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

Optimal. Leaf size=211 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-2 n+1)-B c (m-n+1)))}{c^2 e (m+1) n (b c-a d)^2}+\frac{b (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (B c-A d)}{c e n (b c-a d) \left (c+d x^n\right )} \]

[Out]

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

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Rubi [A]  time = 1.41038, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-2 n+1)-B c (m-n+1)))}{c^2 e (m+1) n (b c-a d)^2}+\frac{b (e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (B c-A d)}{c e n (b c-a d) \left (c+d x^n\right )} \]

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

[Out]

Timed out

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Mathematica [A]  time = 0.513531, size = 177, normalized size = 0.84 \[ \frac{x (e x)^m \left (-\frac{\, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right ) (a d (A d (m-n+1)-B c (m+1))+b c (B c (m-n+1)-A d (m-2 n+1)))}{c^2 (m+1) n}+\frac{(b c-a d) (B c-A d)}{c n \left (c+d x^n\right )}+\frac{b (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)}\right )}{(b c-a d)^2} \]

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 - a*d)*(B*c - A*d))/(c*n*(c + d*x^n)) + (b*(A*b - a*B)*Hyperge
ometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m)) - ((b*c*(-(A*
d*(1 + m - 2*n)) + B*c*(1 + m - n)) + a*d*(-(B*c*(1 + m)) + A*d*(1 + m - n)))*Hy
pergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c^2*(1 + m)*n)))/(b
*c - a*d)^2

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Maple [F]  time = 0.112, 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. \[ \frac{{\left (B c e^{m} - A d e^{m}\right )} x x^{m}}{b c^{3} n - a c^{2} d n +{\left (b c^{2} d n - a c d^{2} n\right )} x^{n}} -{\left ({\left (a d^{2} e^{m}{\left (m - n + 1\right )} - b c d e^{m}{\left (m - 2 \, n + 1\right )}\right )} A +{\left (b c^{2} e^{m}{\left (m - n + 1\right )} - a c d e^{m}{\left (m + 1\right )}\right )} B\right )} \int \frac{x^{m}}{b^{2} c^{4} n - 2 \, a b c^{3} d n + a^{2} c^{2} d^{2} n +{\left (b^{2} c^{3} d n - 2 \, a b c^{2} d^{2} n + a^{2} c d^{3} n\right )} x^{n}}\,{d x} -{\left (B a b e^{m} - A b^{2} e^{m}\right )} \int \frac{x^{m}}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{n}}\,{d x} \]

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)^2),x, algorithm="maxima")

[Out]

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

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

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)^2),x, algorithm="fricas")

[Out]

integral((B*x^n + A)*(e*x)^m/(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), 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)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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