3.282 \(\int \frac{a x^m+b x^n}{c x^m+d x^n} \, dx\)

Optimal. Leaf size=54 \[ \frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{m-n};1+\frac{1}{m-n};-\frac{c x^{m-n}}{d}\right )}{c d}+\frac{a x}{c} \]

[Out]

(a*x)/c + ((b*c - a*d)*x*Hypergeometric2F1[1, (m - n)^(-1), 1 + (m - n)^(-1), -(
(c*x^(m - n))/d)])/(c*d)

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Rubi [A]  time = 0.0939208, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{m-n};1+\frac{1}{m-n};-\frac{c x^{m-n}}{d}\right )}{c d}+\frac{a x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^m + b*x^n)/(c*x^m + d*x^n),x]

[Out]

(a*x)/c + ((b*c - a*d)*x*Hypergeometric2F1[1, (m - n)^(-1), 1 + (m - n)^(-1), -(
(c*x^(m - n))/d)])/(c*d)

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Rubi in Sympy [A]  time = 10.4547, size = 36, normalized size = 0.67 \[ \frac{a x}{c} - \frac{x \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{m - n} \\ 1 + \frac{1}{m - n} \end{matrix}\middle |{- \frac{c x^{m - n}}{d}} \right )}}{c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**m+b*x**n)/(c*x**m+d*x**n),x)

[Out]

a*x/c - x*(a*d - b*c)*hyper((1, 1/(m - n)), (1 + 1/(m - n),), -c*x**(m - n)/d)/(
c*d)

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Mathematica [A]  time = 0.259487, size = 0, normalized size = 0. \[ \int \frac{a x^m+b x^n}{c x^m+d x^n} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(a*x^m + b*x^n)/(c*x^m + d*x^n),x]

[Out]

Integrate[(a*x^m + b*x^n)/(c*x^m + d*x^n), x]

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \[ \int{\frac{a{x}^{m}+b{x}^{n}}{c{x}^{m}+d{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^m+b*x^n)/(c*x^m+d*x^n),x)

[Out]

int((a*x^m+b*x^n)/(c*x^m+d*x^n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -{\left (b c - a d\right )} \int \frac{x^{m}}{c d x^{m} + d^{2} x^{n}}\,{d x} + \frac{b x}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^m + b*x^n)/(c*x^m + d*x^n),x, algorithm="maxima")

[Out]

-(b*c - a*d)*integrate(x^m/(c*d*x^m + d^2*x^n), x) + b*x/d

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{a x^{m} + b x^{n}}{c x^{m} + d x^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^m + b*x^n)/(c*x^m + d*x^n),x, algorithm="fricas")

[Out]

integral((a*x^m + b*x^n)/(c*x^m + d*x^n), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a x^{m} + b x^{n}}{c x^{m} + d x^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**m+b*x**n)/(c*x**m+d*x**n),x)

[Out]

Integral((a*x**m + b*x**n)/(c*x**m + d*x**n), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a x^{m} + b x^{n}}{c x^{m} + d x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^m + b*x^n)/(c*x^m + d*x^n),x, algorithm="giac")

[Out]

integrate((a*x^m + b*x^n)/(c*x^m + d*x^n), x)