Optimal. Leaf size=122 \[ 2 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {139}{10} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {23 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}} \]
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Rubi [A]
time = 0.02, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 159, 164,
114, 120} \begin {gather*} \frac {23 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}}+\frac {139}{10} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{3/2}}{\sqrt {1-2 x}}+2 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {\sqrt {2+3 x} \left (\frac {37}{2}+30 x\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=2 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{15} \int \frac {-660-\frac {2085 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=2 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {23}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {139}{10} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=2 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {(2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {139}{10} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {23 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 5.75, size = 103, normalized size = 0.84 \begin {gather*} \frac {-30 (-4+x) \sqrt {2+3 x} \sqrt {3+5 x}-139 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+70 \sqrt {2-4 x} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{30 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 138, normalized size = 1.13
method | result | size |
default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (69 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-139 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+450 x^{3}-1230 x^{2}-2100 x -720\right )}{900 x^{3}+690 x^{2}-210 x -180}\) | \(138\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {7 \left (-30 x^{2}-38 x -12\right )}{4 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {44 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {139 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{42 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(220\) |
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 = 0.17, size = 32, normalized size = 0.26 \begin {gather*} \frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (x - 4\right )} \sqrt {-2 \, x + 1}}{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 (3 x + 2\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, 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 (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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