Optimal. Leaf size=149 \[ -\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^2}-\frac {6401 \sqrt {1-2 x} \sqrt {5 x+3}}{10584 (3 x+2)}-\frac {50}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {250433 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{31752 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \begin {gather*} -\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^3}-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^2}-\frac {6401 \sqrt {1-2 x} \sqrt {5 x+3}}{10584 (3 x+2)}-\frac {50}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {250433 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{31752 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 97
Rule 149
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {1}{378} \int \frac {\left (\frac {801}{4}-2100 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {\int \frac {-\frac {141567}{8}-73500 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{7938}\\ &=-\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}-\frac {250}{81} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {250433 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{63504}\\ &=-\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac {250433 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{31752}-\frac {1}{81} \left (100 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{10584 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^3}-\frac {50}{81} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {250433 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{31752 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 134, normalized size = 0.90 \begin {gather*} \frac {-21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (124179 x^2+159174 x+51056\right )-250433 \sqrt {14 x-7} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+137200 \sqrt {10-20 x} (3 x+2)^3 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{222264 \sqrt {2 x-1} (3 x+2)^3} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.23, size = 142, normalized size = 0.95 \begin {gather*} -\frac {11 \sqrt {1-2 x} \left (\frac {6401 (1-2 x)^2}{(5 x+3)^2}+\frac {116872 (1-2 x)}{5 x+3}+646751\right )}{10584 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^3}+\frac {50}{81} \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {250433 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{31752 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.61, size = 156, normalized size = 1.05 \begin {gather*} -\frac {250433 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 ) - 137200 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (124179 \, x^{2} + 159174 \, x + 51056\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{444528 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.45, size = 377, normalized size = 2.53 \begin {gather*} \frac {250433}{4445280} \, \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 {25}{81} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (6401 \, {\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} + 4674880 \, {\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 {1034801600 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {4139206400 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{5292 \, {\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 )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 253, normalized size = 1.70 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-3704400 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+6761691 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-7408800 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+13523382 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-5215518 \sqrt {-10 x^{2}-x +3}\, x^{2}-4939200 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+9015588 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-6685308 \sqrt {-10 x^{2}-x +3}\, x -1097600 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2003464 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2144352 \sqrt {-10 x^{2}-x +3}\right )}{444528 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 132, normalized size = 0.89 \begin {gather*} -\frac {25}{81} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {250433}{444528} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {515}{2646} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{63 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {103 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{588 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {5989 \, \sqrt {-10 \, x^{2} - x + 3}}{10584 \, {\left (3 \, x + 2\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 {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^4} \,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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