Optimal. Leaf size=150 \[ -\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {1922677 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{777600 \sqrt {10}}-\frac {98}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {103, 159, 163,
56, 222, 95, 210} \begin {gather*} \frac {1922677 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{777600 \sqrt {10}}-\frac {98}{243} \sqrt {7} \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {1}{12} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {181 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1080}+\frac {7093 (5 x+3)^{3/2} \sqrt {1-2 x}}{21600}-\frac {390869 \sqrt {5 x+3} \sqrt {1-2 x}}{259200} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 56
Rule 95
Rule 103
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{2+3 x} \, dx &=\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {1}{12} \int \frac {\left (-51-\frac {181 x}{2}\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx\\ &=\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {1}{540} \int \frac {\left (-\frac {5133}{2}-\frac {21279 x}{4}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{2+3 x} \, dx\\ &=\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {\int \frac {\left (-\frac {116469}{4}-\frac {1172607 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{16200}\\ &=-\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {\int \frac {\frac {3020277}{8}+\frac {5768031 x}{16}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{97200}\\ &=-\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {1922677 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1555200}+\frac {343}{243} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {686}{243} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {1922677 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{777600 \sqrt {5}}\\ &=-\frac {390869 \sqrt {1-2 x} \sqrt {3+5 x}}{259200}+\frac {7093 \sqrt {1-2 x} (3+5 x)^{3/2}}{21600}+\frac {181 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1080}+\frac {1}{12} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {1922677 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{777600 \sqrt {10}}-\frac {98}{243} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.24, size = 119, normalized size = 0.79 \begin {gather*} \frac {30 \sqrt {1-2 x} \left (178797+990815 x-666900 x^2-1740000 x^3+2160000 x^4\right )-1922677 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )-3136000 \sqrt {7} \sqrt {3+5 x} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{7776000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 132, normalized size = 0.88
method | result | size |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (25920000 x^{3} \sqrt {-10 x^{2}-x +3}-36432000 x^{2} \sqrt {-10 x^{2}-x +3}+1922677 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3136000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13856400 x \sqrt {-10 x^{2}-x +3}+3575940 \sqrt {-10 x^{2}-x +3}\right )}{15552000 \sqrt {-10 x^{2}-x +3}}\) | \(132\) |
risch | \(-\frac {\left (432000 x^{3}-607200 x^{2}+230940 x +59599\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{259200 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {1922677 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{15552000}-\frac {49 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{243}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 98, normalized size = 0.65 \begin {gather*} -\frac {1}{6} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {271}{1080} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {7093}{4320} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {1922677}{15552000} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {49}{243} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {135521}{259200} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.82, size = 112, normalized size = 0.75 \begin {gather*} \frac {1}{259200} \, {\left (432000 \, x^{3} - 607200 \, x^{2} + 230940 \, x + 59599\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {49}{243} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {1922677}{15552000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3 x + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.06, size = 199, normalized size = 1.33 \begin {gather*} \frac {49}{2430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{1296000} \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 577 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 23769 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 390869 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1922677}{15552000} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \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 \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{3\,x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________