Optimal. Leaf size=199 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.100251, antiderivative size = 199, normalized size of antiderivative = 1., 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 = {130, 484} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 130
Rule 484
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{x} \sqrt{c+d x} (4 c+d x)} \, dx &=3 \operatorname{Subst}\left (\int \frac{x}{\sqrt{c+d x^3} \left (4 c+d x^3\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{3} \sqrt{c}}\right )}{2^{2/3} \sqrt{3} c^{5/6} d^{2/3}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{d} \sqrt [3]{x}\right )}{\sqrt{c+d x}}\right )}{2^{2/3} c^{5/6} d^{2/3}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c}}\right )}{3\ 2^{2/3} c^{5/6} d^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0368889, size = 61, normalized size = 0.31 \[ \frac{3 x^{2/3} \sqrt{\frac{c+d x}{c}} F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x}{c},-\frac{d x}{4 c}\right )}{8 c \sqrt{c+d x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+4\,c}{\frac{1}{\sqrt [3]{x}}}{\frac{1}{\sqrt{dx+c}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + 4 \, c\right )} \sqrt{d x + c} x^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{x} \sqrt{c + d x} \left (4 c + d x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + 4 \, c\right )} \sqrt{d x + c} x^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]