Optimal. Leaf size=256 \[ -\frac {2}{45} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {202}{351} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {13318 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac {\sqrt {2 x+3} (629153 x+534271) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac {\sqrt {2 x+3} (7817373 x+6006884) \sqrt {3 x^2+5 x+2}}{21891870}+\frac {1015187 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {207851 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ -\frac {2}{45} (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {202}{351} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac {13318 \sqrt {2 x+3} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac {\sqrt {2 x+3} (629153 x+534271) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac {\sqrt {2 x+3} (7817373 x+6006884) \sqrt {3 x^2+5 x+2}}{21891870}+\frac {1015187 \sqrt {-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {3 x^2+5 x+2}}-\frac {207851 \sqrt {-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {x+1}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 419
Rule 424
Rule 718
Rule 814
Rule 832
Rule 843
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2}{45} \int (3+2 x)^{3/2} \left (385+\frac {505 x}{2}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {4 \int \sqrt {3+2 x} \left (\frac {46155}{4}+\frac {33295 x}{4}\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx}{1755}\\ &=\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {8 \int \frac {\left (242380+\frac {1348185 x}{8}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt {3+2 x}} \, dx}{57915}\\ &=\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {4 \int \frac {\left (\frac {17298645}{8}+\frac {13028955 x}{8}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {3+2 x}} \, dx}{3648645}\\ &=-\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}+\frac {2 \int \frac {\frac {1333245}{2}-\frac {21824355 x}{8}}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{164189025}\\ &=-\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {207851 \int \frac {\sqrt {3+2 x}}{\sqrt {2+5 x+3 x^2}} \, dx}{12509640}+\frac {1015187 \int \frac {1}{\sqrt {3+2 x} \sqrt {2+5 x+3 x^2}} \, dx}{17513496}\\ &=-\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {\left (207851 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {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 )}{6254820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {\left (1015187 \sqrt {-2-5 x-3 x^2}\right ) \operatorname {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 )}{8756748 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ &=-\frac {\sqrt {3+2 x} (6006884+7817373 x) \sqrt {2+5 x+3 x^2}}{21891870}+\frac {\sqrt {3+2 x} (534271+629153 x) \left (2+5 x+3 x^2\right )^{3/2}}{243243}+\frac {13318 \sqrt {3+2 x} \left (2+5 x+3 x^2\right )^{5/2}}{11583}+\frac {202}{351} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {2}{45} (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}-\frac {207851 \sqrt {-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{6254820 \sqrt {3} \sqrt {2+5 x+3 x^2}}+\frac {1015187 \sqrt {-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt {3} \sqrt {1+x}\right )|-\frac {2}{3}\right )}{8756748 \sqrt {3} \sqrt {2+5 x+3 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.35, size = 218, normalized size = 0.85 \[ -\frac {2 \left (630485856 x^9+1907623872 x^8-11776907520 x^7-82311172272 x^6-217661096106 x^5-319887585072 x^4-283276026729 x^3-150475882830 x^2-44206631441 x-5523159638\right ) \sqrt {2 x+3}+1590604 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )+1454957 \sqrt {5} \sqrt {\frac {x+1}{2 x+3}} \sqrt {\frac {3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{3}}}{\sqrt {2 x+3}}\right )|\frac {3}{5}\right )}{131351220 (2 x+3) \sqrt {3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (12 \, x^{5} - 4 \, x^{4} - 185 \, x^{3} - 406 \, x^{2} - 327 \, x - 90\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {2 \, x + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 166, normalized size = 0.65 \[ \frac {\sqrt {2 x +3}\, \sqrt {3 x^{2}+5 x +2}\, \left (-12609717120 x^{9}-38152477440 x^{8}+235538150400 x^{7}+1646223445440 x^{6}+4353221922120 x^{5}+6397751701440 x^{4}+5665520534580 x^{3}+3009604954020 x^{2}+884278124520 x +1454957 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticE \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+3620978 \sqrt {2 x +3}\, \sqrt {15}\, \sqrt {-2 x -2}\, \sqrt {-30 x -20}\, \EllipticF \left (\frac {\sqrt {30 x +45}}{5}, \frac {\sqrt {15}}{3}\right )+110521391040\right )}{7881073200 x^{3}+24956731800 x^{2}+24956731800 x +7881073200} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x + 3\right )}^{\frac {5}{2}} {\left (x - 5\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int {\left (2\,x+3\right )}^{5/2}\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- 90 \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 327 x \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 406 x^{2} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 185 x^{3} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 4 x^{4} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 12 x^{5} \sqrt {2 x + 3} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________