Optimal. Leaf size=160 \[ -\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {46 (1-2 x)^{3/2} \sqrt {2+3 x}}{75 \sqrt {3+5 x}}-\frac {76}{375} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {338 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1875}+\frac {992 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1875} \]
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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 {992 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1875}+\frac {338 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1875}-\frac {2 \sqrt {3 x+2} (1-2 x)^{5/2}}{15 (5 x+3)^{3/2}}-\frac {46 \sqrt {3 x+2} (1-2 x)^{3/2}}{75 \sqrt {5 x+3}}-\frac {76}{375} \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x} \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)^{5/2} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (-\frac {17}{2}-18 x\right ) (1-2 x)^{3/2}}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {46 (1-2 x)^{3/2} \sqrt {2+3 x}}{75 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (-78-\frac {171 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {46 (1-2 x)^{3/2} \sqrt {2+3 x}}{75 \sqrt {3+5 x}}-\frac {76}{375} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {8 \int \frac {-\frac {5823}{4}-\frac {1521 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3375}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {46 (1-2 x)^{3/2} \sqrt {2+3 x}}{75 \sqrt {3+5 x}}-\frac {76}{375} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {338 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1875}-\frac {5456 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1875}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {46 (1-2 x)^{3/2} \sqrt {2+3 x}}{75 \sqrt {3+5 x}}-\frac {76}{375} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {338 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1875}+\frac {992 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1875}\\ \end {align*}
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Mathematica [A]
time = 7.54, size = 102, normalized size = 0.64 \begin {gather*} \frac {2 \left (\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \left (-712-925 x+100 x^2\right )}{(3+5 x)^{3/2}}-169 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-8015 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{5625} \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 | \(\frac {2 \left (40920 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-845 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+24552 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-507 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9000 x^{4}-81750 x^{3}-80955 x^{2}+17070 x +21360\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{5625 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(220\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9375 \left (x +\frac {3}{5}\right )^{2}}-\frac {418 \left (-30 x^{2}-5 x +10\right )}{1875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {1294 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {338 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{375}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(244\) |
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 = 0.21, size = 45, normalized size = 0.28 \begin {gather*} \frac {2 \, {\left (100 \, x^{2} - 925 \, x - 712\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{375 \, {\left (25 \, x^{2} + 30 \, x + 9\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 )}^{5/2}\,\sqrt {3\,x+2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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