Optimal. Leaf size=218 \[ -\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}-\frac {1654421 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1913625 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164,
114, 120} \begin {gather*} -\frac {1654421 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1913625 \sqrt {33}}-\frac {146222113 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {9698 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{93555}-\frac {146963 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{467775}-\frac {1654421 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{4209975} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx &=\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{33} \int \frac {\left (-50-\frac {185 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {9245}{4}-\frac {24245 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx}{4455}\\ &=\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (\frac {84395}{4}-\frac {2204445 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{467775}\\ &=-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {44328915}{16}+\frac {24816315 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7016625}\\ &=-\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {685297455}{8}-\frac {2193331695 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{63149625}\\ &=-\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {1654421 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3827250}+\frac {146222113 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42099750}\\ &=-\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}-\frac {1654421 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1913625 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 8.28, size = 107, normalized size = 0.49 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (3748468+9143865 x-16381350 x^2-12379500 x^3+25515000 x^4\right )+146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-91626220 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{63149625 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 158, normalized size = 0.72
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-22963500000 x^{7}-6463800000 x^{6}+54595893 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-146222113 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+28643220000 x^{5}+5066658000 x^{4}-15351281550 x^{3}-3614874270 x^{2}+2433073980 x +674724240\right )}{126299250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
risch | \(-\frac {\left (25515000 x^{4}-12379500 x^{3}-16381350 x^{2}+9143865 x +3748468\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{4209975 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {45686497 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{231548625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {146222113 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{463097250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{33}-\frac {2620 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{891}-\frac {72806 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18711}+\frac {203197 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{93555}+\frac {3748468 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4209975}+\frac {45686497 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{88409475 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {146222113 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{176818950 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(278\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.17, size = 43, normalized size = 0.20 \begin {gather*} \frac {1}{4209975} \, {\left (25515000 \, x^{4} - 12379500 \, x^{3} - 16381350 \, x^{2} + 9143865 \, x + 3748468\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {3\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________