Optimal. Leaf size=67 \[ -\frac {\sqrt [3]{\frac {b x^3}{a}+1} F_1\left (-\frac {4}{3};\frac {4}{3},1;-\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a c x^4 \sqrt [3]{a+b x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 67, 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} \[ -\frac {\sqrt [3]{\frac {b x^3}{a}+1} F_1\left (-\frac {4}{3};\frac {4}{3},1;-\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a c x^4 \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {1}{x^5 \left (1+\frac {b x^3}{a}\right )^{4/3} \left (c+d x^3\right )} \, dx}{a \sqrt [3]{a+b x^3}}\\ &=-\frac {\sqrt [3]{1+\frac {b x^3}{a}} F_1\left (-\frac {4}{3};\frac {4}{3},1;-\frac {1}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a c x^4 \sqrt [3]{a+b x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.37, size = 264, normalized size = 3.94 \[ \frac {-2 b d x^9 \sqrt [3]{\frac {b x^3}{a}+1} \left (2 a^2 d^2+a b c d-5 b^2 c^2\right ) F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+5 x^6 \sqrt [3]{\frac {b x^3}{a}+1} \left (2 a^3 d^3-2 a^2 b c d^2-a b^2 c^2 d+5 b^3 c^3\right ) F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+5 c \left (a^3 d \left (4 d x^3-c\right )+a^2 b \left (c^2+c d x^3+4 d^2 x^6\right )+a b^2 c x^3 \left (2 d x^3-5 c\right )-10 b^3 c^2 x^6\right )}{20 a^3 c^3 x^4 \sqrt [3]{a+b x^3} (a d-b c)} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right ) x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {4}{3}} {\left (d x^{3} + c\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^5\,{\left (b\,x^3+a\right )}^{4/3}\,\left (d\,x^3+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{5} \left (a + b x^{3}\right )^{\frac {4}{3}} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________