Optimal. Leaf size=131 \[ -\frac{1045 \sqrt{1-2 x}}{14 (5 x+3)}+\frac{52 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)}-\frac{7209}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+1000 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0451106, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {99, 151, 156, 63, 206} \[ -\frac{1045 \sqrt{1-2 x}}{14 (5 x+3)}+\frac{52 \sqrt{1-2 x}}{7 (3 x+2) (5 x+3)}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)}-\frac{7209}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+1000 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 99
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^2} \, dx &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)}-\frac{1}{2} \int \frac{-18+25 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)}+\frac{52 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}-\frac{1}{14} \int \frac{-1363+1560 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)}+\frac{52 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}+\frac{1}{154} \int \frac{-56309+34485 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)}+\frac{52 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}+\frac{21627}{14} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-2500 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)}+\frac{52 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}-\frac{21627}{14} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+2500 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{1045 \sqrt{1-2 x}}{14 (3+5 x)}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)}+\frac{52 \sqrt{1-2 x}}{7 (2+3 x) (3+5 x)}-\frac{7209}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+1000 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0954411, size = 94, normalized size = 0.72 \[ -\frac{\sqrt{1-2 x} \left (9405 x^2+12228 x+3965\right )}{14 (3 x+2)^2 (5 x+3)}-\frac{7209}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+1000 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 82, normalized size = 0.6 \begin{align*} 162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{139\, \left ( 1-2\,x \right ) ^{3/2}}{126}}-{\frac{47\,\sqrt{1-2\,x}}{18}} \right ) }-{\frac{7209\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+10\,{\frac{\sqrt{1-2\,x}}{-2\,x-6/5}}+{\frac{1000\,\sqrt{55}}{11}{\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 = 2.60693, size = 173, normalized size = 1.32 \begin{align*} -\frac{500}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{7209}{98} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9405 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 43266 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49721 \, \sqrt{-2 \, x + 1}}{7 \,{\left (45 \,{\left (2 \, x - 1\right )}^{3} + 309 \,{\left (2 \, x - 1\right )}^{2} + 1414 \, x - 168\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.65404, size = 423, normalized size = 3.23 \begin{align*} \frac{49000 \, \sqrt{11} \sqrt{5}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 79299 \, \sqrt{7} \sqrt{3}{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (9405 \, x^{2} + 12228 \, x + 3965\right )} \sqrt{-2 \, x + 1}}{1078 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 88.5384, size = 474, normalized size = 3.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 2.55973, size = 166, normalized size = 1.27 \begin{align*} -\frac{500}{11} \, \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{7209}{98} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) - \frac{25 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} + \frac{9 \,{\left (139 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 329 \, \sqrt{-2 \, x + 1}\right )}}{28 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]