Optimal. Leaf size=160 \[ -\frac {1150}{81} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{9 \sqrt {2+3 x}}+\frac {592}{81} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {230}{81} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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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 = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 159,
164, 114, 120} \begin {gather*} -\frac {230}{81} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {592}{81} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{9 \sqrt {3 x+2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}-\frac {1150}{81} \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 155
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{9 \sqrt {2+3 x}}-\frac {4}{27} \int \frac {\left (126-\frac {1725 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {1150}{81} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{9 \sqrt {2+3 x}}+\frac {4}{243} \int \frac {-\frac {1533}{4}-2220 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1150}{81} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{9 \sqrt {2+3 x}}-\frac {592}{81} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx+\frac {1265}{81} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1150}{81} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{9 \sqrt {2+3 x}}+\frac {592}{81} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {230}{81} \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 = 8.41, size = 102, normalized size = 0.64 \begin {gather*} \frac {1}{243} \left (-\frac {6 \sqrt {1-2 x} \sqrt {3+5 x} \left (329+564 x+90 x^2\right )}{(2+3 x)^{3/2}}-592 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+4387 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 220, normalized size = 1.38
method | result | size |
default | \(-\frac {\left (11385 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1776 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+7590 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1184 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+5400 x^{4}+34380 x^{3}+21504 x^{2}-8178 x -5922\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{243 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\) | \(220\) |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {296 \left (-30 x^{2}-3 x +9\right )}{243 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {73 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{243 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2960 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1701 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {20 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81}+\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729 \left (\frac {2}{3}+x \right )^{2}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(255\) |
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 = 0.18, size = 45, normalized size = 0.28 \begin {gather*} -\frac {2 \, {\left (90 \, x^{2} + 564 \, x + 329\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{81 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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