Optimal. Leaf size=156 \[ -\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {3}{5} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {1}{5} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 156, 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 {1}{5} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {3}{5} \sqrt {33} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2 \sqrt {1-2 x} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}+\frac {4}{3} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\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 {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{3} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {2}{45} \int \frac {\left (-\frac {45}{2}-\frac {405 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {2}{405} \int \frac {\frac {5265}{4}+\frac {3645 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}+\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {9}{5} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {4}{3} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2 \sqrt {1-2 x} (3+5 x)^{5/2}}{3 \sqrt {2+3 x}}-\frac {3}{5} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {1}{5} \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 = 3.79, size = 112, normalized size = 0.72 \begin {gather*} \frac {10 \sqrt {1-2 x} x \sqrt {2+3 x} \sqrt {3+5 x} (7+10 x)+18 \sqrt {2} (2+3 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+15 \sqrt {2} (2+3 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{60+90 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 142, normalized size = 0.91
method | result | size |
default | \(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-18 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1000 x^{4}-800 x^{3}+230 x^{2}+210 x \right )}{30 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(142\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {10 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9}+\frac {\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}+\frac {13 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{42 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {3 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-3 x +9\right )}{81 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(240\) |
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 = 0.12, size = 32, normalized size = 0.21 \begin {gather*} \frac {{\left (10 \, x^{2} + 7 \, x\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3 \, \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 {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\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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