Optimal. Leaf size=211 \[ \frac {(B c-A d) (e x)^{1+m}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac {b (A b-a B) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a (b c-a d)^2 e (1+m)}+\frac {(b c (A d (1+m-2 n)-B c (1+m-n))+a d (B c (1+m)-A d (1+m-n))) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c^2 (b c-a d)^2 e (1+m) n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.35, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {609, 611, 371}
\begin {gather*} \frac {(e x)^{m+1} \, _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-2 n+1)-B c (m-n+1)))}{c^2 e (m+1) n (b c-a d)^2}+\frac {b (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)^2}+\frac {(e x)^{m+1} (B c-A d)}{c e n (b c-a d) \left (c+d x^n\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 609
Rule 611
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 )^2} \, dx &=\frac {(B c-A d) (e x)^{1+m}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac {\int \frac {(e x)^m \left (-a (B c-A d) (1+m)+A (b c-a d) n-b (B c-A d) (1+m-n) x^n\right )}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{c (b c-a d) n}\\ &=\frac {(B c-A d) (e x)^{1+m}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac {\int \left (\frac {b (A b-a B) c n (e x)^m}{(b c-a d) \left (a+b x^n\right )}+\frac {(b c (A d (1+m-2 n)-B c (1+m-n))+a d (B c (1+m)-A d (1+m-n))) (e x)^m}{(b c-a d) \left (c+d x^n\right )}\right ) \, dx}{c (b c-a d) n}\\ &=\frac {(B c-A d) (e x)^{1+m}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac {(b (A b-a B)) \int \frac {(e x)^m}{a+b x^n} \, dx}{(b c-a d)^2}+\frac {(b c (A d (1+m-2 n)-B c (1+m-n))+a d (B c (1+m)-A d (1+m-n))) \int \frac {(e x)^m}{c+d x^n} \, dx}{c (b c-a d)^2 n}\\ &=\frac {(B c-A d) (e x)^{1+m}}{c (b c-a d) e n \left (c+d x^n\right )}+\frac {b (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)^2 e (1+m)}+\frac {(b c (A d (1+m-2 n)-B c (1+m-n))+a d (B c (1+m)-A d (1+m-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 (b c-a d)^2 e (1+m) n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.36, size = 177, normalized size = 0.84 \begin {gather*} \frac {x (e x)^m \left (\frac {(b c-a d) (B c-A d)}{c n \left (c+d x^n\right )}+\frac {b (A b-a B) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a (1+m)}-\frac {(b c (-A d (1+m-2 n)+B c (1+m-n))+a d (-B c (1+m)+A d (1+m-n))) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c^2 (1+m) n}\right )}{(b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right )}{\left (a +b \,x^{n}\right ) \left (c +d \,x^{n}\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )\,{\left (c+d\,x^n\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________