Optimal. Leaf size=68 \[ -\frac{69 \sqrt{1-2 x}}{1210 (5 x+3)}-\frac{\sqrt{1-2 x}}{110 (5 x+3)^2}-\frac{69 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0147039, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac{69 \sqrt{1-2 x}}{1210 (5 x+3)}-\frac{\sqrt{1-2 x}}{110 (5 x+3)^2}-\frac{69 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)^3} \, dx &=-\frac{\sqrt{1-2 x}}{110 (3+5 x)^2}+\frac{69}{110} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^2} \, dx\\ &=-\frac{\sqrt{1-2 x}}{110 (3+5 x)^2}-\frac{69 \sqrt{1-2 x}}{1210 (3+5 x)}+\frac{69 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{1210}\\ &=-\frac{\sqrt{1-2 x}}{110 (3+5 x)^2}-\frac{69 \sqrt{1-2 x}}{1210 (3+5 x)}-\frac{69 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1210}\\ &=-\frac{\sqrt{1-2 x}}{110 (3+5 x)^2}-\frac{69 \sqrt{1-2 x}}{1210 (3+5 x)}-\frac{69 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}}\\ \end{align*}
Mathematica [A] time = 0.0295189, size = 53, normalized size = 0.78 \[ -\frac{\sqrt{1-2 x} (345 x+218)}{1210 (5 x+3)^2}-\frac{69 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.008, size = 48, normalized size = 0.7 \begin{align*} -100\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{69\, \left ( 1-2\,x \right ) ^{3/2}}{12100}}+{\frac{71\,\sqrt{1-2\,x}}{5500}} \right ) }-{\frac{69\,\sqrt{55}}{33275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 3.9379, size = 100, normalized size = 1.47 \begin{align*} \frac{69}{66550} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{345 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 781 \, \sqrt{-2 \, x + 1}}{605 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62925, size = 200, normalized size = 2.94 \begin{align*} \frac{69 \, \sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (345 \, x + 218\right )} \sqrt{-2 \, x + 1}}{66550 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.39818, size = 92, normalized size = 1.35 \begin{align*} \frac{69}{66550} \, \sqrt{55} \log \left (\frac{{\left | -2 \, \sqrt{55} + 10 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{345 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 781 \, \sqrt{-2 \, x + 1}}{2420 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]