Optimal. Leaf size=65 \[ \frac {c x^2 \sqrt {c+d x^3} F_1\left (\frac {2}{3};2,-\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 \sqrt {\frac {d x^3}{c}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {511, 510} \[ \frac {c x^2 \sqrt {c+d x^3} F_1\left (\frac {2}{3};2,-\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 \sqrt {\frac {d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx &=\frac {\left (c \sqrt {c+d x^3}\right ) \int \frac {x \left (1+\frac {d x^3}{c}\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx}{\sqrt {1+\frac {d x^3}{c}}}\\ &=\frac {c x^2 \sqrt {c+d x^3} F_1\left (\frac {2}{3};2,-\frac {3}{2};\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 \sqrt {1+\frac {d x^3}{c}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.26, size = 177, normalized size = 2.72 \[ \frac {x^2 \left (-d x^3 \left (a+b x^3\right ) \sqrt {\frac {d x^3}{c}+1} (b c-7 a d) F_1\left (\frac {5}{3};\frac {1}{2},1;\frac {8}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+5 c \left (a+b x^3\right ) \sqrt {\frac {d x^3}{c}+1} (2 a d+b c) F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )-10 a \left (c+d x^3\right ) (a d-b c)\right )}{30 a^2 b \left (a+b x^3\right ) \sqrt {c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.36, size = 955, normalized size = 14.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,{\left (d\,x^3+c\right )}^{3/2}}{{\left (b\,x^3+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (c + d x^{3}\right )^{\frac {3}{2}}}{\left (a + b x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________