Optimal. Leaf size=200 \[ -\frac {412}{189} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {13688 (3 x+2) \sqrt {x}}{2835 \sqrt {3 x^2+5 x+2}}+\frac {412 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{189 \sqrt {3 x^2+5 x+2}}-\frac {13688 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2835 \sqrt {3 x^2+5 x+2}}-\frac {10}{21} \sqrt {3 x^2+5 x+2} x^{5/2}+\frac {128}{105} \sqrt {3 x^2+5 x+2} x^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {832, 839, 1189, 1100, 1136} \[ -\frac {10}{21} \sqrt {3 x^2+5 x+2} x^{5/2}+\frac {128}{105} \sqrt {3 x^2+5 x+2} x^{3/2}-\frac {412}{189} \sqrt {3 x^2+5 x+2} \sqrt {x}+\frac {13688 (3 x+2) \sqrt {x}}{2835 \sqrt {3 x^2+5 x+2}}+\frac {412 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{189 \sqrt {3 x^2+5 x+2}}-\frac {13688 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2835 \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 832
Rule 839
Rule 1100
Rule 1136
Rule 1189
Rubi steps
\begin {align*} \int \frac {(2-5 x) x^{5/2}}{\sqrt {2+5 x+3 x^2}} \, dx &=-\frac {10}{21} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {2}{21} \int \frac {x^{3/2} (25+96 x)}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {128}{105} x^{3/2} \sqrt {2+5 x+3 x^2}-\frac {10}{21} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {4}{315} \int \frac {\left (-288-\frac {1545 x}{2}\right ) \sqrt {x}}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {412}{189} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {128}{105} x^{3/2} \sqrt {2+5 x+3 x^2}-\frac {10}{21} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {8 \int \frac {\frac {1545}{2}+\frac {5133 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{2835}\\ &=-\frac {412}{189} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {128}{105} x^{3/2} \sqrt {2+5 x+3 x^2}-\frac {10}{21} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {16 \operatorname {Subst}\left (\int \frac {\frac {1545}{2}+\frac {5133 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{2835}\\ &=-\frac {412}{189} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {128}{105} x^{3/2} \sqrt {2+5 x+3 x^2}-\frac {10}{21} x^{5/2} \sqrt {2+5 x+3 x^2}+\frac {824}{189} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {13688}{945} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {13688 \sqrt {x} (2+3 x)}{2835 \sqrt {2+5 x+3 x^2}}-\frac {412}{189} \sqrt {x} \sqrt {2+5 x+3 x^2}+\frac {128}{105} x^{3/2} \sqrt {2+5 x+3 x^2}-\frac {10}{21} x^{5/2} \sqrt {2+5 x+3 x^2}-\frac {13688 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2835 \sqrt {2+5 x+3 x^2}}+\frac {412 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{189 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.17, size = 160, normalized size = 0.80 \[ \frac {-7508 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+13688 i \sqrt {2} \sqrt {\frac {1}{x}+1} \sqrt {\frac {2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-4050 x^5+3618 x^4-3960 x^3+17076 x^2+56080 x+27376}{2835 \sqrt {x} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (5 \, x^{3} - 2 \, x^{2}\right )} \sqrt {x}}{\sqrt {3 \, x^{2} + 5 \, x + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (5 \, x - 2\right )} x^{\frac {5}{2}}}{\sqrt {3 \, x^{2} + 5 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.15, size = 122, normalized size = 0.61 \[ -\frac {2 \left (6075 x^{5}-5427 x^{4}+5940 x^{3}+35982 x^{2}+18540 x -3422 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+7176 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{8505 \sqrt {3 x^{2}+5 x +2}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (5 \, x - 2\right )} x^{\frac {5}{2}}}{\sqrt {3 \, x^{2} + 5 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {x^{5/2}\,\left (5\,x-2\right )}{\sqrt {3\,x^2+5\,x+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {2 x^{\frac {5}{2}}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {5 x^{\frac {7}{2}}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________