Optimal. Leaf size=128 \[ -\frac{9145 \sqrt{1-2 x}}{57624 (3 x+2)}-\frac{9145 \sqrt{1-2 x}}{24696 (3 x+2)^2}-\frac{1829 \sqrt{1-2 x}}{1764 (3 x+2)^3}-\frac{2179 \sqrt{1-2 x}}{588 (3 x+2)^4}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^4}-\frac{9145 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0390836, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 51, 63, 206} \[ -\frac{9145 \sqrt{1-2 x}}{57624 (3 x+2)}-\frac{9145 \sqrt{1-2 x}}{24696 (3 x+2)^2}-\frac{1829 \sqrt{1-2 x}}{1764 (3 x+2)^3}-\frac{2179 \sqrt{1-2 x}}{588 (3 x+2)^4}+\frac{121}{14 \sqrt{1-2 x} (3 x+2)^4}-\frac{9145 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 89
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^2}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^4}-\frac{1}{14} \int \frac{-1336+175 x}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^4}-\frac{2179 \sqrt{1-2 x}}{588 (2+3 x)^4}+\frac{1829}{84} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^4}-\frac{2179 \sqrt{1-2 x}}{588 (2+3 x)^4}-\frac{1829 \sqrt{1-2 x}}{1764 (2+3 x)^3}+\frac{9145 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{1764}\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^4}-\frac{2179 \sqrt{1-2 x}}{588 (2+3 x)^4}-\frac{1829 \sqrt{1-2 x}}{1764 (2+3 x)^3}-\frac{9145 \sqrt{1-2 x}}{24696 (2+3 x)^2}+\frac{9145 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{8232}\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^4}-\frac{2179 \sqrt{1-2 x}}{588 (2+3 x)^4}-\frac{1829 \sqrt{1-2 x}}{1764 (2+3 x)^3}-\frac{9145 \sqrt{1-2 x}}{24696 (2+3 x)^2}-\frac{9145 \sqrt{1-2 x}}{57624 (2+3 x)}+\frac{9145 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{57624}\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^4}-\frac{2179 \sqrt{1-2 x}}{588 (2+3 x)^4}-\frac{1829 \sqrt{1-2 x}}{1764 (2+3 x)^3}-\frac{9145 \sqrt{1-2 x}}{24696 (2+3 x)^2}-\frac{9145 \sqrt{1-2 x}}{57624 (2+3 x)}-\frac{9145 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{57624}\\ &=\frac{121}{14 \sqrt{1-2 x} (2+3 x)^4}-\frac{2179 \sqrt{1-2 x}}{588 (2+3 x)^4}-\frac{1829 \sqrt{1-2 x}}{1764 (2+3 x)^3}-\frac{9145 \sqrt{1-2 x}}{24696 (2+3 x)^2}-\frac{9145 \sqrt{1-2 x}}{57624 (2+3 x)}-\frac{9145 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{28812 \sqrt{21}}\\ \end{align*}
Mathematica [C] time = 0.0175316, size = 60, normalized size = 0.47 \[ \frac{29264 (2 x-1) (3 x+2)^4 \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )+747397 (2 x-1)+1743126}{201684 \sqrt{1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 75, normalized size = 0.6 \begin{align*}{\frac{648}{16807\, \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{29167}{288} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2001923}{2592} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{15060395}{7776} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{12452615}{7776}\sqrt{1-2\,x}} \right ) }-{\frac{9145\,\sqrt{21}}{605052}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{968}{16807}{\frac{1}{\sqrt{1-2\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 3.21841, size = 161, normalized size = 1.26 \begin{align*} \frac{9145}{1210104} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{246915 \,{\left (2 \, x - 1\right )}^{4} + 2112495 \,{\left (2 \, x - 1\right )}^{3} + 6542333 \,{\left (2 \, x - 1\right )}^{2} + 17218306 \, x - 4624865}{28812 \,{\left (81 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 756 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 2646 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 4116 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2401 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57696, size = 348, normalized size = 2.72 \begin{align*} \frac{9145 \, \sqrt{21}{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (493830 \, x^{4} + 1124835 \, x^{3} + 843169 \, x^{2} + 218578 \, x + 6486\right )} \sqrt{-2 \, x + 1}}{1210104 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\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 = 1.78096, size = 147, normalized size = 1.15 \begin{align*} \frac{9145}{1210104} \, \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{968}{16807 \, \sqrt{-2 \, x + 1}} - \frac{787509 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 6005769 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 15060395 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 12452615 \, \sqrt{-2 \, x + 1}}{3226944 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]