Optimal. Leaf size=160 \[ \frac {494}{567} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 \sqrt {3 x+2}}+\frac {2470}{567} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {2209}{567} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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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 = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac {118 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 \sqrt {3 x+2}}+\frac {2470}{567} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}+\frac {494}{567} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2209}{567} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx\\ &=-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{63 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {4}{189} \int \frac {\left (\frac {1395}{4}-\frac {3705 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {2470}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{63 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}-\frac {4 \int \frac {-\frac {5865}{2}-\frac {33135 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1701}\\ &=\frac {2470}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{63 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}+\frac {2209}{567} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx-\frac {2717}{567} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2470}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {118 \sqrt {1-2 x} (3+5 x)^{3/2}}{63 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}-\frac {2209}{567} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {494}{567} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A] time = 0.33, size = 102, normalized size = 0.64 \[ \frac {-10360 \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {6 \sqrt {1-2 x} \sqrt {5 x+3} \left (1575 x^2+2841 x+1187\right )}{(3 x+2)^{3/2}}+2209 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{1701} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 224, normalized size = 1.40 \[ \frac {\left (94500 x^{4}+179910 x^{3}+59916 x^{2}-6627 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+31080 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-44016 x -4418 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+20720 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-21366\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{1701 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{5/2}} \,d x \]
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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