Optimal. Leaf size=113 \[ \frac {3 (c+d x)^{2/3} \sqrt [3]{-\frac {b c+a d+2 b d x}{b c-a d}} F_1\left (\frac {2}{3};\frac {4}{3},1;\frac {5}{3};\frac {2 b (c+d x)}{b c-a d},\frac {b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^2 \sqrt [3]{b c+a d+2 b d x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {142, 141}
\begin {gather*} \frac {3 (c+d x)^{2/3} \sqrt [3]{-\frac {a d+b c+2 b d x}{b c-a d}} F_1\left (\frac {2}{3};\frac {4}{3},1;\frac {5}{3};\frac {2 b (c+d x)}{b c-a d},\frac {b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^2 \sqrt [3]{a d+b c+2 b d x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 141
Rule 142
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx &=\frac {\left (d \sqrt [3]{\frac {d (b c+a d+2 b d x)}{-2 b c d+d (b c+a d)}}\right ) \int \frac {1}{(a+b x) \sqrt [3]{c+d x} \left (\frac {d (b c+a d)}{-2 b c d+d (b c+a d)}+\frac {2 b d^2 x}{-2 b c d+d (b c+a d)}\right )^{4/3}} \, dx}{(-2 b c d+d (b c+a d)) \sqrt [3]{b c+a d+2 b d x}}\\ &=\frac {3 (c+d x)^{2/3} \sqrt [3]{-\frac {b c+a d+2 b d x}{b c-a d}} F_1\left (\frac {2}{3};\frac {4}{3},1;\frac {5}{3};\frac {2 b (c+d x)}{b c-a d},\frac {b (c+d x)}{b c-a d}\right )}{2 (b c-a d)^2 \sqrt [3]{b c+a d+2 b d x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 20.66, size = 145, normalized size = 1.28 \begin {gather*} \frac {3 \left (2 b (c+d x)-\frac {(a d+b (c+2 d x)) F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};\frac {b c-a d}{2 b c+2 b d x},\frac {b c-a d}{b c+b d x}\right )}{\left (\frac {b c+a d+2 b d x}{2 b c+2 b d x}\right )^{2/3}}\right )}{2 b^2 (b c-a d) (c+d x)^{4/3} \sqrt [3]{a d+b (c+2 d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right ) \left (d x +c \right )^{\frac {1}{3}} \left (2 b d x +a d +b c \right )^{\frac {4}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right ) \sqrt [3]{c + d x} \left (a d + b c + 2 b d x\right )^{\frac {4}{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{1/3}\,{\left (a\,d+b\,c+2\,b\,d\,x\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________