Optimal. Leaf size=137 \[ -\frac {57595 \sqrt {5 x+3}}{249018 \sqrt {1-2 x}}+\frac {51 \sqrt {5 x+3}}{28 (1-2 x)^{3/2} (3 x+2)}-\frac {1735 \sqrt {5 x+3}}{3234 (1-2 x)^{3/2}}+\frac {3 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}-\frac {5805 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {57595 \sqrt {5 x+3}}{249018 \sqrt {1-2 x}}+\frac {51 \sqrt {5 x+3}}{28 (1-2 x)^{3/2} (3 x+2)}-\frac {1735 \sqrt {5 x+3}}{3234 (1-2 x)^{3/2}}+\frac {3 \sqrt {5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}-\frac {5805 \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 \sqrt {3+5 x}} \, dx &=\frac {3 \sqrt {3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {1}{14} \int \frac {-\frac {1}{2}-90 x}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {3 \sqrt {3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {51 \sqrt {3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}+\frac {1}{98} \int \frac {-\frac {5005}{4}-3570 x}{(1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {1735 \sqrt {3+5 x}}{3234 (1-2 x)^{3/2}}+\frac {3 \sqrt {3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {51 \sqrt {3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}-\frac {\int \frac {\frac {38815}{8}+\frac {182175 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{11319}\\ &=-\frac {1735 \sqrt {3+5 x}}{3234 (1-2 x)^{3/2}}-\frac {57595 \sqrt {3+5 x}}{249018 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {51 \sqrt {3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}+\frac {2 \int \frac {14750505}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{871563}\\ &=-\frac {1735 \sqrt {3+5 x}}{3234 (1-2 x)^{3/2}}-\frac {57595 \sqrt {3+5 x}}{249018 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {51 \sqrt {3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}+\frac {5805 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {1735 \sqrt {3+5 x}}{3234 (1-2 x)^{3/2}}-\frac {57595 \sqrt {3+5 x}}{249018 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {51 \sqrt {3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}+\frac {5805 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {1735 \sqrt {3+5 x}}{3234 (1-2 x)^{3/2}}-\frac {57595 \sqrt {3+5 x}}{249018 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac {51 \sqrt {3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}-\frac {5805 \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 = 0.06, size = 95, normalized size = 0.69 \begin {gather*} -\frac {-7 \sqrt {5 x+3} \left (2073420 x^3-676860 x^2-945629 x+391476\right )-2107215 \sqrt {7-14 x} (2 x-1) (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{3486252 (1-2 x)^{3/2} (3 x+2)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 122, normalized size = 0.89 \begin {gather*} \frac {(5 x+3)^{3/2} \left (\frac {3037785 (1-2 x)^3}{(5 x+3)^3}+\frac {14174825 (1-2 x)^2}{(5 x+3)^2}+\frac {181888 (1-2 x)}{5 x+3}+6272\right )}{498036 (1-2 x)^{3/2} \left (\frac {1-2 x}{5 x+3}+7\right )^2}-\frac {5805 \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 = 1.18, size = 116, normalized size = 0.85 \begin {gather*} -\frac {2107215 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 (2073420 \, x^{3} - 676860 \, x^{2} - 945629 \, x + 391476\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6972504 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.58, size = 291, normalized size = 2.12 \begin {gather*} \frac {1161}{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 {32 \, {\left (367 \, \sqrt {5} {\left (5 \, x + 3\right )} - 2211 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{21789075 \, {\left (2 \, x - 1\right )}^{2}} + \frac {297 \, \sqrt {10} {\left (197 \, {\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 {36680 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {146720 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\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 = 257, normalized size = 1.88 \begin {gather*} \frac {\left (75859740 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+25286580 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+29027880 \sqrt {-10 x^{2}-x +3}\, x^{3}-48465945 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-9476040 \sqrt {-10 x^{2}-x +3}\, x^{2}-8428860 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-13238806 \sqrt {-10 x^{2}-x +3}\, x +8428860 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+5480664 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{6972504 \left (3 x +2\right )^{2} \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \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\,\sqrt {5\,x+3}} \,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} \sqrt {5 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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