Optimal. Leaf size=173 \[ -\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {5 x+3}}+\frac {1403963 \sqrt {1-2 x}}{3136 (3 x+2) \sqrt {5 x+3}}+\frac {8063 \sqrt {1-2 x}}{224 (3 x+2)^2 \sqrt {5 x+3}}+\frac {33 \sqrt {1-2 x}}{8 (3 x+2)^3 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}+\frac {145708761 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 151, 152, 12, 93, 204} \begin {gather*} -\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {5 x+3}}+\frac {1403963 \sqrt {1-2 x}}{3136 (3 x+2) \sqrt {5 x+3}}+\frac {8063 \sqrt {1-2 x}}{224 (3 x+2)^2 \sqrt {5 x+3}}+\frac {33 \sqrt {1-2 x}}{8 (3 x+2)^3 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x}}{12 (3 x+2)^4 \sqrt {5 x+3}}+\frac {145708761 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {1}{12} \int \frac {\frac {341}{2}-264 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {1}{252} \int \frac {\frac {86163}{4}-31185 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {\int \frac {\frac {15937383}{8}-2539845 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx}{3528}\\ &=\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}+\frac {\int \frac {\frac {1880555061}{16}-\frac {442248345 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{24696}\\ &=-\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {100976171373}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{135828}\\ &=-\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}-\frac {145708761 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272}\\ &=-\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}-\frac {145708761 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136}\\ &=-\frac {63678595 \sqrt {1-2 x}}{9408 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{12 (2+3 x)^4 \sqrt {3+5 x}}+\frac {33 \sqrt {1-2 x}}{8 (2+3 x)^3 \sqrt {3+5 x}}+\frac {8063 \sqrt {1-2 x}}{224 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1403963 \sqrt {1-2 x}}{3136 (2+3 x) \sqrt {3+5 x}}+\frac {145708761 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 84, normalized size = 0.49 \begin {gather*} \frac {145708761 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {7 \sqrt {1-2 x} \left (1719322065 x^4+4546951839 x^3+4508028900 x^2+1985778980 x+327908240\right )}{(3 x+2)^4 \sqrt {5 x+3}}}{21952} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.38, size = 214, normalized size = 1.24 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (-343864413 \sqrt {5} (5 x+3)^4-420578883 \sqrt {5} (5 x+3)^3-186256251 \sqrt {5} (5 x+3)^2-33950549 \sqrt {5} (5 x+3)-1724800 \sqrt {5}\right )}{3136 \sqrt {5 x+3} (3 (5 x+3)+1)^4}+\frac {145708761 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{3136 \sqrt {7}}+\frac {145708761 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{3136 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.81, size = 131, normalized size = 0.76 \begin {gather*} \frac {145708761 \, \sqrt {7} {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\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 (1719322065 \, x^{4} + 4546951839 \, x^{3} + 4508028900 \, x^{2} + 1985778980 \, x + 327908240\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{43904 \, {\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.97, size = 438, normalized size = 2.53 \begin {gather*} -\frac {145708761}{439040} \, \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 {275}{2} \, \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 {11 \, {\left (13252949 \, \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 )}^{7} + 8830442040 \, \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 )}^{5} + 2086818820800 \, \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} + 170309125952000 \, \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 )}\right )}}{1568 \, {\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 )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 298, normalized size = 1.72 \begin {gather*} -\frac {\left (59012048205 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+192772690803 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+24070508910 \sqrt {-10 x^{2}-x +3}\, x^{4}+251784739008 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+63657325746 \sqrt {-10 x^{2}-x +3}\, x^{3}+164359482408 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+63112404600 \sqrt {-10 x^{2}-x +3}\, x^{2}+53620824048 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+27800905720 \sqrt {-10 x^{2}-x +3}\, x +6994020528 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4590715360 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{43904 \left (3 x +2\right )^{4} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.17, size = 296, normalized size = 1.71 \begin {gather*} -\frac {145708761}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {63678595 \, x}{4704 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {66486521}{9408 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{36 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {665}{72 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {7799}{96 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {457237}{448 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^5\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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