Optimal. Leaf size=156 \[ -\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {458 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1375}-\frac {169 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {496 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}} \]
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Rubi [A]
time = 0.04, antiderivative size = 156, 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 {496 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {169 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {2 \sqrt {1-2 x} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}-\frac {326 \sqrt {1-2 x} (3 x+2)^{3/2}}{825 \sqrt {5 x+3}}+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{1375} \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 {\sqrt {1-2 x} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {\left (\frac {11}{2}-18 x\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {4}{825} \int \frac {\left (\frac {675}{4}-\frac {2061 x}{2}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {458 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1375}-\frac {4 \int \frac {-\frac {5823}{4}-\frac {1521 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12375}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {458 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1375}+\frac {169 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{6875}+\frac {248}{625} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {326 \sqrt {1-2 x} (2+3 x)^{3/2}}{825 \sqrt {3+5 x}}+\frac {458 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1375}-\frac {169 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}-\frac {496 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{625 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 4.39, size = 102, normalized size = 0.65 \begin {gather*} \frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (193+1825 x+2475 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 )}{20625} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 220, normalized size = 1.41
method | result | size |
default | \(-\frac {\left (40920 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-845 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+24552 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-507 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-148500 x^{4}-134250 x^{3}+19670 x^{2}+34570 x +3860\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{20625 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(220\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9375 \left (x +\frac {3}{5}\right )^{2}}-\frac {458 \left (-30 x^{2}-5 x +10\right )}{20625 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}+\frac {647 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{28875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {169 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{28875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {6 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(244\) |
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 = 0.12, size = 45, normalized size = 0.29 \begin {gather*} \frac {2 \, {\left (2475 \, x^{2} + 1825 \, x + 193\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4125 \, {\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 {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}}{{\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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