Optimal. Leaf size=137 \[ -\frac {3125575 \sqrt {1-2 x}}{166012 \sqrt {5 x+3}}-\frac {6205}{7546 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {555}{196 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}+\frac {177255 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]
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Rubi [A] time = 0.05, 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} \[ -\frac {3125575 \sqrt {1-2 x}}{166012 \sqrt {5 x+3}}-\frac {6205}{7546 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {555}{196 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}+\frac {3}{14 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}+\frac {177255 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \]
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)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {1}{14} \int \frac {\frac {65}{2}-90 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {555}{196 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {1}{98} \int \frac {\frac {4895}{4}-5550 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {6205}{7546 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {555}{196 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {-\frac {401735}{8}+\frac {93075 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{3773}\\ &=-\frac {6205}{7546 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {3125575 \sqrt {1-2 x}}{166012 \sqrt {3+5 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {555}{196 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {2 \int -\frac {21447855}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{41503}\\ &=-\frac {6205}{7546 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {3125575 \sqrt {1-2 x}}{166012 \sqrt {3+5 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {555}{196 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {177255 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=-\frac {6205}{7546 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {3125575 \sqrt {1-2 x}}{166012 \sqrt {3+5 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {555}{196 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}-\frac {177255 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=-\frac {6205}{7546 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {3125575 \sqrt {1-2 x}}{166012 \sqrt {3+5 x}}+\frac {3}{14 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}}+\frac {555}{196 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}}+\frac {177255 \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 = 0.08, size = 79, normalized size = 0.58 \[ \frac {\frac {7 \left (56260350 x^3+45655035 x^2-12730165 x-12072596\right )}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}+21447855 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1162084} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 116, normalized size = 0.85 \[ \frac {21447855 \, \sqrt {7} {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (56260350 \, x^{3} + 45655035 \, x^{2} - 12730165 \, x - 12072596\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2324168 \, {\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.34, size = 342, normalized size = 2.50 \[ -\frac {35451}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {125}{242} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{207515 \, {\left (2 \, x - 1\right )}} - \frac {297 \, {\left (47 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 10520 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
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 \[ -\frac {\sqrt {-2 x +1}\, \left (1930306950 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2766773295 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+787644900 \sqrt {-10 x^{2}-x +3}\, x^{3}+536196375 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+639170490 \sqrt {-10 x^{2}-x +3}\, x^{2}-686331360 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-178222310 \sqrt {-10 x^{2}-x +3}\, x -257374260 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-169016344 \sqrt {-10 x^{2}-x +3}\right )}{2324168 \left (3 x +2\right )^{2} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 143, normalized size = 1.04 \[ -\frac {177255}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {3125575 \, x}{83006 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3262085}{166012 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3}{14 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {555}{196 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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