Optimal. Leaf size=407 \[ \frac{(e x)^{m+1} (A b (a d (m-4 n+1)-b c (m-2 n+1))+a B (b c (m+1)-a d (m-2 n+1)))}{2 a^2 e n^2 (b c-a d)^2 \left (a+b x^n\right )}+\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+m (2-5 n)+6 n^2-5 n+1\right )-2 a b c d \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )+b^2 c^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )+a B \left (-a^2 d^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )+2 a b c d (m+1) (m-2 n+1)-b^2 c^2 (m+1) (m-n+1)\right )\right )}{2 a^3 e (m+1) n^2 (b c-a d)^3}+\frac{d^2 (e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (A b-a B)}{2 a e n (b c-a d) \left (a+b x^n\right )^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 3.26471, antiderivative size = 407, 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 \[ \frac{(e x)^{m+1} (A b (a d (m-4 n+1)-b c (m-2 n+1))+a B (b c (m+1)-a d (m-2 n+1)))}{2 a^2 e n^2 (b c-a d)^2 \left (a+b x^n\right )}+\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+m (2-5 n)+6 n^2-5 n+1\right )-2 a b c d \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )+b^2 c^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )+a B \left (-a^2 d^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )+2 a b c d (m+1) (m-2 n+1)-b^2 c^2 (m+1) (m-n+1)\right )\right )}{2 a^3 e (m+1) n^2 (b c-a d)^3}+\frac{d^2 (e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)^3}+\frac{(e x)^{m+1} (A b-a B)}{2 a e n (b c-a d) \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(A + B*x^n))/((a + b*x^n)^3*(c + d*x^n)),x]
[Out]
_______________________________________________________________________________________
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)**3/(c+d*x**n),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 2.97657, size = 402, normalized size = 0.99 \[ \frac{x (e x)^m \left (2 a^3 d^2 n^2 \left (a+b x^n\right )^2 (A d-B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )-c \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2+m (2-5 n)+6 n^2-5 n+1\right )-2 a b c d \left (m^2+m (2-4 n)+3 n^2-4 n+1\right )+b^2 c^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )\right )-a B \left (a^2 d^2 \left (m^2+m (2-3 n)+2 n^2-3 n+1\right )-2 a b c d (m+1) (m-2 n+1)+b^2 c^2 (m+1) (m-n+1)\right )\right )+a c (m+1) (b c-a d) \left (a^3 B d (m-3 n+1)-a^2 b \left (A d (m-5 n+1)+B c (m-n+1)-B d (m-2 n+1) x^n\right )+a b^2 \left (A c (m-3 n+1)-A d (m-4 n+1) x^n-B c (m+1) x^n\right )+A b^3 c (m-2 n+1) x^n\right )\right )}{2 a^3 c (m+1) n^2 (a d-b c)^3 \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)^3*(c + d*x^n)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.132, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) ^{3} \left ( c+d{x}^{n} \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(A+B*x^n)/(a+b*x^n)^3/(c+d*x^n),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -{\left ({\left ({\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} b^{3} c^{2} e^{m} - 2 \,{\left (m^{2} - 2 \, m{\left (2 \, n - 1\right )} + 3 \, n^{2} - 4 \, n + 1\right )} a b^{2} c d e^{m} +{\left (m^{2} - m{\left (5 \, n - 2\right )} + 6 \, n^{2} - 5 \, 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 )} - 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 )} \int -\frac{x^{m}}{2 \,{\left (a^{3} b^{3} c^{3} n^{2} - 3 \, a^{4} b^{2} c^{2} d n^{2} + 3 \, a^{5} b c d^{2} n^{2} - a^{6} d^{3} n^{2} +{\left (a^{2} b^{4} c^{3} n^{2} - 3 \, a^{3} b^{3} c^{2} d n^{2} + 3 \, a^{4} b^{2} c d^{2} n^{2} - a^{5} b d^{3} n^{2}\right )} x^{n}\right )}}\,{d x} -{\left (B c d^{2} e^{m} - A d^{3} e^{m}\right )} \int -\frac{x^{m}}{b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3} +{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x^{n}}\,{d x} - \frac{{\left ({\left (a b^{2} c e^{m}{\left (m - 3 \, n + 1\right )} - a^{2} b d e^{m}{\left (m - 5 \, n + 1\right )}\right )} A -{\left (a^{2} b c e^{m}{\left (m - n + 1\right )} - a^{3} d e^{m}{\left (m - 3 \, n + 1\right )}\right )} B\right )} x x^{m} +{\left ({\left (b^{3} c e^{m}{\left (m - 2 \, n + 1\right )} - a b^{2} d e^{m}{\left (m - 4 \, n + 1\right )}\right )} A +{\left (a^{2} b d e^{m}{\left (m - 2 \, n + 1\right )} - a b^{2} c 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 (a^{4} b^{2} c^{2} n^{2} - 2 \, a^{5} b c d n^{2} + a^{6} d^{2} n^{2} +{\left (a^{2} b^{4} c^{2} n^{2} - 2 \, a^{3} b^{3} c d n^{2} + a^{4} b^{2} d^{2} n^{2}\right )} x^{2 \, n} + 2 \,{\left (a^{3} b^{3} c^{2} n^{2} - 2 \, a^{4} b^{2} c d n^{2} + a^{5} b d^{2} 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)^3*(d*x^n + c)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b^{3} d x^{4 \, n} + a^{3} c +{\left (b^{3} c + 3 \, a b^{2} d\right )} x^{3 \, n} + 3 \,{\left (a b^{2} c + a^{2} b d\right )} x^{2 \, n} +{\left (3 \, a^{2} b c + a^{3} d\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)^3*(d*x^n + c)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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)**3/(c+d*x**n),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)^3*(d*x^n + c)),x, algorithm="giac")
[Out]