Optimal. Leaf size=191 \[ \frac {1112}{35} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {36968 \sqrt {1-2 x} \sqrt {3 x+2}}{21 \sqrt {5 x+3}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {44 \sqrt {1-2 x}}{5 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {36968}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, 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 = {98, 152, 158, 113, 119} \[ -\frac {36968 \sqrt {1-2 x} \sqrt {3 x+2}}{21 \sqrt {5 x+3}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {44 \sqrt {1-2 x}}{5 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {14 \sqrt {1-2 x}}{15 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {1112}{35} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {36968}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 98
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {2}{15} \int \frac {121-165 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {4}{315} \int \frac {\frac {18249}{2}-10395 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {8 \int \frac {389235-\frac {481635 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2205}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}-\frac {16 \int \frac {\frac {20273715}{4}+\frac {16011765 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{24255}\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}-\frac {6116}{35} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {36968}{35} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{15 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {44 \sqrt {1-2 x}}{5 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {6116 \sqrt {1-2 x}}{35 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{21 \sqrt {3+5 x}}+\frac {36968}{35} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {1112}{35} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 105, normalized size = 0.55 \[ \frac {2}{105} \left (-2 \sqrt {2} \left (9242 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-4655 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )-\frac {3 \sqrt {1-2 x} \left (831780 x^3+1636038 x^2+1071882 x+233897\right )}{(3 x+2)^{5/2} \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 314, normalized size = 1.64 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-4990680 x^{4}-7320888 x^{3}+166356 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-83790 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1523178 x^{2}+221808 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-111720 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1812264 x +73936 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-37240 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+701691\right )}{105 \left (3 x +2\right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________