Optimal. Leaf size=46 \[ -\frac{2 \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{3 \sqrt{c} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.115931, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2138, 206} \[ -\frac{2 \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{3 \sqrt{c} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2138
Rule 206
Rubi steps
\begin{align*} \int \frac{c-2 d x}{(c+d x) \sqrt{c^3-8 d^3 x^3}} \, dx &=-\frac{(2 c) \operatorname{Subst}\left (\int \frac{1}{9-c^3 x^2} \, dx,x,\frac{\left (1-\frac{2 d x}{c}\right )^2}{\sqrt{c^3-8 d^3 x^3}}\right )}{d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{3 \sqrt{c} d}\\ \end{align*}
Mathematica [A] time = 0.0244973, size = 46, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{(c-2 d x)^2}{3 \sqrt{c} \sqrt{c^3-8 d^3 x^3}}\right )}{3 \sqrt{c} d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.143, size = 650, normalized size = 14.1 \begin{align*} \text{result too large to display} \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{2 \, d x - c}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.6736, size = 651, normalized size = 14.15 \begin{align*} \left [\frac{\log \left (\frac{8 \, d^{6} x^{6} - 240 \, c d^{5} x^{5} + 408 \, c^{2} d^{4} x^{4} + 88 \, c^{3} d^{3} x^{3} + 156 \, c^{4} d^{2} x^{2} + 12 \, c^{5} d x + 17 \, c^{6} - 3 \,{\left (8 \, d^{4} x^{4} - 52 \, c d^{3} x^{3} + 12 \, c^{2} d^{2} x^{2} - 4 \, c^{3} d x + 5 \, c^{4}\right )} \sqrt{-8 \, d^{3} x^{3} + c^{3}} \sqrt{c}}{d^{6} x^{6} + 6 \, c d^{5} x^{5} + 15 \, c^{2} d^{4} x^{4} + 20 \, c^{3} d^{3} x^{3} + 15 \, c^{4} d^{2} x^{2} + 6 \, c^{5} d x + c^{6}}\right )}{6 \, \sqrt{c} d}, -\frac{\sqrt{-c} \arctan \left (\frac{{\left (4 \, d^{3} x^{3} - 24 \, c d^{2} x^{2} - 6 \, c^{2} d x - 5 \, c^{3}\right )} \sqrt{-8 \, d^{3} x^{3} + c^{3}} \sqrt{-c}}{3 \,{\left (16 \, c d^{4} x^{4} - 8 \, c^{2} d^{3} x^{3} - 2 \, c^{4} d x + c^{5}\right )}}\right )}{3 \, c d}\right ] \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{c}{c \sqrt{c^{3} - 8 d^{3} x^{3}} + d x \sqrt{c^{3} - 8 d^{3} x^{3}}}\, dx - \int \frac{2 d x}{c \sqrt{c^{3} - 8 d^{3} x^{3}} + d x \sqrt{c^{3} - 8 d^{3} x^{3}}}\, 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{2 \, d x - c}{\sqrt{-8 \, d^{3} x^{3} + c^{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]