Optimal. Leaf size=159 \[ \frac {22 \sqrt {x} (3 x+2)}{9 \sqrt {3 x^2+5 x+2}}-\frac {2 (5 x+6) \sqrt {3 x^2+5 x+2}}{3 \sqrt {x}}+\frac {10 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {22 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {812, 839, 1189, 1100, 1136} \[ \frac {22 \sqrt {x} (3 x+2)}{9 \sqrt {3 x^2+5 x+2}}-\frac {2 (5 x+6) \sqrt {3 x^2+5 x+2}}{3 \sqrt {x}}+\frac {10 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}-\frac {22 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 812
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {(2-5 x) \sqrt {2+5 x+3 x^2}}{x^{3/2}} \, dx &=-\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}-\frac {2}{3} \int \frac {-5-\frac {11 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {-5-\frac {11 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}+\frac {20}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {22}{3} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {22 \sqrt {x} (2+3 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {2 (6+5 x) \sqrt {2+5 x+3 x^2}}{3 \sqrt {x}}-\frac {22 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{9 \sqrt {2+5 x+3 x^2}}+\frac {10 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.17, size = 153, normalized size = 0.96 \[ \frac {8 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+22 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-2 \left (45 x^3+96 x^2+65 x+14\right )}{9 \sqrt {x} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 113, normalized size = 0.71 \[ -\frac {270 x^{3}+774 x^{2}+720 x -11 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+3 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+216}{27 \sqrt {3 x^{2}+5 x +2}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )}}{x^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {\left (5\,x-2\right )\,\sqrt {3\,x^2+5\,x+2}}{x^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {2 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {3}{2}}}\right )\, dx - \int \frac {5 \sqrt {3 x^{2} + 5 x + 2}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________