Optimal. Leaf size=191 \[ -\frac {7536 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{12005}+\frac {733812 \sqrt {1-2 x} \sqrt {5 x+3}}{132055 \sqrt {3 x+2}}+\frac {10308 \sqrt {1-2 x} \sqrt {5 x+3}}{18865 (3 x+2)^{3/2}}+\frac {138 \sqrt {1-2 x} \sqrt {5 x+3}}{2695 (3 x+2)^{5/2}}+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}-\frac {244604 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005} \]
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Rubi [A] time = 0.07, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac {733812 \sqrt {1-2 x} \sqrt {5 x+3}}{132055 \sqrt {3 x+2}}+\frac {10308 \sqrt {1-2 x} \sqrt {5 x+3}}{18865 (3 x+2)^{3/2}}+\frac {138 \sqrt {1-2 x} \sqrt {5 x+3}}{2695 (3 x+2)^{5/2}}+\frac {4 \sqrt {5 x+3}}{77 \sqrt {1-2 x} (3 x+2)^{5/2}}-\frac {7536 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {244604 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005} \]
Antiderivative was successfully verified.
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Rule 104
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {2}{77} \int \frac {-\frac {123}{2}-75 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}-\frac {4 \int \frac {-\frac {1887}{2}+\frac {1035 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{2695}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}-\frac {8 \int \frac {-\frac {131913}{4}+\frac {38655 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{56595}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}+\frac {733812 \sqrt {1-2 x} \sqrt {3+5 x}}{132055 \sqrt {2+3 x}}-\frac {16 \int \frac {-\frac {1744335}{4}-\frac {2751795 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{396165}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}+\frac {733812 \sqrt {1-2 x} \sqrt {3+5 x}}{132055 \sqrt {2+3 x}}+\frac {11304 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12005}+\frac {733812 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{132055}\\ &=\frac {4 \sqrt {3+5 x}}{77 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {138 \sqrt {1-2 x} \sqrt {3+5 x}}{2695 (2+3 x)^{5/2}}+\frac {10308 \sqrt {1-2 x} \sqrt {3+5 x}}{18865 (2+3 x)^{3/2}}+\frac {733812 \sqrt {1-2 x} \sqrt {3+5 x}}{132055 \sqrt {2+3 x}}-\frac {244604 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}-\frac {7536 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12005}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 106, normalized size = 0.55 \[ \frac {4 \left (\sqrt {2} \left (61151 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-30065 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {5 x+3} \left (-6604308 x^3-5720058 x^2+1424784 x+1546591\right )}{2 \sqrt {1-2 x} (3 x+2)^{5/2}}\right )}{132055} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1620 \, x^{7} + 3672 \, x^{6} + 2025 \, x^{5} - 1077 \, x^{4} - 1312 \, x^{3} - 152 \, x^{2} + 176 \, x + 48}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 314, normalized size = 1.64 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (33021540 x^{4}+48413214 x^{3}-1100718 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+541170 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+10036254 x^{2}-1467624 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+721560 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-12007307 x -489208 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+240520 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-4639773\right )}{132055 \left (3 x +2\right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {1}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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