Optimal. Leaf size=101 \[ \frac {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}-\frac {55 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac {385 \sqrt {1-2 x}}{\sqrt {5 x+3}}-385 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}-\frac {55 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac {385 \sqrt {1-2 x}}{\sqrt {5 x+3}}-385 \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)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx &=\frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac {55}{2} \int \frac {(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}-\frac {385}{2} \int \frac {\sqrt {1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac {385 \sqrt {1-2 x}}{\sqrt {3+5 x}}+\frac {2695}{2} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac {385 \sqrt {1-2 x}}{\sqrt {3+5 x}}+2695 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac {(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac {385 \sqrt {1-2 x}}{\sqrt {3+5 x}}-385 \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 = 0.04, size = 92, normalized size = 0.91 \begin {gather*} \frac {\sqrt {1-2 x} \left (17667 x^2+21988 x+6823\right )-1155 \sqrt {7} \sqrt {5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3 (3 x+2) (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.23, size = 171, normalized size = 1.69 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (17667 (5 x+3)^2+3938 (5 x+3)-242\right )}{15 \sqrt {5} (5 x+3)^{3/2} (3 (5 x+3)+1)}-385 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )-385 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 101, normalized size = 1.00 \begin {gather*} -\frac {1155 \, \sqrt {7} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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 (17667 \, x^{2} + 21988 \, x + 6823\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.43, size = 309, normalized size = 3.06 \begin {gather*} \frac {77}{4} \, \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 {11}{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 {1680 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {6720 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {1078 \, \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 )}}{{\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 202, normalized size = 2.00 \begin {gather*} \frac {\left (86625 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+161700 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+35334 \sqrt {-10 x^{2}-x +3}\, x^{2}+100485 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+43976 \sqrt {-10 x^{2}-x +3}\, x +20790 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13646 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{6 \left (3 x +2\right ) \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 = 1.18, size = 138, normalized size = 1.37 \begin {gather*} \frac {385}{2} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {3926 \, x}{5 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {16 \, x^{2}}{45 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {30743}{75 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {133642 \, x}{675 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{81 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {217433}{2025 \, {\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 )}^{5/2}}{{\left (3\,x+2\right )}^2\,{\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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