Optimal. Leaf size=77 \[ \frac {(d x)^{m+1} \left (\frac {b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac {m+1}{3},-2 p;\frac {m+4}{3};-\frac {b x^3}{a}\right )}{d (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1356, 364} \[ \frac {(d x)^{m+1} \left (\frac {b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac {m+1}{3},-2 p;\frac {m+4}{3};-\frac {b x^3}{a}\right )}{d (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 1356
Rubi steps
\begin {align*} \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx &=\left (\left (1+\frac {b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p\right ) \int (d x)^m \left (1+\frac {b x^3}{a}\right )^{2 p} \, dx\\ &=\frac {(d x)^{1+m} \left (1+\frac {b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac {1+m}{3},-2 p;\frac {4+m}{3};-\frac {b x^3}{a}\right )}{d (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 66, normalized size = 0.86 \[ \frac {x (d x)^m \left (\left (a+b x^3\right )^2\right )^p \left (\frac {b x^3}{a}+1\right )^{-2 p} \, _2F_1\left (\frac {m+1}{3},-2 p;\frac {m+1}{3}+1;-\frac {b x^3}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} \left (d x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________