Optimal. Leaf size=160 \[ -\frac {487 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13125}+\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {46159 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625}-\frac {2281 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625} \]
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Rubi [A]
time = 0.04, 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 = {103, 159, 164,
114, 120} \begin {gather*} -\frac {2281 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625}-\frac {46159 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625}+\frac {2}{35} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{3/2}+\frac {62}{875} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {487 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{13125} \end {gather*}
Antiderivative was successfully verified.
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx &=\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {2}{35} \int \frac {\left (-\frac {69}{2}-\frac {93 x}{2}\right ) \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {4 \int \frac {\left (-\frac {1425}{2}-\frac {1461 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2625}\\ &=-\frac {487 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13125}+\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {4 \int \frac {\frac {181227}{8}+\frac {138477 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{39375}\\ &=-\frac {487 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13125}+\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {25091 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{131250}+\frac {46159 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{65625}\\ &=-\frac {487 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13125}+\frac {62}{875} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {2}{35} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {46159 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625}-\frac {2281 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{65625}\\ \end {align*}
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Mathematica [A]
time = 5.94, size = 97, normalized size = 0.61 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (2873+2040 x-4500 x^2\right )+92318 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-17045 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{196875 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 148, normalized size = 0.92
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (75273 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-92318 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+4050000 x^{5}+1269000 x^{4}-4938300 x^{3}-2363970 x^{2}+970530 x +517140\right )}{393750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(148\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {12 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35}+\frac {136 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}+\frac {2873 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{13125}+\frac {60409 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{551250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {46159 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{275625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
risch | \(\frac {\left (4500 x^{2}-2040 x -2873\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{13125 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {60409 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{1443750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {46159 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{721875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(251\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.22, size = 33, normalized size = 0.21 \begin {gather*} -\frac {1}{13125} \, {\left (4500 \, x^{2} - 2040 \, x - 2873\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{\frac {3}{2}}}{\sqrt {5 x + 3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}}{\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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