Optimal. Leaf size=112 \[ \frac {(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac {(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]
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Rubi [A] time = 0.06, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 364} \[ \frac {(e x)^{m+1} (B c (m+1)-A d (m-2 n+1)) \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{2 c^3 d e (m+1) n}-\frac {(e x)^{m+1} (B c-A d)}{2 c d e n \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 457
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right )}{\left (c+d x^n\right )^3} \, dx &=-\frac {(B c-A d) (e x)^{1+m}}{2 c d e n \left (c+d x^n\right )^2}+\frac {(B c (1+m)-A d (1+m-2 n)) \int \frac {(e x)^m}{\left (c+d x^n\right )^2} \, dx}{2 c d n}\\ &=-\frac {(B c-A d) (e x)^{1+m}}{2 c d e n \left (c+d x^n\right )^2}+\frac {(B c (1+m)-A d (1+m-2 n)) (e x)^{1+m} \, _2F_1\left (2,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{2 c^3 d e (1+m) n}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 83, normalized size = 0.74 \[ \frac {x (e x)^m \left ((A d-B c) \, _2F_1\left (3,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )+B c \, _2F_1\left (2,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )\right )}{c^3 d (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d^{3} x^{3 \, n} + 3 \, c d^{2} x^{2 \, n} + 3 \, c^{2} d x^{n} + c^{3}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (e x \right )^{m}}{\left (d \,x^{n}+c \right )^{3}}\, dx \]
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 \[ -{\left ({\left (m^{2} - m {\left (n - 2\right )} - n + 1\right )} B c e^{m} - {\left (m^{2} - m {\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} A d e^{m}\right )} \int \frac {x^{m}}{2 \, {\left (c^{2} d^{2} n^{2} x^{n} + c^{3} d n^{2}\right )}}\,{d x} + \frac {{\left (B c^{2} e^{m} {\left (m - n + 1\right )} - A c d e^{m} {\left (m - 3 \, n + 1\right )}\right )} x x^{m} - {\left (A d^{2} e^{m} {\left (m - 2 \, n + 1\right )} - B c d e^{m} {\left (m + 1\right )}\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{2 \, {\left (c^{2} d^{3} n^{2} x^{2 \, n} + 2 \, c^{3} d^{2} n^{2} x^{n} + c^{4} d n^{2}\right )}} \]
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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{{\left (c+d\,x^n\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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