Optimal. Leaf size=212 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (a d (m-2 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-n+1)))}{a^2 e (m+1) n (b c-a d)^2}-\frac{d (e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right )} \]
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Rubi [A] time = 1.38523, antiderivative size = 212, 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 \[ \frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (a d (m-2 n+1)-b c (m-n+1))+a B (b c (m+1)-a d (m-n+1)))}{a^2 e (m+1) n (b c-a d)^2}-\frac{d (e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{a e n (b c-a d) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(A + B*x^n))/((a + b*x^n)^2*(c + d*x^n)),x]
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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)**2/(c+d*x**n),x)
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Mathematica [A] time = 0.518965, size = 177, normalized size = 0.83 \[ \frac{x (e x)^m \left (-\frac{\, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (b c (m-n+1)-a d (m-2 n+1))+a B (a d (m-n+1)-b c (m+1)))}{a^2 (m+1) n}+\frac{(a B-A b) (a d-b c)}{a n \left (a+b x^n\right )}+\frac{d (A d-B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c (m+1)}\right )}{(b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)^2*(c + d*x^n)),x]
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Maple [F] time = 0.113, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(A+B*x^n)/(a+b*x^n)^2/(c+d*x^n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{{\left (B a e^{m} - A b e^{m}\right )} x x^{m}}{a^{2} b c n - a^{3} d n +{\left (a b^{2} c n - a^{2} b d n\right )} x^{n}} -{\left ({\left (b^{2} c e^{m}{\left (m - n + 1\right )} - a b d e^{m}{\left (m - 2 \, n + 1\right )}\right )} A +{\left (a^{2} d e^{m}{\left (m - n + 1\right )} - a b c e^{m}{\left (m + 1\right )}\right )} B\right )} \int \frac{x^{m}}{a^{2} b^{2} c^{2} n - 2 \, a^{3} b c d n + a^{4} d^{2} n +{\left (a b^{3} c^{2} n - 2 \, a^{2} b^{2} c d n + a^{3} b d^{2} n\right )} x^{n}}\,{d x} -{\left (B c d e^{m} - A d^{2} e^{m}\right )} \int \frac{x^{m}}{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b^{2} d x^{3 \, n} + a^{2} c +{\left (b^{2} c + 2 \, a b d\right )} x^{2 \, n} +{\left (2 \, a b c + a^{2} d\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="fricas")
[Out]
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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)**2/(c+d*x**n),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{2}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)^2*(d*x^n + c)),x, algorithm="giac")
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