Optimal. Leaf size=160 \[ -\frac {12758 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3375}-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{3 \sqrt {3 x+2}}-\frac {8}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {1076}{675} \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {31588 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375} \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {97, 154, 158, 113, 119} \[ -\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{3 \sqrt {3 x+2}}-\frac {8}{15} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {1076}{675} \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {12758 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}+\frac {31588 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{225} \int \frac {\left (-\frac {2895}{4}-\frac {4035 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {1076}{675} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {8 \int \frac {-\frac {73785}{8}-\frac {118455 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10125}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {1076}{675} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {31588 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3375}+\frac {70169 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3375}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {1076}{675} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {8}{15} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {31588 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}-\frac {12758 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3375}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 102, normalized size = 0.64 \[ \frac {242095 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {30 \sqrt {1-2 x} \sqrt {5 x+3} \left (180 x^2-534 x-1661\right )}{\sqrt {3 x+2}}-31588 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{10125} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, x\right ) \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 145, normalized size = 0.91 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \left (-54000 x^{4}+154800 x^{3}+530520 x^{2}+1770 x -31588 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+242095 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-149490\right )}{10125 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{3/2}} \,d x \]
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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