Optimal. Leaf size=78 \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d e (m+1)} \]
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Rubi [A] time = 0.118703, antiderivative size = 78, 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 \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(A + B*x^n))/(c + d*x^n),x]
[Out]
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Rubi in Sympy [A] time = 13.5056, size = 54, normalized size = 0.69 \[ \frac{B \left (e x\right )^{m + 1}}{d e \left (m + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (A d - B c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c d e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(A+B*x**n)/(c+d*x**n),x)
[Out]
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Mathematica [A] time = 0.0719696, size = 59, normalized size = 0.76 \[ -\frac{x (e x)^m \left ((B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )-B c\right )}{c d (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(A + B*x^n))/(c + d*x^n),x]
[Out]
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Maple [F] time = 0.1, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{c+d{x}^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(A+B*x^n)/(c+d*x^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{B e^{m} x x^{m}}{d{\left (m + 1\right )}} -{\left (B c e^{m} - A d e^{m}\right )} \int \frac{x^{m}}{d^{2} x^{n} + c d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/(d*x^n + c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d x^{n} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/(d*x^n + c),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(A+B*x**n)/(c+d*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d x^{n} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/(d*x^n + c),x, algorithm="giac")
[Out]