Optimal. Leaf size=54 \[ \frac {a x}{c}+\frac {(b c-a d) x \, _2F_1\left (1,\frac {1}{m-n};1+\frac {1}{m-n};-\frac {c x^{m-n}}{d}\right )}{c d} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1607, 1598,
396, 251} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 396
Rule 1598
Rule 1607
Rubi steps
\begin {align*} \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx &=\int \frac {x^n \left (b+a x^{m-n}\right )}{c x^m+d x^n} \, dx\\ &=\int \frac {b+a x^{m-n}}{d+c x^{m-n}} \, dx\\ &=\frac {a x}{c}-\frac {(-b c+a d) \int \frac {1}{d+c x^{m-n}} \, dx}{c}\\ &=\frac {a x}{c}+\frac {(b c-a d) x \, _2F_1\left (1,\frac {1}{m-n};1+\frac {1}{m-n};-\frac {c x^{m-n}}{d}\right )}{c d}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 52, normalized size = 0.96 \begin {gather*} \frac {x \left (a d+(b c-a d) \, _2F_1\left (1,\frac {1}{m-n};1+\frac {1}{m-n};-\frac {c x^{m-n}}{d}\right )\right )}{c d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{m}+b \,x^{n}}{c \,x^{m}+d \,x^{n}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{m} + b x^{n}}{c x^{m} + d x^{n}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a\,x^m+b\,x^n}{c\,x^m+d\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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