Optimal. Leaf size=267 \[ -\frac {b^2 (A d (1+m+n)-B c (1+m+2 n)) x^{1+n} (e x)^m}{c d^2 n (1+m+n)}-\frac {b (2 a d (A d (1+m)-B c (1+m+n))-b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^{1+m}}{c d^3 e (1+m) n}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{c d e n \left (c+d x^n\right )}-\frac {(b c-a d) (a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c^2 d^3 e (1+m) n} \]
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Rubi [A]
time = 0.45, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {608, 584, 20,
30, 371} \begin {gather*} -\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 ) (a d (B c (m+1)-A d (m-n+1))+b c (A d (m+n+1)-B c (m+2 n+1)))}{c^2 d^3 e (m+1) n}-\frac {b (e x)^{m+1} (2 a d (A d (m+1)-B c (m+n+1))-b c (A d (m+n+1)-B c (m+2 n+1)))}{c d^3 e (m+1) n}-\frac {(e x)^{m+1} \left (a+b x^n\right )^2 (B c-A d)}{c d e n \left (c+d x^n\right )}-\frac {b^2 x^{n+1} (e x)^m (A d (m+n+1)-B c (m+2 n+1))}{c d^2 n (m+n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 30
Rule 371
Rule 584
Rule 608
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx &=-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{c d e n \left (c+d x^n\right )}-\frac {\int \frac {(e x)^m \left (a+b x^n\right ) \left (-a (B c (1+m)-A d (1+m-n))+b (A d (1+m+n)-B c (1+m+2 n)) x^n\right )}{c+d x^n} \, dx}{c d n}\\ &=-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{c d e n \left (c+d x^n\right )}-\frac {\int \left (\frac {b (2 a d (A d (1+m)-B c (1+m+n))-b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^m}{d^2}+\frac {b^2 (A d (1+m+n)-B c (1+m+2 n)) x^n (e x)^m}{d}+\frac {(b c-a d) (a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^m}{d^2 \left (c+d x^n\right )}\right ) \, dx}{c d n}\\ &=-\frac {b (2 a d (A d (1+m)-B c (1+m+n))-b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^{1+m}}{c d^3 e (1+m) n}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{c d e n \left (c+d x^n\right )}-\frac {\left (b^2 (A d (1+m+n)-B c (1+m+2 n))\right ) \int x^n (e x)^m \, dx}{c d^2 n}-\frac {((b c-a d) (a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m+n)-B c (1+m+2 n)))) \int \frac {(e x)^m}{c+d x^n} \, dx}{c d^3 n}\\ &=-\frac {b (2 a d (A d (1+m)-B c (1+m+n))-b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^{1+m}}{c d^3 e (1+m) n}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{c d e n \left (c+d x^n\right )}-\frac {(b c-a d) (a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c^2 d^3 e (1+m) n}-\frac {\left (b^2 (A d (1+m+n)-B c (1+m+2 n)) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx}{c d^2 n}\\ &=-\frac {b^2 (A d (1+m+n)-B c (1+m+2 n)) x^{1+n} (e x)^m}{c d^2 n (1+m+n)}-\frac {b (2 a d (A d (1+m)-B c (1+m+n))-b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^{1+m}}{c d^3 e (1+m) n}-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^2}{c d e n \left (c+d x^n\right )}-\frac {(b c-a d) (a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m+n)-B c (1+m+2 n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c^2 d^3 e (1+m) n}\\ \end {align*}
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Mathematica [A]
time = 0.82, size = 249, normalized size = 0.93 \begin {gather*} x (e x)^m \left (\frac {2 a b \left (-A d (1+m)+B c (1+m+n)+B d n x^n\right )}{d^2 (1+m) n \left (c+d x^n\right )}-\frac {a^2 B c-a^2 A d}{c^2 d n+c d^2 n x^n}+\frac {b^2 \left (A d \left (\frac {1}{1+m}+\frac {c}{c n+d n x^n}\right )+B \left (-\frac {2 c}{1+m}+\frac {d x^n}{1+m+n}-\frac {c^2}{c n+d n x^n}\right )\right )}{d^3}+\frac {(b c-a d) (a d (-B c (1+m)+A d (1+m-n))+b c (-A d (1+m+n)+B c (1+m+2 n))) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c^2 d^3 (1+m) n}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (a +b \,x^{n}\right )^{2} \left (A +B \,x^{n}\right )}{\left (c +d \,x^{n}\right )^{2}}\, dx\]
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 {\left (e x\right )^{m} \left (A + B x^{n}\right ) \left (a + b x^{n}\right )^{2}}{\left (c + d x^{n}\right )^{2}}\, 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 = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^2}{{\left (c+d\,x^n\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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