Optimal. Leaf size=188 \[ \frac {3683}{210} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {(3 x+2)^{3/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}+\frac {12}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {167}{14} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {3683}{42} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}+\frac {244879}{420} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 188, 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 = {97, 154, 158, 113, 119} \[ \frac {(3 x+2)^{3/2} (5 x+3)^{5/2}}{\sqrt {1-2 x}}+\frac {12}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {167}{14} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}+\frac {3683}{42} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}+\frac {3683}{210} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {244879}{420} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {77}{2}+60 x\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {1}{35} \int \frac {\left (-4105-\frac {12525 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {1}{525} \int \frac {\sqrt {3+5 x} \left (\frac {1077075}{4}+\frac {828675 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {3683}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\int \frac {-\frac {17440875}{2}-\frac {55097775 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4725}\\ &=\frac {3683}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {40513}{420} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {244879}{420} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {3683}{42} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {167}{14} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {12}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {244879}{420} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {3683}{210} \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 = 0.27, size = 115, normalized size = 0.61 \[ \frac {123340 \sqrt {2-4 x} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-30 \sqrt {3 x+2} \sqrt {5 x+3} \left (450 x^3+1650 x^2+3349 x-6590\right )-244879 \sqrt {2-4 x} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{1260 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 150, normalized size = 0.80 \[ -\frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-202500 x^{5}-999000 x^{4}-2528550 x^{3}+759570 x^{2}+3153480 x -244879 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+123340 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1186200\right )}{1260 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]
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]
________________________________________________________________________________________