Optimal. Leaf size=127 \[ \frac {(e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a e (m+1) (b c-a d)}+\frac {(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]
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Rubi [A] time = 0.14, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {597, 364} \[ \frac {(e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a e (m+1) (b c-a d)}+\frac {(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 597
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx &=\int \left (\frac {(A b-a B) (e x)^m}{(b c-a d) \left (a+b x^n\right )}+\frac {(B c-A d) (e x)^m}{(b c-a d) \left (c+d x^n\right )}\right ) \, dx\\ &=\frac {(A b-a B) \int \frac {(e x)^m}{a+b x^n} \, dx}{b c-a d}+\frac {(B c-A d) \int \frac {(e x)^m}{c+d x^n} \, dx}{b c-a d}\\ &=\frac {(A b-a B) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a (b c-a d) e (1+m)}+\frac {(B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c (b c-a d) e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 102, normalized size = 0.80 \[ \frac {x (e x)^m \left ((a B c-A b c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )+a (A d-B c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )\right )}{a c (m+1) (a d-b c)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b d x^{2 \, n} + a c + {\left (b c + a d\right )} x^{n}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.99, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (e x \right )^{m}}{\left (b \,x^{n}+a \right ) \left (d \,x^{n}+c \right )}\, dx \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{\left (a+b\,x^n\right )\,\left (c+d\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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