Optimal. Leaf size=63 \[ 2 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {104, 21, 113} \[ 2 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 104
Rule 113
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}-\frac {2}{11} \int \frac {9+15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}-\frac {6}{11} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+2 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 106, normalized size = 1.68 \[ -\frac {2 \left (-\sqrt {2} (5 x+3) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+5 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}+\sqrt {2} (5 x+3) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )}{55 x+33} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 134, normalized size = 2.13 \[ -\frac {2 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {3 x +2}\, \left (30 x^{2}+5 x -\sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+\sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-10\right )}{11 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {1-2\,x}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - 2 x} \sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________