Optimal. Leaf size=84 \[ \frac {b x \left (c+d x^3\right )^{q+1}}{d (3 q+4)}-\frac {x \left (c+d x^3\right )^{q+1} (b c-a d (3 q+4)) \, _2F_1\left (1,q+\frac {4}{3};\frac {4}{3};-\frac {d x^3}{c}\right )}{c d (3 q+4)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {388, 246, 245} \[ x \left (c+d x^3\right )^q \left (\frac {d x^3}{c}+1\right )^{-q} \left (a-\frac {b c}{3 d q+4 d}\right ) \, _2F_1\left (\frac {1}{3},-q;\frac {4}{3};-\frac {d x^3}{c}\right )+\frac {b x \left (c+d x^3\right )^{q+1}}{d (3 q+4)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 246
Rule 388
Rubi steps
\begin {align*} \int \left (a+b x^3\right ) \left (c+d x^3\right )^q \, dx &=\frac {b x \left (c+d x^3\right )^{1+q}}{d (4+3 q)}-\left (-a+\frac {b c}{4 d+3 d q}\right ) \int \left (c+d x^3\right )^q \, dx\\ &=\frac {b x \left (c+d x^3\right )^{1+q}}{d (4+3 q)}-\left (\left (-a+\frac {b c}{4 d+3 d q}\right ) \left (c+d x^3\right )^q \left (1+\frac {d x^3}{c}\right )^{-q}\right ) \int \left (1+\frac {d x^3}{c}\right )^q \, dx\\ &=\frac {b x \left (c+d x^3\right )^{1+q}}{d (4+3 q)}+\left (a-\frac {b c}{4 d+3 d q}\right ) x \left (c+d x^3\right )^q \left (1+\frac {d x^3}{c}\right )^{-q} \, _2F_1\left (\frac {1}{3},-q;\frac {4}{3};-\frac {d x^3}{c}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 90, normalized size = 1.07 \[ \frac {x \left (c+d x^3\right )^q \left (\frac {d x^3}{c}+1\right )^{-q} \left ((a d (3 q+4)-b c) \, _2F_1\left (\frac {1}{3},-q;\frac {4}{3};-\frac {d x^3}{c}\right )+b \left (c+d x^3\right ) \left (\frac {d x^3}{c}+1\right )^q\right )}{d (3 q+4)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{3} + a\right )} {\left (d x^{3} + c\right )}^{q}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{3} + a\right )} {\left (d x^{3} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{3}+a \right ) \left (d \,x^{3}+c \right )^{q}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{3} + a\right )} {\left (d x^{3} + c\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (b\,x^3+a\right )\,{\left (d\,x^3+c\right )}^q \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 82.86, size = 75, normalized size = 0.89 \[ \frac {a c^{q} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, - q \\ \frac {4}{3} \end {matrix}\middle | {\frac {d x^{3} e^{i \pi }}{c}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {b c^{q} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, - q \\ \frac {7}{3} \end {matrix}\middle | {\frac {d x^{3} e^{i \pi }}{c}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________