Optimal. Leaf size=160 \[ \frac {494}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {8}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2209}{675} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {494}{675} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, 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 = {99, 159, 164,
114, 120} \begin {gather*} \frac {494}{675} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2209}{675} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {8}{15} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3 \sqrt {3 x+2}}+\frac {494}{135} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \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 {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {8}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {4}{225} \int \frac {\left (\frac {1395}{4}-\frac {3705 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {494}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {8}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4 \int \frac {-\frac {5865}{2}-\frac {33135 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2025}\\ &=\frac {494}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {8}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {2209}{675} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx-\frac {2717}{675} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {494}{135} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3 \sqrt {2+3 x}}-\frac {8}{15} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2209}{675} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {494}{675} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 6.36, size = 102, normalized size = 0.64 \begin {gather*} \frac {-\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (-143-102 x+90 x^2\right )}{\sqrt {2+3 x}}+2209 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-10360 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{2025} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 143, normalized size = 0.89
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (8151 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2209 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-27000 x^{4}+27900 x^{3}+54060 x^{2}-4890 x -12870\right )}{60750 x^{3}+46575 x^{2}-14175 x -12150}\) | \(143\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {4 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9}+\frac {4 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5}+\frac {782 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2835 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2209 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2835 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {140}{27} x^{2}-\frac {14}{27} x +\frac {14}{9}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(240\) |
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 = 0.23, size = 33, normalized size = 0.21 \begin {gather*} -\frac {2 \, {\left (90 \, x^{2} - 102 \, x - 143\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{135 \, \sqrt {3 \, x + 2}} \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 \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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