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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]