Optimal. Leaf size=280 \[ -\frac {23763809947 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13682418750}-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {1580201444291 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12438562500 \sqrt {33}}-\frac {23763809947 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6219281250 \sqrt {33}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps
used = 10, 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 {23763809947 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6219281250 \sqrt {33}}-\frac {1580201444291 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12438562500 \sqrt {33}}+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac {106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}}{4875}+\frac {8038 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}+\frac {364267 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{36196875}-\frac {26534891 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{760134375}-\frac {359748241 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{1520268750}-\frac {23763809947 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{13682418750} \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} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx &=\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {2}{75} \int \left (-\frac {113}{2}-\frac {159 x}{2}\right ) (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx\\ &=\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {4 \int \left (-3084-\frac {12057 x}{4}\right ) \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx}{14625}\\ &=\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {8 \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2} \left (-\frac {1010595}{8}+\frac {1092801 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{2413125}\\ &=\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {8 \int \frac {(3+5 x)^{5/2} \left (\frac {108689433}{16}+\frac {79604673 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{108590625}\\ &=-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {5294426955}{8}-\frac {16188670845 x}{16}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2280403125}\\ &=-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {1390090964715}{32}+\frac {1069371447615 x}{16}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{34206046875}\\ &=-\frac {23763809947 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13682418750}-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {8 \int \frac {-\frac {22509028090305}{16}-\frac {71109064993095 x}{32}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{307854421875}\\ &=-\frac {23763809947 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13682418750}-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {23763809947 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12438562500}+\frac {1580201444291 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{136824187500}\\ &=-\frac {23763809947 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13682418750}-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {1580201444291 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12438562500 \sqrt {33}}-\frac {23763809947 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6219281250 \sqrt {33}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 8.26, size = 119, normalized size = 0.42 \begin {gather*} \frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (9093216326+157612390605 x+59959633500 x^2-487924998750 x^3-352885207500 x^4+579573225000 x^5+547296750000 x^6\right )+\sqrt {2} \left (1580201444291 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-795995716040 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{410472562500} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 168, normalized size = 0.60
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-492567075000000 x^{9}-899250660000000 x^{8}+32623479000000 x^{7}+902847084300000 x^{6}+784205728251 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1580201444291 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+312921865912500 x^{5}-349206885747000 x^{4}-192171420950850 x^{3}+37617016792110 x^{2}+30279805737360 x +1636778938680\right )}{410472562500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(168\) |
risch | \(-\frac {\left (547296750000 x^{6}+579573225000 x^{5}-352885207500 x^{4}-487924998750 x^{3}+59959633500 x^{2}+157612390605 x +9093216326\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{13682418750 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {500200624229 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{752533031250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1580201444291 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{1505066062500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(272\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (40 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{6}+\frac {1652 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{39}-\frac {55322 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{2145}-\frac {2065291 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{57915}+\frac {2049902 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{467775}+\frac {3502497569 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{304053750}+\frac {4546608163 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6841209375}+\frac {500200624229 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{287330793750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1580201444291 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{574661587500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(334\) |
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 = 53, normalized size = 0.19 \begin {gather*} \frac {1}{13682418750} \, {\left (547296750000 \, x^{6} + 579573225000 \, x^{5} - 352885207500 \, x^{4} - 487924998750 \, x^{3} + 59959633500 \, x^{2} + 157612390605 \, x + 9093216326\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}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________