Optimal. Leaf size=187 \[ -\frac {663409 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{13750 \sqrt {33}}+\frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}-\frac {910 \sqrt {5 x+3} (3 x+2)^{5/2}}{363 \sqrt {1-2 x}}-\frac {27271 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{6050}-\frac {317384 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{15125}-\frac {44109377 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27500 \sqrt {33}} \]
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Rubi [A] time = 0.07, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {98, 150, 154, 158, 113, 119} \[ \frac {7 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 (1-2 x)^{3/2}}-\frac {910 \sqrt {5 x+3} (3 x+2)^{5/2}}{363 \sqrt {1-2 x}}-\frac {27271 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{6050}-\frac {317384 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{15125}-\frac {663409 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{13750 \sqrt {33}}-\frac {44109377 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27500 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^{5/2} \left (\frac {499}{2}+411 x\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {910 (2+3 x)^{5/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-24996-\frac {81813 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {27271 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6050}-\frac {910 (2+3 x)^{5/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {\int \frac {\sqrt {2+3 x} \left (\frac {7044525}{4}+2856456 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{9075}\\ &=-\frac {317384 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{15125}-\frac {27271 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6050}-\frac {910 (2+3 x)^{5/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {\int \frac {-\frac {125663067}{2}-\frac {396984393 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{136125}\\ &=-\frac {317384 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{15125}-\frac {27271 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6050}-\frac {910 (2+3 x)^{5/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {663409 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{27500}+\frac {44109377 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{302500}\\ &=-\frac {317384 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{15125}-\frac {27271 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{6050}-\frac {910 (2+3 x)^{5/2} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {44109377 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27500 \sqrt {33}}-\frac {663409 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{13750 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 125, normalized size = 0.67 \[ -\frac {-22216880 \sqrt {2-4 x} (2 x-1) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+10 \sqrt {3 x+2} \sqrt {5 x+3} \left (294030 x^3+1528956 x^2-9445541 x+3478434\right )+44109377 \sqrt {2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{907500 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{40 \, x^{4} - 36 \, x^{3} - 6 \, x^{2} + 13 \, x - 3}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 238, normalized size = 1.27 \[ \frac {\left (-44104500 x^{5}-285209100 x^{4}+1108687710 x^{3}+1181150330 x^{2}-88218754 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+44433760 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-94170000 x +44109377 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-22216880 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-208706040\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {3 x +2}}{907500 \left (2 x -1\right )^{2} \left (15 x^{2}+19 x +6\right )} \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]
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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