Optimal. Leaf size=180 \[ \frac{137735775 \sqrt{1-2 x}}{83006 (5 x+3)}-\frac{2076675 \sqrt{1-2 x}}{7546 (5 x+3)^2}+\frac{12555 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)^2}+\frac{90 \sqrt{1-2 x}}{49 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0713214, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {103, 151, 156, 63, 206} \[ \frac{137735775 \sqrt{1-2 x}}{83006 (5 x+3)}-\frac{2076675 \sqrt{1-2 x}}{7546 (5 x+3)^2}+\frac{12555 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)^2}+\frac{90 \sqrt{1-2 x}}{49 (3 x+2)^2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 156
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^3} \, dx &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{1}{21} \int \frac{90-135 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{1}{294} \int \frac{12510-18900 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{\int \frac{1362060-1883250 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx}{2058}\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}-\frac{\int \frac{97998660-112140450 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{45276}\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{137735775 \sqrt{1-2 x}}{83006 (3+5 x)}+\frac{\int \frac{4048216380-2479243950 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{498036}\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{137735775 \sqrt{1-2 x}}{83006 (3+5 x)}-\frac{11779020}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{13449375}{242} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{137735775 \sqrt{1-2 x}}{83006 (3+5 x)}+\frac{11779020}{343} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{13449375}{242} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{2076675 \sqrt{1-2 x}}{7546 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)^2}+\frac{90 \sqrt{1-2 x}}{49 (2+3 x)^2 (3+5 x)^2}+\frac{12555 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)^2}+\frac{137735775 \sqrt{1-2 x}}{83006 (3+5 x)}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0988242, size = 106, normalized size = 0.59 \[ \frac{\sqrt{1-2 x} \left (18594329625 x^4+47728484550 x^3+45899434890 x^2+19599448500 x+3135381218\right )}{83006 (3 x+2)^3 (5 x+3)^2}+\frac{7852680}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2689875}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 103, normalized size = 0.6 \begin{align*} -2916\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{3755\, \left ( 1-2\,x \right ) ^{5/2}}{1029}}-{\frac{22690\, \left ( 1-2\,x \right ) ^{3/2}}{1323}}+{\frac{3809\,\sqrt{1-2\,x}}{189}} \right ) }+{\frac{7852680\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+312500\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{261\, \left ( 1-2\,x \right ) ^{3/2}}{12100}}+{\frac{259\,\sqrt{1-2\,x}}{5500}} \right ) }-{\frac{2689875\,\sqrt{55}}{1331}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.8111, size = 221, normalized size = 1.23 \begin{align*} \frac{2689875}{2662} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3926340}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{18594329625 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 169834287600 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 581534624610 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 884739292920 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 504610725773 \, \sqrt{-2 \, x + 1}}{41503 \,{\left (675 \,{\left (2 \, x - 1\right )}^{5} + 7695 \,{\left (2 \, x - 1\right )}^{4} + 35082 \,{\left (2 \, x - 1\right )}^{3} + 79954 \,{\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86304, size = 621, normalized size = 3.45 \begin{align*} \frac{6458389875 \, \sqrt{11} \sqrt{5}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 10451917080 \, \sqrt{7} \sqrt{3}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (18594329625 \, x^{4} + 47728484550 \, x^{3} + 45899434890 \, x^{2} + 19599448500 \, x + 3135381218\right )} \sqrt{-2 \, x + 1}}{6391462 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 = 1.42145, size = 204, normalized size = 1.13 \begin{align*} \frac{2689875}{2662} \, \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{3926340}{2401} \, \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{625 \,{\left (1305 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2849 \, \sqrt{-2 \, x + 1}\right )}}{484 \,{\left (5 \, x + 3\right )}^{2}} + \frac{27 \,{\left (33795 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 158830 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 186641 \, \sqrt{-2 \, x + 1}\right )}}{686 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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