Optimal. Leaf size=160 \[ -\frac {28174 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{28125}-\frac {2 \sqrt {3 x+2} (1-2 x)^{5/2}}{5 \sqrt {5 x+3}}-\frac {24}{125} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {3028 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{5625}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125} \]
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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} \[ -\frac {2 \sqrt {3 x+2} (1-2 x)^{5/2}}{5 \sqrt {5 x+3}}-\frac {24}{125} \sqrt {3 x+2} \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {3028 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{5625}-\frac {28174 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125} \]
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 {2+3 x}}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {\left (-\frac {17}{2}-18 x\right ) (1-2 x)^{3/2}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{375} \int \frac {\left (-\frac {1887}{4}-\frac {2271 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {8 \int \frac {-\frac {53121}{8}-\frac {60873 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{16875}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {81164 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{28125}+\frac {154957 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{28125}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {3028 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5625}-\frac {24}{125} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {81164 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}-\frac {28174 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{28125}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 102, normalized size = 0.64 \[ \frac {546035 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {30 \sqrt {1-2 x} \sqrt {3 x+2} \left (900 x^2-2530 x-7287\right )}{\sqrt {5 x+3}}-81164 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{84375} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, x\right ) \]
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 \[ \int \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-162000 x^{4}+428400 x^{3}+1441560 x^{2}+66810 x -81164 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+546035 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-437220\right )}{84375 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
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 \[ \int \frac {\sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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