Optimal. Leaf size=202 \[ -\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {11732 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{125 \sqrt {2+5 x+3 x^2}}+\frac {3212 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{25 \sqrt {2+5 x+3 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {836, 848, 857,
732, 435, 430} \begin {gather*} \frac {3212 \sqrt {3} \sqrt {-3 x^2-5 x-2} F\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{25 \sqrt {3 x^2+5 x+2}}-\frac {11732 \sqrt {3} \sqrt {-3 x^2-5 x-2} E\left (\text {ArcSin}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{125 \sqrt {3 x^2+5 x+2}}-\frac {2 (47 x+37)}{5 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{3/2}}+\frac {23464 \sqrt {3 x^2+5 x+2}}{125 \sqrt {2 x+3}}+\frac {4 (2409 x+2054)}{25 \sqrt {2 x+3} \sqrt {3 x^2+5 x+2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 435
Rule 732
Rule 836
Rule 848
Rule 857
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {918+705 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {6441+7227 x}{(3+2 x)^{3/2} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {8}{375} \int \frac {10764+\frac {26397 x}{2}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {17598}{125} \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx+\frac {4818}{25} \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {\left (11732 \sqrt {3} \sqrt {-2-5 x-3 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 x^2}{3}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{125 \sqrt {2+5 x+3 x^2}}+\frac {\left (3212 \sqrt {3} \sqrt {-2-5 x-3 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 x^2}{3}}} \, dx,x,\frac {\sqrt {6+6 x}}{\sqrt {2}}\right )}{25 \sqrt {2+5 x+3 x^2}}\\ &=-\frac {2 (37+47 x)}{5 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2054+2409 x)}{25 \sqrt {3+2 x} \sqrt {2+5 x+3 x^2}}+\frac {23464 \sqrt {2+5 x+3 x^2}}{125 \sqrt {3+2 x}}-\frac {11732 \sqrt {3} \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{125 \sqrt {2+5 x+3 x^2}}+\frac {3212 \sqrt {3} \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{25 \sqrt {2+5 x+3 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 30.29, size = 215, normalized size = 1.06 \begin {gather*} \frac {2 \left (5 \left (8031+29941 x+36414 x^2+14454 x^3\right )-5866 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} \sqrt {3+2 x} \sqrt {\frac {2+3 x}{3+2 x}} \left (6+19 x+19 x^2+6 x^3\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )+1048 \sqrt {5} \sqrt {\frac {1+x}{3+2 x}} \sqrt {3+2 x} \sqrt {\frac {2+3 x}{3+2 x}} \left (6+19 x+19 x^2+6 x^3\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {3+2 x}}\right )|\frac {3}{5}\right )\right )}{125 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 308, normalized size = 1.52
method | result | size |
elliptic | \(\frac {\sqrt {\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (-\frac {302}{225}-\frac {134 x}{75}\right ) \sqrt {6 x^{3}+19 x^{2}+19 x +6}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 \left (9+6 x \right ) \left (-\frac {1002}{25}-\frac {1194 x}{25}\right )}{\sqrt {\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right ) \left (9+6 x \right )}}-\frac {208 \left (6 x^{2}+10 x +4\right )}{125 \sqrt {\left (x +\frac {3}{2}\right ) \left (6 x^{2}+10 x +4\right )}}-\frac {9568 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{625 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}-\frac {11732 \sqrt {45+30 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )}{2}-\EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )\right )}{625 \sqrt {6 x^{3}+19 x^{2}+19 x +6}}\right )}{\sqrt {3+2 x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(259\) |
default | \(\frac {2 \left (8799 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}+3246 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x^{2} \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}+14665 \sqrt {15}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}+5410 \sqrt {15}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right ) x \sqrt {-20-30 x}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}+5866 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticE \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )+2164 \sqrt {15}\, \sqrt {3+2 x}\, \sqrt {-2-2 x}\, \sqrt {-20-30 x}\, \EllipticF \left (\frac {\sqrt {45+30 x}}{5}, \frac {\sqrt {15}}{3}\right )+527940 x^{4}+2121150 x^{3}+3080770 x^{2}+1921725 x +435415\right ) \sqrt {3 x^{2}+5 x +2}}{625 \left (2+3 x \right )^{2} \left (1+x \right )^{2} \sqrt {3+2 x}}\) | \(308\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.51, size = 146, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (12671 \, \sqrt {6} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right ) + 52794 \, \sqrt {6} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} {\rm weierstrassZeta}\left (\frac {19}{27}, -\frac {28}{729}, {\rm weierstrassPInverse}\left (\frac {19}{27}, -\frac {28}{729}, x + \frac {19}{18}\right )\right ) + 9 \, {\left (105588 \, x^{4} + 424230 \, x^{3} + 616154 \, x^{2} + 384345 \, x + 87083\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}\right )}}{1125 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x}{18 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 68 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 12 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{18 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 68 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2} + 12 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x-5}{{\left (2\,x+3\right )}^{3/2}\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________