Optimal. Leaf size=126 \[ -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac {175 (6 x+5) \sqrt {3 x^2+5 x+2}}{82944}-\frac {175 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{165888 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 612, 621, 206} \[ -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac {175 (6 x+5) \sqrt {3 x^2+5 x+2}}{82944}-\frac {175 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{165888 \sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 612
Rule 621
Rule 640
Rubi steps
\begin {align*} \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac {35}{6} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175}{432} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac {175 \int \sqrt {2+5 x+3 x^2} \, dx}{6912}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{165888}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{82944}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{165888 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 108, normalized size = 0.86 \[ -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 \left (\sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )+6 \sqrt {3 x^2+5 x+2} \left (144 x^3+360 x^2+290 x+75\right )\right )}{497664} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.74, size = 83, normalized size = 0.66 \[ -\frac {1}{580608} \, {\left (746496 \, x^{6} - 1347840 \, x^{5} - 13454208 \, x^{4} - 26388720 \, x^{3} - 23110872 \, x^{2} - 9651790 \, x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{995328} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 79, normalized size = 0.63 \[ -\frac {1}{580608} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, x - 65\right )} x - 3893\right )} x - 61085\right )} x - 962953\right )} x - 4825895\right )} x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{497664} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 102, normalized size = 0.81 \[ -\frac {175 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{497664}+\frac {35 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{216}-\frac {175 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{10368}+\frac {175 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{82944}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.21, size = 130, normalized size = 1.03 \[ -\frac {1}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {35}{36} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {175}{216} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {175}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {875}{10368} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {175}{13824} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {175}{497664} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {875}{82944} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- 96 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 165 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 113 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 15 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 9 x^{5} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 20 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________