Optimal. Leaf size=160 \[ -\frac {(b c-a d) x \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2}+\frac {(b c-a d) (a d (1-2 n)-b c (1+n)) x}{2 c^2 d^2 n^2 \left (c+d x^n\right )}-\frac {\left (2 a b c d (1-n)-b^2 c^2 (1+n)-a^2 d^2 \left (1-3 n+2 n^2\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 d^2 n^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {424, 393, 251}
\begin {gather*} -\frac {x \left (-a^2 d^2 \left (2 n^2-3 n+1\right )+2 a b c d (1-n)-b^2 c^2 (n+1)\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 d^2 n^2}+\frac {x (b c-a d) (a d (1-2 n)-b c (n+1))}{2 c^2 d^2 n^2 \left (c+d x^n\right )}-\frac {x (b c-a d) \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 251
Rule 393
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (a+b x^n\right )^2}{\left (c+d x^n\right )^3} \, dx &=-\frac {(b c-a d) x \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2}+\frac {\int \frac {a (b c-a d (1-2 n))-b (a d (1-n)-b c (1+n)) x^n}{\left (c+d x^n\right )^2} \, dx}{2 c d n}\\ &=-\frac {(b c-a d) x \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2}+\frac {(b c-a d) (a d (1-2 n)-b c (1+n)) x}{2 c^2 d^2 n^2 \left (c+d x^n\right )}-\frac {\left (2 a b c d (1-n)-b^2 c^2 (1+n)-a^2 d^2 \left (1-3 n+2 n^2\right )\right ) \int \frac {1}{c+d x^n} \, dx}{2 c^2 d^2 n^2}\\ &=-\frac {(b c-a d) x \left (a+b x^n\right )}{2 c d n \left (c+d x^n\right )^2}+\frac {(b c-a d) (a d (1-2 n)-b c (1+n)) x}{2 c^2 d^2 n^2 \left (c+d x^n\right )}-\frac {\left (2 a b c d (1-n)-b^2 c^2 (1+n)-a^2 d^2 \left (1-3 n+2 n^2\right )\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )}{2 c^3 d^2 n^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 133, normalized size = 0.83 \begin {gather*} \frac {x \left (\frac {c^2 (b c-a d)^2 n}{\left (c+d x^n\right )^2}-\frac {c (b c-a d) (a d (-1+2 n)+b (c+2 c n))}{c+d x^n}+\left (2 a b c d (-1+n)+b^2 c^2 (1+n)+a^2 d^2 \left (1-3 n+2 n^2\right )\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {d x^n}{c}\right )\right )}{2 c^3 d^2 n^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,x^{n}\right )^{2}}{\left (c +d \,x^{n}\right )^{3}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\left (a+b\,x^n\right )}^2}{{\left (c+d\,x^n\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________