Optimal. Leaf size=166 \[ -\frac{15987390}{456533 \sqrt{1-2 x}}+\frac{1176400}{5929 \sqrt{1-2 x} (5 x+3)}-\frac{35825}{1078 \sqrt{1-2 x} (5 x+3)^2}+\frac{435}{98 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^2}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0730113, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {103, 151, 152, 156, 63, 206} \[ -\frac{15987390}{456533 \sqrt{1-2 x}}+\frac{1176400}{5929 \sqrt{1-2 x} (5 x+3)}-\frac{35825}{1078 \sqrt{1-2 x} (5 x+3)^2}+\frac{435}{98 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^2}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 103
Rule 151
Rule 152
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{1}{14} \int \frac{55-135 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1}{98} \int \frac{5195-15225 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}-\frac{\int \frac{296270-1074750 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx}{2156}\\ &=-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}+\frac{\int \frac{5187810-42350400 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{23716}\\ &=-\frac{15987390}{456533 \sqrt{1-2 x}}-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}-\frac{\int \frac{-391579365+239810850 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{913066}\\ &=-\frac{15987390}{456533 \sqrt{1-2 x}}-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}-\frac{1242945}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{7805625 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac{15987390}{456533 \sqrt{1-2 x}}-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}+\frac{1242945}{686} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{7805625 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2662}\\ &=-\frac{15987390}{456533 \sqrt{1-2 x}}-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}
Mathematica [C] time = 0.0547425, size = 83, normalized size = 0.5 \[ \frac{-1102906530 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+1070931750 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+\frac{77 \left (105876000 x^3+201146925 x^2+127185805 x+26765111\right )}{(3 x+2)^2 (5 x+3)^2}}{913066 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 103, normalized size = 0.6 \begin{align*} -{\frac{8748}{343\, \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{217}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{511}{36}\sqrt{1-2\,x}} \right ) }+{\frac{414315\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{64}{456533}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{312500}{1331\, \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{191}{100} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2079}{500}\sqrt{1-2\,x}} \right ) }-{\frac{1561125\,\sqrt{55}}{14641}{\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 = 1.76648, size = 209, normalized size = 1.26 \begin{align*} \frac{1561125}{29282} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{414315}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1798581375 \,{\left (2 \, x - 1\right )}^{4} + 12230911800 \,{\left (2 \, x - 1\right )}^{3} + 27711289905 \,{\left (2 \, x - 1\right )}^{2} + 41836111240 \, x - 20918245348\right )}}{456533 \,{\left (225 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 2040 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 6934 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 10472 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 5929 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6253, size = 599, normalized size = 3.61 \begin{align*} \frac{3748261125 \, \sqrt{11} \sqrt{5}{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 6065985915 \, \sqrt{7} \sqrt{3}{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (7194325500 \, x^{4} + 10073172600 \, x^{3} + 1810042755 \, x^{2} - 2503057145 \, x - 909821467\right )} \sqrt{-2 \, x + 1}}{70306082 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\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 = 2.38461, size = 212, normalized size = 1.28 \begin{align*} \frac{1561125}{29282} \, \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{414315}{4802} \, \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{64}{456533 \, \sqrt{-2 \, x + 1}} + \frac{2 \,{\left (256941225 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 1747282440 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 3958787399 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2988341532 \, \sqrt{-2 \, x + 1}\right )}}{65219 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]