Optimal. Leaf size=106 \[ \frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {107}{180} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {4091 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{540 \sqrt {10}}+\frac {14}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {101, 154, 157, 54, 216, 93, 204} \begin {gather*} \frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {107}{180} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {4091 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{540 \sqrt {10}}+\frac {14}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 101
Rule 154
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx &=\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{6} \int \frac {\left (-31-\frac {107 x}{2}\right ) \sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{90} \int \frac {-\frac {1037}{2}-\frac {4091 x}{4}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {49}{27} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {4091 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1080}\\ &=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {98}{27} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {4091 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{540 \sqrt {5}}\\ &=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {4091 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{540 \sqrt {10}}+\frac {14}{27} \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 = 0.12, size = 108, normalized size = 1.02 \begin {gather*} \frac {30 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} (137-60 x)+2800 \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-4091 \sqrt {10-20 x} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{5400 \sqrt {2 x-1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.18, size = 127, normalized size = 1.20 \begin {gather*} \frac {11 \sqrt {1-2 x} \left (\frac {865 (1-2 x)}{5 x+3}+214\right )}{180 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {4091 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{540 \sqrt {10}}+\frac {14}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.44, size = 102, normalized size = 0.96 \begin {gather*} -\frac {1}{180} \, {\left (60 \, x - 137\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} + \frac {7}{27} \, \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 {4091}{10800} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.24, size = 173, normalized size = 1.63 \begin {gather*} -\frac {7}{270} \, \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 {1}{900} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 173 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {4091}{10800} \, \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 98, normalized size = 0.92 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-3600 \sqrt {-10 x^{2}-x +3}\, x +4091 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8220 \sqrt {-10 x^{2}-x +3}\right )}{10800 \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 69, normalized size = 0.65 \begin {gather*} -\frac {1}{3} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {4091}{10800} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7}{27} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {137}{180} \, \sqrt {-10 \, x^{2} - x + 3} \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 {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{3\,x+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 {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{3 x + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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