Optimal. Leaf size=122 \[ \frac {b x}{a (b c-a d) n \left (a+b x^n\right )}+\frac {b (a d (1-2 n)-b c (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^2 n}+\frac {d^2 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c (b c-a d)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {425, 536, 251}
\begin {gather*} \frac {b x (a d (1-2 n)-b c (1-n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 n (b c-a d)^2}+\frac {d^2 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c (b c-a d)^2}+\frac {b x}{a n (b c-a d) \left (a+b x^n\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 251
Rule 425
Rule 536
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^n\right )^2 \left (c+d x^n\right )} \, dx &=\frac {b x}{a (b c-a d) n \left (a+b x^n\right )}-\frac {\int \frac {a d n+b (c-c n)+b d (1-n) x^n}{\left (a+b x^n\right ) \left (c+d x^n\right )} \, dx}{a (b c-a d) n}\\ &=\frac {b x}{a (b c-a d) n \left (a+b x^n\right )}+\frac {d^2 \int \frac {1}{c+d x^n} \, dx}{(b c-a d)^2}+\frac {(b (a d (1-2 n)-b c (1-n))) \int \frac {1}{a+b x^n} \, dx}{a (b c-a d)^2 n}\\ &=\frac {b x}{a (b c-a d) n \left (a+b x^n\right )}+\frac {b (a d (1-2 n)-b c (1-n)) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 (b c-a d)^2 n}+\frac {d^2 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c (b c-a d)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 108, normalized size = 0.89 \begin {gather*} \frac {x \left (\frac {b^2 c-a b d}{a^2 n+a b n x^n}+\frac {b (a d (1-2 n)+b c (-1+n)) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 n}+\frac {d^2 \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{c}\right )}{(b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,x^{n}\right )^{2} \left (c +d \,x^{n}\right )}\, dx\]
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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,x^n\right )}^2\,\left (c+d\,x^n\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________