Optimal. Leaf size=151 \[ \frac {3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{28 (3 x+2)^4}+\frac {181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{168 (3 x+2)^3}+\frac {1991 (5 x+3)^{3/2} \sqrt {1-2 x}}{224 (3 x+2)^2}-\frac {21901 \sqrt {5 x+3} \sqrt {1-2 x}}{3136 (3 x+2)}-\frac {240911 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3136 \sqrt {7}} \]
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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} \begin {gather*} \frac {3 (5 x+3)^{3/2} (1-2 x)^{5/2}}{28 (3 x+2)^4}+\frac {181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{168 (3 x+2)^3}+\frac {1991 (5 x+3)^{3/2} \sqrt {1-2 x}}{224 (3 x+2)^2}-\frac {21901 \sqrt {5 x+3} \sqrt {1-2 x}}{3136 (3 x+2)}-\frac {240911 \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 93
Rule 94
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^5} \, dx &=\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181}{56} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^4} \, dx\\ &=\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991}{112} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {21901}{448} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {240911 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{6272}\\ &=-\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac {240911 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{3136}\\ &=-\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{3136 (2+3 x)}+\frac {3 (1-2 x)^{5/2} (3+5 x)^{3/2}}{28 (2+3 x)^4}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{168 (2+3 x)^3}+\frac {1991 \sqrt {1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}-\frac {240911 \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 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (1705089 x^3+3485960 x^2+2381420 x+541680\right )}{(3 x+2)^4}-722733 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{65856} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 122, normalized size = 0.81 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {543 (1-2 x)^3}{(5 x+3)^3}-\frac {17423 (1-2 x)^2}{(5 x+3)^2}-\frac {97559 (1-2 x)}{5 x+3}-186249\right )}{9408 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^4}-\frac {240911 \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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fricas [A] time = 0.99, size = 116, normalized size = 0.77 \begin {gather*} -\frac {722733 \, \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 (1705089 \, x^{3} + 3485960 \, x^{2} + 2381420 \, x + 541680\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{131712 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.60, size = 368, normalized size = 2.44 \begin {gather*} \frac {240911}{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 {1331 \, \sqrt {10} {\left (543 \, {\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} - 696920 \, {\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} - 156094400 \, {\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 {11919936000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {47679744000 \, \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 250, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (58541373 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+156110328 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+23871246 \sqrt {-10 x^{2}-x +3}\, x^{3}+156110328 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+48803440 \sqrt {-10 x^{2}-x +3}\, x^{2}+69382368 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+33339880 \sqrt {-10 x^{2}-x +3}\, x +11563728 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+7583520 \sqrt {-10 x^{2}-x +3}\right )}{131712 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 157, normalized size = 1.04 \begin {gather*} \frac {240911}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {9955}{2352} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {169 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{168 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {5973 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {73667 \, \sqrt {-10 \, x^{2} - x + 3}}{9408 \, {\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 {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^5} \,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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