Optimal. Leaf size=86 \[ -\frac {2}{15} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {103}{45} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {14}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {102, 157, 54, 216, 93, 204} \[ -\frac {2}{15} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {103}{45} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {14}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 102
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x) \sqrt {3+5 x}} \, dx &=-\frac {2}{15} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{15} \int \frac {13-103 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {2}{15} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {103}{45} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {49}{9} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {2}{15} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {98}{9} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {206 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{45 \sqrt {5}}\\ &=-\frac {2}{15} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {103}{45} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {14}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 103, normalized size = 1.20 \[ \frac {-30 \sqrt {-(2 x-1)^2} \sqrt {5 x+3}-350 \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+103 \sqrt {10-20 x} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{225 \sqrt {2 x-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.67, size = 103, normalized size = 1.20 \[ \frac {103}{450} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {7}{9} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {2}{15} \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.22, size = 160, normalized size = 1.86 \[ \frac {7}{90} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {103}{450} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {2}{75} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 83, normalized size = 0.97 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (103 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-350 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+60 \sqrt {-10 x^{2}-x +3}\right )}{450 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 54, normalized size = 0.63 \[ -\frac {103}{450} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {7}{9} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2}{15} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{\left (3\,x+2\right )\,\sqrt {5\,x+3}} \,d x \]
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 \[ \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right ) \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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