3.25 \(\int \frac{(e x)^m \left (A+B x^n\right )}{c+d x^n} \, dx\)

Optimal. Leaf size=78 \[ \frac{B (e x)^{m+1}}{d e (m+1)}-\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d e (m+1)} \]

[Out]

(B*(e*x)^(1 + m))/(d*e*(1 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1
, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*d*e*(1 + m))

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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]

(B*(e*x)^(1 + m))/(d*e*(1 + m)) - ((B*c - A*d)*(e*x)^(1 + m)*Hypergeometric2F1[1
, (1 + m)/n, (1 + m + n)/n, -((d*x^n)/c)])/(c*d*e*(1 + m))

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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]

B*(e*x)**(m + 1)/(d*e*(m + 1)) + (e*x)**(m + 1)*(A*d - B*c)*hyper((1, (m + 1)/n)
, ((m + n + 1)/n,), -d*x**n/c)/(c*d*e*(m + 1))

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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]

-((x*(e*x)^m*(-(B*c) + (B*c - A*d)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n
, -((d*x^n)/c)]))/(c*d*(1 + m)))

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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]

int((e*x)^m*(A+B*x^n)/(c+d*x^n),x)

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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]

B*e^m*x*x^m/(d*(m + 1)) - (B*c*e^m - A*d*e^m)*integrate(x^m/(d^2*x^n + c*d), x)

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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]

integral((B*x^n + A)*(e*x)^m/(d*x^n + c), x)

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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]

Exception raised: TypeError

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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]

integrate((B*x^n + A)*(e*x)^m/(d*x^n + c), x)