Optimal. Leaf size=249 \[ -\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8120161139 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}-\frac {486785077 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{248771250 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {486785077 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{248771250 \sqrt {33}}-\frac {8120161139 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}}{10725}+\frac {2314 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{111375}-\frac {121031 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{30405375}-\frac {3872003 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{30405375}-\frac {486785077 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{547296750} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx &=\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {2}{65} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {5653}{2}-\frac {15041 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx}{10725}\\ &=\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {518563}{8}-\frac {121031 x}{8}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1447875}\\ &=-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {8 \int \frac {(3+5 x)^{3/2} \left (\frac {71027395}{16}+\frac {58080045 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{30405375}\\ &=-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {2382196995}{8}-\frac {7301776155 x}{16}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{456080625}\\ &=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {8 \int \frac {\frac {308389708545}{32}+\frac {121802417085 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4104725625}\\ &=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {486785077 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{497542500}+\frac {8120161139 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1368241875}\\ &=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8120161139 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}-\frac {486785077 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{248771250 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 8.33, size = 112, normalized size = 0.45 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (495379991+2923422930 x-1730459250 x^2-7942630500 x^3+2577015000 x^4+8419950000 x^5\right )+32480644556 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-16416737015 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{8209451250 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 163, normalized size = 0.65
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-7577955000000 x^{8}-8129079000000 x^{7}+7138416600000 x^{6}+16063907541 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-32480644556 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+9094592520000 x^{5}-2641153459500 x^{4}-4256073746100 x^{3}-39376043490 x^{2}+630245925510 x +89168398380\right )}{16418902500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(163\) |
risch | \(-\frac {\left (8419950000 x^{5}+2577015000 x^{4}-7942630500 x^{3}-1730459250 x^{2}+2923422930 x +495379991\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{547296750 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {20559313903 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{60202642500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {8120161139 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{15050660625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {200 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}+\frac {2020 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{429}-\frac {168098 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11583}-\frac {59161 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{18711}+\frac {32482477 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6081075}+\frac {495379991 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{547296750}+\frac {20559313903 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{22986463500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {8120161139 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{5746615875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(300\) |
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 = 0.22, size = 48, normalized size = 0.19 \begin {gather*} \frac {1}{547296750} \, {\left (8419950000 \, x^{5} + 2577015000 \, x^{4} - 7942630500 \, x^{3} - 1730459250 \, x^{2} + 2923422930 \, x + 495379991\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 {\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.
[In]
[Out]
________________________________________________________________________________________