Optimal. Leaf size=65 \[ -\frac {c \sqrt {c+d x^3} F_1\left (-\frac {2}{3};1,-\frac {3}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a x^2 \sqrt {\frac {d x^3}{c}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {511, 510} \[ -\frac {c \sqrt {c+d x^3} F_1\left (-\frac {2}{3};1,-\frac {3}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a x^2 \sqrt {\frac {d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^{3/2}}{x^3 \left (a+b x^3\right )} \, dx &=\frac {\left (c \sqrt {c+d x^3}\right ) \int \frac {\left (1+\frac {d x^3}{c}\right )^{3/2}}{x^3 \left (a+b x^3\right )} \, dx}{\sqrt {1+\frac {d x^3}{c}}}\\ &=-\frac {c \sqrt {c+d x^3} F_1\left (-\frac {2}{3};1,-\frac {3}{2};\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a x^2 \sqrt {1+\frac {d x^3}{c}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.33, size = 343, normalized size = 5.28 \[ -\frac {d x^6 \sqrt {\frac {d x^3}{c}+1} (b c-4 a d) F_1\left (\frac {4}{3};\frac {1}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+\frac {8 a c \left (3 x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (2 b c F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )-4 a c \left (2 a c-5 a d x^3+6 b c x^3+2 b d x^6\right ) F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac {4}{3};\frac {1}{2},2;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d F_1\left (\frac {4}{3};\frac {3}{2},1;\frac {7}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )-8 a c F_1\left (\frac {1}{3};\frac {1}{2},1;\frac {4}{3};-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )}}{16 a^2 x^2 \sqrt {c+d x^3}} \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.30, size = 1096, normalized size = 16.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,x^3+c\right )}^{3/2}}{x^3\,\left (b\,x^3+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d x^{3}\right )^{\frac {3}{2}}}{x^{3} \left (a + b x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________