Optimal. Leaf size=130 \[ -\frac{351 \sqrt{1-2 x}}{19208 (3 x+2)}-\frac{117 \sqrt{1-2 x}}{2744 (3 x+2)^2}-\frac{117 \sqrt{1-2 x}}{980 (3 x+2)^3}+\frac{341 \sqrt{1-2 x}}{8820 (3 x+2)^4}-\frac{\sqrt{1-2 x}}{315 (3 x+2)^5}-\frac{117 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604} \]
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Rubi [A] time = 0.0382197, antiderivative size = 130, 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} \[ -\frac{351 \sqrt{1-2 x}}{19208 (3 x+2)}-\frac{117 \sqrt{1-2 x}}{2744 (3 x+2)^2}-\frac{117 \sqrt{1-2 x}}{980 (3 x+2)^3}+\frac{341 \sqrt{1-2 x}}{8820 (3 x+2)^4}-\frac{\sqrt{1-2 x}}{315 (3 x+2)^5}-\frac{117 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604} \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{(3+5 x)^2}{\sqrt{1-2 x} (2+3 x)^6} \, dx &=-\frac{\sqrt{1-2 x}}{315 (2+3 x)^5}+\frac{1}{315} \int \frac{1409+2625 x}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=-\frac{\sqrt{1-2 x}}{315 (2+3 x)^5}+\frac{341 \sqrt{1-2 x}}{8820 (2+3 x)^4}+\frac{351}{140} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{\sqrt{1-2 x}}{315 (2+3 x)^5}+\frac{341 \sqrt{1-2 x}}{8820 (2+3 x)^4}-\frac{117 \sqrt{1-2 x}}{980 (2+3 x)^3}+\frac{117}{196} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{\sqrt{1-2 x}}{315 (2+3 x)^5}+\frac{341 \sqrt{1-2 x}}{8820 (2+3 x)^4}-\frac{117 \sqrt{1-2 x}}{980 (2+3 x)^3}-\frac{117 \sqrt{1-2 x}}{2744 (2+3 x)^2}+\frac{351 \int \frac{1}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{2744}\\ &=-\frac{\sqrt{1-2 x}}{315 (2+3 x)^5}+\frac{341 \sqrt{1-2 x}}{8820 (2+3 x)^4}-\frac{117 \sqrt{1-2 x}}{980 (2+3 x)^3}-\frac{117 \sqrt{1-2 x}}{2744 (2+3 x)^2}-\frac{351 \sqrt{1-2 x}}{19208 (2+3 x)}+\frac{351 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{19208}\\ &=-\frac{\sqrt{1-2 x}}{315 (2+3 x)^5}+\frac{341 \sqrt{1-2 x}}{8820 (2+3 x)^4}-\frac{117 \sqrt{1-2 x}}{980 (2+3 x)^3}-\frac{117 \sqrt{1-2 x}}{2744 (2+3 x)^2}-\frac{351 \sqrt{1-2 x}}{19208 (2+3 x)}-\frac{351 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{19208}\\ &=-\frac{\sqrt{1-2 x}}{315 (2+3 x)^5}+\frac{341 \sqrt{1-2 x}}{8820 (2+3 x)^4}-\frac{117 \sqrt{1-2 x}}{980 (2+3 x)^3}-\frac{117 \sqrt{1-2 x}}{2744 (2+3 x)^2}-\frac{351 \sqrt{1-2 x}}{19208 (2+3 x)}-\frac{117 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{9604}\\ \end{align*}
Mathematica [C] time = 0.0178667, size = 47, normalized size = 0.36 \[ \frac{\sqrt{1-2 x} \left (\frac{1029 (341 x+218)}{(3 x+2)^5}-50544 \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};\frac{3}{7}-\frac{6 x}{7}\right )\right )}{3025260} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 75, normalized size = 0.6 \begin{align*} -3888\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ( -{\frac{117\, \left ( 1-2\,x \right ) ^{9/2}}{153664}}+{\frac{13\, \left ( 1-2\,x \right ) ^{7/2}}{1568}}-{\frac{26\, \left ( 1-2\,x \right ) ^{5/2}}{735}}+{\frac{77587\, \left ( 1-2\,x \right ) ^{3/2}}{1143072}}-{\frac{5287\,\sqrt{1-2\,x}}{108864}} \right ) }-{\frac{117\,\sqrt{21}}{67228}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.12548, size = 173, normalized size = 1.33 \begin{align*} \frac{117}{134456} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{426465 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 4643730 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 19813248 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 38017630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 27201615 \, \sqrt{-2 \, x + 1}}{144060 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63692, size = 379, normalized size = 2.92 \begin{align*} \frac{1755 \, \sqrt{7} \sqrt{3}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \,{\left (426465 \, x^{4} + 1468935 \, x^{3} + 2110212 \, x^{2} + 1327058 \, x + 298748\right )} \sqrt{-2 \, x + 1}}{2016840 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 2.19724, size = 157, normalized size = 1.21 \begin{align*} \frac{117}{134456} \, \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{426465 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 4643730 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 19813248 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 38017630 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 27201615 \, \sqrt{-2 \, x + 1}}{4609920 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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