Optimal. Leaf size=191 \[ -\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2911577 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {175111 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1181250} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {175111 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1181250}-\frac {2911577 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}+\frac {2}{45} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac {23 \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}}{1575}-\frac {1244 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{13125}-\frac {175111 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{236250} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x} \, dx &=\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2}{45} \int \frac {\left (-\frac {27}{2}-\frac {23 x}{2}\right ) (2+3 x)^{3/2} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {2 \int \frac {\sqrt {2+3 x} \sqrt {3+5 x} \left (\frac {4815}{4}+1866 x\right )}{\sqrt {1-2 x}} \, dx}{1575}\\ &=-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2 \int \frac {\left (-\frac {170757}{2}-\frac {525333 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{39375}\\ &=-\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {2 \int \frac {\frac {22119087}{8}+\frac {8734731 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{354375}\\ &=-\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {1926221 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2362500}+\frac {2911577 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{590625}\\ &=-\frac {175111 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{236250}-\frac {1244 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{13125}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{1575}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {2911577 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{590625}-\frac {175111 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1181250}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.55, size = 102, normalized size = 0.53 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-136987+410490 x+861750 x^2+472500 x^3\right )+11646308 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5867645 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{3543750 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 153, normalized size = 0.80
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-425250000 x^{6}+5778663 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-11646308 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1101600000 x^{5}-864823500 x^{4}+106067700 x^{3}+335838930 x^{2}+45120930 x -24657660\right )}{7087500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(153\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (2 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}+\frac {383 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{105}+\frac {4561 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2625}-\frac {136987 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{236250}+\frac {7373029 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{9922500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2911577 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2480625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
risch | \(-\frac {\left (472500 x^{3}+861750 x^{2}+410490 x -136987\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{236250 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {7373029 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{25987500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2911577 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{6496875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(257\) |
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 = 0.25, size = 38, normalized size = 0.20 \begin {gather*} \frac {1}{236250} \, {\left (472500 \, x^{3} + 861750 \, x^{2} + 410490 \, x - 136987\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(-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.
[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 = -0.01 \begin {gather*} \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}\,\sqrt {5\,x+3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________