Optimal. Leaf size=108 \[ \frac {84235 \sqrt {1-2 x}}{2541 \sqrt {5 x+3}}-\frac {845 \sqrt {1-2 x}}{231 (5 x+3)^{3/2}}+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac {1593 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {103, 152, 12, 93, 204} \[ \frac {84235 \sqrt {1-2 x}}{2541 \sqrt {5 x+3}}-\frac {845 \sqrt {1-2 x}}{231 (5 x+3)^{3/2}}+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^{3/2}}-\frac {1593 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 103
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx &=\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {1}{7} \int \frac {\frac {97}{2}-60 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {845 \sqrt {1-2 x}}{231 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^{3/2}}-\frac {2}{231} \int \frac {\frac {10763}{4}-2535 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {845 \sqrt {1-2 x}}{231 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {84235 \sqrt {1-2 x}}{2541 \sqrt {3+5 x}}+\frac {4 \int \frac {578259}{8 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2541}\\ &=-\frac {845 \sqrt {1-2 x}}{231 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {84235 \sqrt {1-2 x}}{2541 \sqrt {3+5 x}}+\frac {1593}{14} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {845 \sqrt {1-2 x}}{231 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {84235 \sqrt {1-2 x}}{2541 \sqrt {3+5 x}}+\frac {1593}{7} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {845 \sqrt {1-2 x}}{231 (3+5 x)^{3/2}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^{3/2}}+\frac {84235 \sqrt {1-2 x}}{2541 \sqrt {3+5 x}}-\frac {1593 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 93, normalized size = 0.86 \[ \frac {7 \sqrt {1-2 x} \left (1263525 x^2+1572580 x+487909\right )-578259 \sqrt {7} \sqrt {5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{17787 (3 x+2) (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 101, normalized size = 0.94 \[ -\frac {578259 \, \sqrt {7} {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1263525 \, x^{2} + 1572580 \, x + 487909\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{35574 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.61, size = 309, normalized size = 2.86 \[ \frac {1593}{980} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {5}{5808} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {1536 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {6144 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {594 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{7 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 202, normalized size = 1.87 \[ \frac {\left (43369425 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+80956260 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+17689350 \sqrt {-10 x^{2}-x +3}\, x^{2}+50308533 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+22016120 \sqrt {-10 x^{2}-x +3}\, x +10408662 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6830726 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{35574 \left (3 x +2\right ) \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{2} \sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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