Optimal. Leaf size=166 \[ \frac {618645 \sqrt {1-2 x}}{56 \sqrt {5 x+3}}-\frac {204595 \sqrt {1-2 x}}{168 (5 x+3)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (3 x+2) (5 x+3)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {4246733 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 151, 152, 12, 93, 204} \[ \frac {618645 \sqrt {1-2 x}}{56 \sqrt {5 x+3}}-\frac {204595 \sqrt {1-2 x}}{168 (5 x+3)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (3 x+2) (5 x+3)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {4246733 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {1}{9} \int \frac {\frac {345}{2}-268 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{126} \int \frac {\frac {87003}{4}-31605 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {1}{882} \int \frac {\frac {16024491}{8}-2569245 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {\frac {1808711289}{16}-\frac {425353005 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{14553}\\ &=-\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}+\frac {2 \int \frac {97118536977}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{160083}\\ &=-\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}+\frac {4246733}{112} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}+\frac {4246733}{56} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {204595 \sqrt {1-2 x}}{168 (3+5 x)^{3/2}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {301 \sqrt {1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {24469 \sqrt {1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac {618645 \sqrt {1-2 x}}{56 \sqrt {3+5 x}}-\frac {4246733 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 84, normalized size = 0.51 \[ \frac {\sqrt {1-2 x} \left (250551225 x^4+645909120 x^3+623901861 x^2+267610802 x+43006496\right )}{168 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {4246733 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 131, normalized size = 0.79 \[ -\frac {12740199 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (250551225 \, x^{4} + 645909120 \, x^{3} + 623901861 \, x^{2} + 267610802 \, x + 43006496\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2352 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.16, size = 434, normalized size = 2.61 \[ \frac {4246733}{7840} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {5}{48} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {3216 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {12864 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {99 \, {\left (21713 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 10391360 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 1283172800 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{28 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 298, normalized size = 1.80 \[ \frac {\left (8599634325 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+27518829840 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3507717150 \sqrt {-10 x^{2}-x +3}\, x^{4}+35201169837 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+9042727680 \sqrt {-10 x^{2}-x +3}\, x^{3}+22499191434 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8734626054 \sqrt {-10 x^{2}-x +3}\, x^{2}+7185472236 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3746551228 \sqrt {-10 x^{2}-x +3}\, x +917294328 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+602090944 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{2352 \left (3 x +2\right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 240, normalized size = 1.45 \[ \frac {4246733}{784} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {618645 \, x}{28 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1937773}{168 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {199895 \, x}{36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {343}{81 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {4655}{108 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {165739}{216 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {1943461}{648 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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