*a*b*c^2*d*e^m + (m^2 - m*(n - 2) - n + 1)*a^2*c*d^2*e^m)*B)*integr
ate(-1/2*x^m/(b^3*c^6*n^2 - 3*a*b^2*c^5*d*n^2 + 3*a^2*b*c^4*d^2*n^2 - a^3*c^3*d^
3*n^2 + (b^3*c^5*d*n^2 - 3*a*b^2*c^4*d^2*n^2 + 3*a^2*b*c^3*d^3*n^2 - a^3*c^2*d^4
*n^2)*x^n), x) + (B*a*b^2*e^m - A*b^3*e^m)*integrate(-x^m/(a*b^3*c^3 - 3*a^2*b^2
*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3 + (b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 -
a^3*b*d^3)*x^n), x) - 1/2*(((a*c*d^2*e^m*(m - 3*n + 1) - b*c^2*d*e^m*(m - 5*n +
1))*A - (a*c^2*d*e^m*(m - n + 1) - b*c^3*e^m*(m - 3*n + 1))*B)*x*x^m + ((a*d^3*e
^m*(m - 2*n + 1) - b*c*d^2*e^m*(m - 4*n + 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*log(x)))/(b^2*c^6*n^2 - 2*a*b*c^5*d*n^2
+ a^2*c^4*d^2*n^2 + (b^2*c^4*d^2*n^2 - 2*a*b*c^3*d^3*n^2 + a^2*c^2*d^4*n^2)*x^(
2*n) + 2*(b^2*c^5*d*n^2 - 2*a*b*c^4*d^2*n^2 + a^2*c^3*d^3*n^2)*x^n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^n + A)*(e*x)^m/(b*d^3*x^(4*n) + a*c^3 + (3*b*c*d^2 + a*d^3)*x^(3*n
) + 3*(b*c^2*d + a*c*d^2)*x^(2*n) + (b*c^3 + 3*a*c^2*d)*x^n), 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. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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