Optimal. Leaf size=112 \[ \frac{(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac{(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]
[Out]
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Rubi [A] time = 0.174511, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (2,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac{(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(A + B*x^n))/(c + d*x^n)^3,x]
[Out]
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Rubi in Sympy [A] time = 16.4091, size = 85, normalized size = 0.76 \[ \frac{\left (e x\right )^{m + 1} \left (A d - B c\right )}{2 c d e n \left (c + d x^{n}\right )^{2}} - \frac{\left (e x\right )^{m + 1} \left (A d \left (m - 2 n + 1\right ) - B c \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c^{3} d e n \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(A+B*x**n)/(c+d*x**n)**3,x)
[Out]
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Mathematica [A] time = 0.271832, size = 127, normalized size = 1.13 \[ \frac{x (e x)^m \left (-\frac{c^2 n (B c-A d)}{\left (c+d x^n\right )^2}-\frac{(m-n+1) (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{m+1}+\frac{c (B c (m+1)-A d (m-2 n+1))}{c+d x^n}\right )}{2 c^3 d n^2} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n)^3,x]
[Out]
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Maple [F] time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(A+B*x^n)/(c+d*x^n)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -{\left ({\left (m^{2} - m{\left (n - 2\right )} - n + 1\right )} B c e^{m} -{\left (m^{2} - m{\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} A d e^{m}\right )} \int \frac{x^{m}}{2 \,{\left (c^{2} d^{2} n^{2} x^{n} + c^{3} d n^{2}\right )}}\,{d x} + \frac{{\left (B c^{2} e^{m}{\left (m - n + 1\right )} - A c d e^{m}{\left (m - 3 \, n + 1\right )}\right )} x x^{m} -{\left (A d^{2} e^{m}{\left (m - 2 \, n + 1\right )} - B c d e^{m}{\left (m + 1\right )}\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{2 \,{\left (c^{2} d^{3} n^{2} x^{2 \, n} + 2 \, c^{3} d^{2} n^{2} x^{n} + c^{4} d n^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^3,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)/(c+d*x**n)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/(d*x^n + c)^3,x, algorithm="giac")
[Out]