Optimal. Leaf size=234 \[ \frac{(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac{(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac{(17833 x+21492) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac{5083 \sqrt{3 x^2+5 x+2}}{247500 \sqrt{2 x+3}}-\frac{9421 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{231000 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{5083 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{165000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
[Out]
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Rubi [A] time = 0.485429, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac{(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac{(17833 x+21492) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac{5083 \sqrt{3 x^2+5 x+2}}{247500 \sqrt{2 x+3}}-\frac{9421 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{231000 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{5083 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{165000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]
[Out]
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Rubi in Sympy [A] time = 68.0175, size = 219, normalized size = 0.94 \[ \frac{\left (- 429 x + 949\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{90090 \left (2 x + 3\right )^{\frac{9}{2}}} + \frac{5083 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{495000 \sqrt{3 x^{2} + 5 x + 2}} - \frac{9421 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{693000 \sqrt{3 x^{2} + 5 x + 2}} - \frac{5083 \sqrt{3 x^{2} + 5 x + 2}}{247500 \sqrt{2 x + 3}} + \frac{\left (231829 x + 279396\right ) \sqrt{3 x^{2} + 5 x + 2}}{4504500 \left (2 x + 3\right )^{\frac{5}{2}}} + \frac{\left (585 x + 520\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{715 \left (2 x + 3\right )^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**(15/2),x)
[Out]
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Mathematica [A] time = 0.801482, size = 232, normalized size = 0.99 \[ -\frac{8 \left (3 x^2+5 x+2\right ) \left (2277184 x^6+6409516 x^5+12953760 x^4+33648370 x^3+54318160 x^2+41339721 x+11865789\right )-4 (2 x+3)^6 \left (71162 \left (3 x^2+5 x+2\right )-7318 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+35581 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )}{13860000 (2 x+3)^{13/2} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]
[Out]
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Maple [B] time = 0.054, size = 710, normalized size = 3. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5-x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^(15/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \sqrt{2 \, x + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5-x)*(3*x**2+5*x+2)**(5/2)/(3+2*x)**(15/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2),x, algorithm="giac")
[Out]