Optimal. Leaf size=137 \[ \frac {20743985 \sqrt {1-2 x}}{71148 \sqrt {5 x+3}}-\frac {207895 \sqrt {1-2 x}}{6468 (5 x+3)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (3 x+2) (5 x+3)^{3/2}}+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {392283 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \]
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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} \[ \frac {20743985 \sqrt {1-2 x}}{71148 \sqrt {5 x+3}}-\frac {207895 \sqrt {1-2 x}}{6468 (5 x+3)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (3 x+2) (5 x+3)^{3/2}}+\frac {3 \sqrt {1-2 x}}{14 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {392283 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \]
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}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {1}{14} \int \frac {\frac {131}{2}-90 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac {1}{98} \int \frac {\frac {23507}{4}-7530 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {207895 \sqrt {1-2 x}}{6468 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {\frac {2651953}{8}-\frac {623685 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{1617}\\ &=-\frac {207895 \sqrt {1-2 x}}{6468 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac {20743985 \sqrt {1-2 x}}{71148 \sqrt {3+5 x}}+\frac {2 \int \frac {142398729}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{17787}\\ &=-\frac {207895 \sqrt {1-2 x}}{6468 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac {20743985 \sqrt {1-2 x}}{71148 \sqrt {3+5 x}}+\frac {392283}{392} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {207895 \sqrt {1-2 x}}{6468 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac {20743985 \sqrt {1-2 x}}{71148 \sqrt {3+5 x}}+\frac {392283}{196} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {207895 \sqrt {1-2 x}}{6468 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {753 \sqrt {1-2 x}}{196 (2+3 x) (3+5 x)^{3/2}}+\frac {20743985 \sqrt {1-2 x}}{71148 \sqrt {3+5 x}}-\frac {392283 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 79, normalized size = 0.58 \[ \frac {\sqrt {1-2 x} \left (933479325 x^3+1784145090 x^2+1135041037 x+240342364\right )}{71148 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {392283 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 116, normalized size = 0.85 \[ -\frac {142398729 \, \sqrt {7} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (933479325 \, x^{3} + 1784145090 \, x^{2} + 1135041037 \, x + 240342364\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{996072 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.84, size = 373, normalized size = 2.72 \[ \frac {392283}{27440} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{5808} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {2328 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {9312 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {297 \, {\left (461 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 110600 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 250, normalized size = 1.82 \[ \frac {\left (32039714025 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+81167275530 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13068710550 \sqrt {-10 x^{2}-x +3}\, x^{3}+77037712389 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+24978031260 \sqrt {-10 x^{2}-x +3}\, x^{2}+32466910212 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+15890574518 \sqrt {-10 x^{2}-x +3}\, x +5126354244 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3364793096 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{996072 \left (3 x +2\right )^{2} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{3} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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