Optimal. Leaf size=86 \[ -\frac {515 \sqrt {1-2 x}}{77 \sqrt {5 x+3}}+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}+\frac {321 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \]
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Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \begin {gather*} -\frac {515 \sqrt {1-2 x}}{77 \sqrt {5 x+3}}+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) \sqrt {5 x+3}}+\frac {321 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}} \end {gather*}
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)^{3/2}} \, dx &=\frac {3 \sqrt {1-2 x}}{7 (2+3 x) \sqrt {3+5 x}}+\frac {1}{7} \int \frac {\frac {67}{2}-30 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {515 \sqrt {1-2 x}}{77 \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) \sqrt {3+5 x}}-\frac {2}{77} \int \frac {3531}{4 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {515 \sqrt {1-2 x}}{77 \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) \sqrt {3+5 x}}-\frac {321}{14} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {515 \sqrt {1-2 x}}{77 \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) \sqrt {3+5 x}}-\frac {321}{7} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {515 \sqrt {1-2 x}}{77 \sqrt {3+5 x}}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) \sqrt {3+5 x}}+\frac {321 \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 = 69, normalized size = 0.80 \begin {gather*} \frac {321 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} (1545 x+997)}{77 (3 x+2) \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 90, normalized size = 1.05 \begin {gather*} \frac {321 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \left (\frac {350 (1-2 x)}{5 x+3}+3539\right )}{77 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.43, size = 86, normalized size = 1.00 \begin {gather*} \frac {3531 \, \sqrt {7} {\left (15 \, x^{2} + 19 \, x + 6\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 (1545 \, x + 997\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1078 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.47, size = 252, normalized size = 2.93 \begin {gather*} -\frac {321}{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}{22} \, \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 {198 \, \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 154, normalized size = 1.79 \begin {gather*} -\frac {\left (52965 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+67089 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+21630 \sqrt {-10 x^{2}-x +3}\, x +21186 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13958 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{1078 \left (3 x +2\right ) \sqrt {-10 x^{2}-x +3}\, \sqrt {5 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}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{2} \sqrt {-2 \, x + 1}}\,{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}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2}} \,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}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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