Optimal. Leaf size=130 \[ \frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{5 x+3}}-\frac{28705 \sqrt{1-2 x}}{17787 (5 x+3)^{3/2}}-\frac{58}{539 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac{4887 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
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Rubi [A] time = 0.0470482, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {103, 152, 12, 93, 204} \[ \frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{5 x+3}}-\frac{28705 \sqrt{1-2 x}}{17787 (5 x+3)^{3/2}}-\frac{58}{539 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}}-\frac{4887 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 103
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx &=\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{1}{7} \int \frac{\frac{61}{2}-90 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}-\frac{2}{539} \int \frac{-\frac{3653}{4}+870 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{4 \int \frac{-\frac{361687}{8}+\frac{86115 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{17787}\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{3+5 x}}-\frac{8 \int -\frac{19513791}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{195657}\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{3+5 x}}+\frac{4887}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{3+5 x}}+\frac{4887}{49} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{58}{539 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{28705 \sqrt{1-2 x}}{17787 (3+5 x)^{3/2}}+\frac{3}{7 \sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}}+\frac{2841815 \sqrt{1-2 x}}{195657 \sqrt{3+5 x}}-\frac{4887 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{49 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0607669, size = 102, normalized size = 0.78 \[ \frac{-7 \left (85254450 x^3+63467215 x^2-20145298 x-16461125\right )-19513791 \sqrt{7-14 x} \sqrt{5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1369599 \sqrt{1-2 x} (3 x+2) (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 257, normalized size = 2. \begin{align*}{\frac{1}{ \left ( 5478396+8217594\,x \right ) \left ( 2\,x-1 \right ) }\sqrt{1-2\,x} \left ( 2927068650\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+4000327155\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+663468894\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1193562300\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-995203341\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+888541010\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-351248238\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -282034172\,x\sqrt{-10\,{x}^{2}-x+3}-230455750\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57356, size = 378, normalized size = 2.91 \begin{align*} -\frac{19513791 \, \sqrt{7}{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \,{\left (85254450 \, x^{3} + 63467215 \, x^{2} - 20145298 \, x - 16461125\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2739198 \,{\left (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\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 [B] time = 3.08536, size = 458, normalized size = 3.52 \begin{align*} -\frac{25}{63888} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + \frac{4887}{6860} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{775}{1331} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{32 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{326095 \,{\left (2 \, x - 1\right )}} + \frac{1782 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{49 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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