Optimal. Leaf size=75 \[ -\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {5 x+3}}-14 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {94, 93, 204} \begin {gather*} -\frac {2 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {5 x+3}}-14 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}-7 \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3+5 x}}+49 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3+5 x}}+98 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {2 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3+5 x}}-14 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.80 \begin {gather*} \frac {2 \sqrt {1-2 x} (107 x+62)}{3 (5 x+3)^{3/2}}-14 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 70, normalized size = 0.93 \begin {gather*} -\frac {2 \sqrt {1-2 x} \left (\frac {1-2 x}{5 x+3}-21\right )}{3 \sqrt {5 x+3}}-14 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.70, size = 86, normalized size = 1.15 \begin {gather*} -\frac {21 \, \sqrt {7} {\left (25 \, x^{2} + 30 \, x + 9\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 ) - 2 \, {\left (107 \, x + 62\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.50, size = 190, normalized size = 2.53 \begin {gather*} \frac {7}{10} \, \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 {1}{1200} \, \sqrt {10} {\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 )}^{3} - \frac {840 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {3360 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 147, normalized size = 1.96 \begin {gather*} \frac {\left (525 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+630 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+214 \sqrt {-10 x^{2}-x +3}\, x +189 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+124 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{3 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 104, normalized size = 1.39 \begin {gather*} 7 \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {428 \, x}{15 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {8 \, x^{2}}{15 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1118}{75 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {488 \, x}{75 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {254}{75 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \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 )\,{\left (5\,x+3\right )}^{5/2}} \,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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