Optimal. Leaf size=191 \[ \frac {6584 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3773}-\frac {1100380 \sqrt {1-2 x} \sqrt {3 x+2}}{41503 \sqrt {5 x+3}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {54 \sqrt {1-2 x}}{539 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {220076 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773} \]
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Rubi [A] time = 0.07, antiderivative size = 191, 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} \[ -\frac {1100380 \sqrt {1-2 x} \sqrt {3 x+2}}{41503 \sqrt {5 x+3}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {54 \sqrt {1-2 x}}{539 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {6584 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773}+\frac {220076 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773} \]
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)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {2}{77} \int \frac {-\frac {127}{2}-75 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {4 \int \frac {-\frac {1659}{2}+\frac {1215 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{1617}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {120615}{4}+\frac {37035 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{11319}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1100380 \sqrt {1-2 x} \sqrt {2+3 x}}{41503 \sqrt {3+5 x}}+\frac {16 \int \frac {-\frac {783495}{2}-\frac {2475855 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{124509}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1100380 \sqrt {1-2 x} \sqrt {2+3 x}}{41503 \sqrt {3+5 x}}-\frac {9876 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3773}-\frac {660228 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{41503}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {54 \sqrt {1-2 x}}{539 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {9876 \sqrt {1-2 x}}{3773 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1100380 \sqrt {1-2 x} \sqrt {2+3 x}}{41503 \sqrt {3+5 x}}+\frac {220076 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773}+\frac {6584 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3773}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 104, normalized size = 0.54 \[ \frac {2 \left (\frac {9903420 x^3+7926942 x^2-2259236 x-2088967}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}-2 \sqrt {2} \left (55019 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-27860 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )\right )}{41503} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72}, x\right ) \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 219, normalized size = 1.15 \[ -\frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (9903420 x^{3}+7926942 x^{2}-330114 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+167160 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2259236 x -220076 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+111440 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2088967\right )}{41503 \left (3 x +2\right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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