Optimal. Leaf size=181 \[ \frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {5 x+3}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (5 x+3)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac {3715}{3234 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}-\frac {538245 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]
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Rubi [A] time = 0.07, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \begin {gather*} \frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {5 x+3}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (5 x+3)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac {3715}{3234 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}-\frac {538245 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{14} \int \frac {\frac {59}{2}-150 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {1}{98} \int \frac {\frac {5075}{4}-15540 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {1484385}{8}+\frac {1170225 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx}{11319}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {2 \int \frac {\frac {83479305}{16}-\frac {12840975 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{871563}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac {4 \int \frac {\frac {9239846595}{32}-\frac {2176739775 x}{8}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{28761579}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}+\frac {8 \int \frac {496468037835}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{316377369}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}+\frac {538245 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}+\frac {538245 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {40765}{83006 \sqrt {1-2 x} (3+5 x)^{3/2}}-\frac {34551425 \sqrt {1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac {3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac {111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac {3443814775 \sqrt {1-2 x}}{60262356 \sqrt {3+5 x}}-\frac {538245 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 89, normalized size = 0.49 \begin {gather*} \frac {619886659500 x^5+564878517900 x^4-276089438305 x^3-297937101390 x^2+28838387211 x+39900939556}{60262356 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}-\frac {538245 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 154, normalized size = 0.85 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {42875000 (1-2 x)^5}{(5 x+3)^5}+\frac {1792175000 (1-2 x)^4}{(5 x+3)^4}+\frac {39393489865 (1-2 x)^3}{(5 x+3)^3}+\frac {165483078425 (1-2 x)^2}{(5 x+3)^2}+\frac {1103872 (1-2 x)}{5 x+3}+25088\right )}{60262356 (1-2 x)^{3/2} \left (\frac {1-2 x}{5 x+3}+7\right )^2}-\frac {538245 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.53, size = 146, normalized size = 0.81 \begin {gather*} -\frac {23641335135 \, \sqrt {7} {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\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 (619886659500 \, x^{5} + 564878517900 \, x^{4} - 276089438305 \, x^{3} - 297937101390 \, x^{2} + 28838387211 \, x + 39900939556\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{843672984 \, {\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.73, size = 412, normalized size = 2.28 \begin {gather*} \frac {107649}{38416} \, \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 {625}{702768} \, \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 {2232 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {8928 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {128 \, {\left (577 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3366 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{2636478075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {8019 \, {\left (159 \, \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 )}^{3} + 38360 \, \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 )}\right )}}{4802 \, {\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 )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 353, normalized size = 1.95 \begin {gather*} \frac {\sqrt {-2 x +1}\, \left (21277201621500 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+32625042486300 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8678413233000 \sqrt {-10 x^{2}-x +3}\, x^{5}+2576905529715 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+7908299250600 \sqrt {-10 x^{2}-x +3}\, x^{4}-16123390562070 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-3865252136270 \sqrt {-10 x^{2}-x +3}\, x^{3}-5366583075645 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-4171119419460 \sqrt {-10 x^{2}-x +3}\, x^{2}+1985872151340 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+403737420954 \sqrt {-10 x^{2}-x +3}\, x +851088064860 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+558613153784 \sqrt {-10 x^{2}-x +3}\right )}{843672984 \left (3 x +2\right )^{2} \left (2 x -1\right )^{2} \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 = 1.38, size = 172, normalized size = 0.95 \begin {gather*} \frac {538245}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {3443814775 \, x}{30131178 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3595841045}{60262356 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1022125 \, x}{35574 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {3}{14 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {111}{28 \, {\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 {1103855}{71148 \, {\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 {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,{\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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