Optimal. Leaf size=160 \[ -\frac {2252 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4725}-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {148831 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47250}-\frac {2252 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{23625} \]
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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 {2252 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{23625}-\frac {148831 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47250}+\frac {2}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {31}{525} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {2252 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{4725} \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 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx &=\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{35} \int \frac {\left (-\frac {23}{2}-\frac {31 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{525} \int \frac {\sqrt {3+5 x} \left (\frac {2907}{4}+1126 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {2252 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4725}-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2 \int \frac {-\frac {94253}{4}-\frac {148831 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4725}\\ &=-\frac {2252 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4725}-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {12386 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{23625}+\frac {148831 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{47250}\\ &=-\frac {2252 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4725}-\frac {31}{525} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2}{35} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {148831 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{47250}-\frac {2252 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{23625}\\ \end {align*}
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Mathematica [A]
time = 2.92, size = 97, normalized size = 0.61 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-659+6705 x+6750 x^2\right )+148831 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-74515 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{70875 \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 | \(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (74316 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-148831 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-6075000 x^{5}-10692000 x^{4}-2615850 x^{3}+3077760 x^{2}+1068510 x -118620\right )}{141750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(148\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {149 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{105}-\frac {659 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4725}+\frac {94253 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{198450 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {148831 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{198450 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {10 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(234\) |
risch | \(-\frac {\left (6750 x^{2}+6705 x -659\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{4725 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {94253 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{519750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {148831 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{519750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(252\) |
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 = 0.15, size = 33, normalized size = 0.21 \begin {gather*} \frac {1}{4725} \, {\left (6750 \, x^{2} + 6705 \, x - 659\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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \sqrt {1-2\,x}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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