Optimal. Leaf size=187 \[ \frac {10628 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{41503 \sqrt {33}}-\frac {2377960 \sqrt {1-2 x} \sqrt {3 x+2}}{1369599 \sqrt {5 x+3}}+\frac {5314 \sqrt {1-2 x}}{41503 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {1088}{17787 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {475592 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}} \]
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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 = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ -\frac {2377960 \sqrt {1-2 x} \sqrt {3 x+2}}{1369599 \sqrt {5 x+3}}+\frac {5314 \sqrt {1-2 x}}{41503 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {1088}{17787 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {4}{231 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {10628 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}+\frac {475592 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}} \]
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)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2}{231} \int \frac {-\frac {197}{2}-75 x}{(1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {1088}{17787 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {4 \int \frac {\frac {18977}{4}+6120 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{17787}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {1088}{17787 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {5314 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {8 \int \frac {\frac {94985}{4}-\frac {39855 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{124509}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {1088}{17787 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {5314 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2377960 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}-\frac {16 \int \frac {\frac {2227845}{8}+\frac {891735 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1369599}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {1088}{17787 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {5314 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2377960 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}-\frac {5314 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{41503}-\frac {475592 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{456533}\\ &=\frac {4}{231 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {1088}{17787 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {5314 \sqrt {1-2 x}}{41503 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2377960 \sqrt {1-2 x} \sqrt {2+3 x}}{1369599 \sqrt {3+5 x}}+\frac {475592 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}+\frac {10628 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{41503 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 103, normalized size = 0.55 \[ \frac {2 \left (\sqrt {2} \left (150115 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-237796 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )+\frac {-14267760 x^3+5106644 x^2+5510400 x-2236533}{(1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}\right )}{1369599} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1800 \, x^{7} + 1860 \, x^{6} - 1162 \, x^{5} - 1473 \, x^{4} + 228 \, x^{3} + 395 \, x^{2} - 12 \, x - 36}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 228, normalized size = 1.22 \[ -\frac {2 \left (14267760 x^{3}-5106644 x^{2}-475592 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+300230 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-5510400 x +237796 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-150115 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2236533\right ) \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}}{1369599 \left (15 x^{2}+19 x +6\right ) \left (2 x -1\right )^{2}} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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