Optimal. Leaf size=304 \[ -\frac {(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-n (1-p)+1)-B c (m+n p+1))) F_1\left (\frac {m+1}{n};-p,1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 d e (m+1) n (b c-a d)}-\frac {b (e x)^{m+1} (m+n p+1) (B c-A d) \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{c d e (m+1) n (b c-a d)}+\frac {(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^{p+1}}{c e n (b c-a d) \left (c+d x^n\right )} \]
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Rubi [A] time = 0.54, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {595, 597, 365, 364, 511, 510} \[ -\frac {(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} (a d (B c (m+1)-A d (m-n+1))+b c (A d (m-n (1-p)+1)-B c (m+n p+1))) F_1\left (\frac {m+1}{n};-p,1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 d e (m+1) n (b c-a d)}-\frac {b (e x)^{m+1} (m+n p+1) (B c-A d) \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{c d e (m+1) n (b c-a d)}+\frac {(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^{p+1}}{c e n (b c-a d) \left (c+d x^n\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 510
Rule 511
Rule 595
Rule 597
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (a+b x^n\right )^p \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 )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac {\int \frac {(e x)^m \left (a+b x^n\right )^p \left (-a (B c-A d) (1+m)+A (b c-a d) n-b (B c-A d) (1+m+n p) x^n\right )}{c+d x^n} \, dx}{c (b c-a d) n}\\ &=\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac {\int \left (-\frac {b (B c-A d) (1+m+n p) (e x)^m \left (a+b x^n\right )^p}{d}+\frac {(d (-a (B c-A d) (1+m)+A (b c-a d) n)+b c (B c-A d) (1+m+n p)) (e x)^m \left (a+b x^n\right )^p}{d \left (c+d x^n\right )}\right ) \, dx}{c (b c-a d) n}\\ &=\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}-\frac {(b (B c-A d) (1+m+n p)) \int (e x)^m \left (a+b x^n\right )^p \, dx}{c d (b c-a d) n}+\frac {(d (-a (B c-A d) (1+m)+A (b c-a d) n)+b c (B c-A d) (1+m+n p)) \int \frac {(e x)^m \left (a+b x^n\right )^p}{c+d x^n} \, dx}{c d (b c-a d) n}\\ &=\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}-\frac {\left (b (B c-A d) (1+m+n p) \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac {b x^n}{a}\right )^p \, dx}{c d (b c-a d) n}+\frac {\left ((d (-a (B c-A d) (1+m)+A (b c-a d) n)+b c (B c-A d) (1+m+n p)) \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int \frac {(e x)^m \left (1+\frac {b x^n}{a}\right )^p}{c+d x^n} \, dx}{c d (b c-a d) n}\\ &=\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}-\frac {(a d (B c-A d) (1+m)-A d (b c-a d) n-b c (B c-A d) (1+m+n p)) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} F_1\left (\frac {1+m}{n};-p,1;\frac {1+m+n}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 d (b c-a d) e (1+m) n}-\frac {b (B c-A d) (1+m+n p) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{n},-p;\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{c d (b c-a d) e (1+m) n}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 138, normalized size = 0.45 \[ \frac {x (e x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (A (m+n+1) F_1\left (\frac {m+1}{n};-p,2;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+B (m+1) x^n F_1\left (\frac {m+n+1}{n};-p,2;\frac {m+2 n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{c^2 (m+1) (m+n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{d^{2} x^{2 \, n} + 2 \, c d x^{n} + c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (e x \right )^{m} \left (b \,x^{n}+a \right )^{p}}{\left (d \,x^{n}+c \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{{\left (d x^{n} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^p}{{\left (c+d\,x^n\right )}^2} \,d x \]
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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