Optimal. Leaf size=151 \[ \frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{24 (3 x+2)^3}-\frac {55 \sqrt {1-2 x} (5 x+3)^{3/2}}{672 (3 x+2)^2}-\frac {605 \sqrt {1-2 x} \sqrt {5 x+3}}{3136 (3 x+2)}-\frac {6655 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{24 (3 x+2)^3}-\frac {55 \sqrt {1-2 x} (5 x+3)^{3/2}}{672 (3 x+2)^2}-\frac {605 \sqrt {1-2 x} \sqrt {5 x+3}}{3136 (3 x+2)}-\frac {6655 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx &=\frac {3 \sqrt {1-2 x} (3+5 x)^{7/2}}{28 (2+3 x)^4}+\frac {7}{8} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{24 (2+3 x)^3}+\frac {3 \sqrt {1-2 x} (3+5 x)^{7/2}}{28 (2+3 x)^4}+\frac {55}{48} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {55 \sqrt {1-2 x} (3+5 x)^{3/2}}{672 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{24 (2+3 x)^3}+\frac {3 \sqrt {1-2 x} (3+5 x)^{7/2}}{28 (2+3 x)^4}+\frac {605}{448} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {605 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}-\frac {55 \sqrt {1-2 x} (3+5 x)^{3/2}}{672 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{24 (2+3 x)^3}+\frac {3 \sqrt {1-2 x} (3+5 x)^{7/2}}{28 (2+3 x)^4}+\frac {6655 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272}\\ &=-\frac {605 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}-\frac {55 \sqrt {1-2 x} (3+5 x)^{3/2}}{672 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{24 (2+3 x)^3}+\frac {3 \sqrt {1-2 x} (3+5 x)^{7/2}}{28 (2+3 x)^4}+\frac {6655 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136}\\ &=-\frac {605 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}-\frac {55 \sqrt {1-2 x} (3+5 x)^{3/2}}{672 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{24 (2+3 x)^3}+\frac {3 \sqrt {1-2 x} (3+5 x)^{7/2}}{28 (2+3 x)^4}-\frac {6655 \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 = 0.06, size = 79, normalized size = 0.52 \[ \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (12945 x^3+6920 x^2-6484 x-3600\right )}{(3 x+2)^4}-19965 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{65856} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 116, normalized size = 0.77 \[ -\frac {19965 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (12945 \, x^{3} + 6920 \, x^{2} - 6484 \, x - 3600\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{131712 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.75, size = 368, normalized size = 2.44 \[ \frac {1331}{87808} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {6655 \, \sqrt {10} {\left (3 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 3080 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1144640 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {2956800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {11827200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4704 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 250, normalized size = 1.66 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (1617165 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4312440 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+181230 \sqrt {-10 x^{2}-x +3}\, x^{3}+4312440 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+96880 \sqrt {-10 x^{2}-x +3}\, x^{2}+1916640 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-90776 \sqrt {-10 x^{2}-x +3}\, x +319440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-50400 \sqrt {-10 x^{2}-x +3}\right )}{131712 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 143, normalized size = 0.95 \[ \frac {6655}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{252 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {83 \, \sqrt {-10 \, x^{2} - x + 3}}{1512 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {1355 \, \sqrt {-10 \, x^{2} - x + 3}}{6048 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {4315 \, \sqrt {-10 \, x^{2} - x + 3}}{84672 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^5} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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