Optimal. Leaf size=160 \[ -\frac{1676975 \sqrt{1-2 x}}{7546 (5 x+3)}+\frac{7585 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)}+\frac{145 \sqrt{1-2 x}}{98 (3 x+2)^2 (5 x+3)}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)}-\frac{1051695}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{32750}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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Rubi [A] time = 0.0609598, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{1676975 \sqrt{1-2 x}}{7546 (5 x+3)}+\frac{7585 \sqrt{1-2 x}}{343 (3 x+2) (5 x+3)}+\frac{145 \sqrt{1-2 x}}{98 (3 x+2)^2 (5 x+3)}+\frac{\sqrt{1-2 x}}{7 (3 x+2)^3 (5 x+3)}-\frac{1051695}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{32750}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
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)^2} \, dx &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac{1}{21} \int \frac{75-105 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac{145 \sqrt{1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac{1}{294} \int \frac{7920-10875 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac{145 \sqrt{1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac{7585 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)}+\frac{\int \frac{596595-682650 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{2058}\\ &=-\frac{1676975 \sqrt{1-2 x}}{7546 (3+5 x)}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac{145 \sqrt{1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac{7585 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)}-\frac{\int \frac{24644085-15092775 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{22638}\\ &=-\frac{1676975 \sqrt{1-2 x}}{7546 (3+5 x)}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac{145 \sqrt{1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac{7585 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)}+\frac{3155085}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{81875}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{1676975 \sqrt{1-2 x}}{7546 (3+5 x)}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac{145 \sqrt{1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac{7585 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)}-\frac{3155085}{686} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{81875}{11} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{1676975 \sqrt{1-2 x}}{7546 (3+5 x)}+\frac{\sqrt{1-2 x}}{7 (2+3 x)^3 (3+5 x)}+\frac{145 \sqrt{1-2 x}}{98 (2+3 x)^2 (3+5 x)}+\frac{7585 \sqrt{1-2 x}}{343 (2+3 x) (3+5 x)}-\frac{1051695}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{32750}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.092276, size = 101, normalized size = 0.63 \[ -\frac{\sqrt{1-2 x} \left (45278325 x^3+89054820 x^2+58335165 x+12724912\right )}{7546 (3 x+2)^3 (5 x+3)}-\frac{1051695}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{32750}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 91, normalized size = 0.6 \begin{align*} 972\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{7565\, \left ( 1-2\,x \right ) ^{5/2}}{4116}}-{\frac{11455\, \left ( 1-2\,x \right ) ^{3/2}}{1323}}+{\frac{7711\,\sqrt{1-2\,x}}{756}} \right ) }-{\frac{1051695\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{250}{11}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{32750\,\sqrt{55}}{121}{\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 = 2.24429, size = 197, normalized size = 1.23 \begin{align*} -\frac{16375}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1051695}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{45278325 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 313944615 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 725394915 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 558527921 \, \sqrt{-2 \, x + 1}}{3773 \,{\left (135 \,{\left (2 \, x - 1\right )}^{4} + 1242 \,{\left (2 \, x - 1\right )}^{3} + 4284 \,{\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63037, size = 522, normalized size = 3.26 \begin{align*} \frac{78632750 \, \sqrt{11} \sqrt{5}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 127255095 \, \sqrt{7} \sqrt{3}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (45278325 \, x^{3} + 89054820 \, x^{2} + 58335165 \, x + 12724912\right )} \sqrt{-2 \, x + 1}}{581042 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 = 2.47164, size = 188, normalized size = 1.18 \begin{align*} -\frac{16375}{121} \, \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{1051695}{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{625 \, \sqrt{-2 \, x + 1}}{11 \,{\left (5 \, x + 3\right )}} - \frac{9 \,{\left (68085 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 320740 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 377839 \, \sqrt{-2 \, x + 1}\right )}}{2744 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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