3.1385 \(\int (5-x) (2+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=83 \[ -\frac {1}{21} \left (3 x^2+2\right )^{7/2}+\frac {5}{6} x \left (3 x^2+2\right )^{5/2}+\frac {25}{12} x \left (3 x^2+2\right )^{3/2}+\frac {25}{4} x \sqrt {3 x^2+2}+\frac {25 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}} \]

[Out]

25/12*x*(3*x^2+2)^(3/2)+5/6*x*(3*x^2+2)^(5/2)-1/21*(3*x^2+2)^(7/2)+25/6*arcsinh(1/2*x*6^(1/2))*3^(1/2)+25/4*x*
(3*x^2+2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {641, 195, 215} \[ -\frac {1}{21} \left (3 x^2+2\right )^{7/2}+\frac {5}{6} x \left (3 x^2+2\right )^{5/2}+\frac {25}{12} x \left (3 x^2+2\right )^{3/2}+\frac {25}{4} x \sqrt {3 x^2+2}+\frac {25 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(2 + 3*x^2)^(5/2),x]

[Out]

(25*x*Sqrt[2 + 3*x^2])/4 + (25*x*(2 + 3*x^2)^(3/2))/12 + (5*x*(2 + 3*x^2)^(5/2))/6 - (2 + 3*x^2)^(7/2)/21 + (2
5*ArcSinh[Sqrt[3/2]*x])/(2*Sqrt[3])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rubi steps

\begin {align*} \int (5-x) \left (2+3 x^2\right )^{5/2} \, dx &=-\frac {1}{21} \left (2+3 x^2\right )^{7/2}+5 \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {5}{6} x \left (2+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+3 x^2\right )^{7/2}+\frac {25}{3} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {25}{12} x \left (2+3 x^2\right )^{3/2}+\frac {5}{6} x \left (2+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+3 x^2\right )^{7/2}+\frac {25}{2} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {25}{4} x \sqrt {2+3 x^2}+\frac {25}{12} x \left (2+3 x^2\right )^{3/2}+\frac {5}{6} x \left (2+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+3 x^2\right )^{7/2}+\frac {25}{2} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {25}{4} x \sqrt {2+3 x^2}+\frac {25}{12} x \left (2+3 x^2\right )^{3/2}+\frac {5}{6} x \left (2+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+3 x^2\right )^{7/2}+\frac {25 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 65, normalized size = 0.78 \[ \frac {1}{84} \left (350 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (108 x^6-630 x^5+216 x^4-1365 x^3+144 x^2-1155 x+32\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(2 + 3*x^2)^(5/2),x]

[Out]

(-(Sqrt[2 + 3*x^2]*(32 - 1155*x + 144*x^2 - 1365*x^3 + 216*x^4 - 630*x^5 + 108*x^6)) + 350*Sqrt[3]*ArcSinh[Sqr
t[3/2]*x])/84

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 70, normalized size = 0.84 \[ -\frac {1}{84} \, {\left (108 \, x^{6} - 630 \, x^{5} + 216 \, x^{4} - 1365 \, x^{3} + 144 \, x^{2} - 1155 \, x + 32\right )} \sqrt {3 \, x^{2} + 2} + \frac {25}{12} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

-1/84*(108*x^6 - 630*x^5 + 216*x^4 - 1365*x^3 + 144*x^2 - 1155*x + 32)*sqrt(3*x^2 + 2) + 25/12*sqrt(3)*log(-sq
rt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 61, normalized size = 0.73 \[ -\frac {1}{84} \, {\left (3 \, {\left ({\left ({\left (6 \, {\left ({\left (6 \, x - 35\right )} x + 12\right )} x - 455\right )} x + 48\right )} x - 385\right )} x + 32\right )} \sqrt {3 \, x^{2} + 2} - \frac {25}{6} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

-1/84*(3*(((6*((6*x - 35)*x + 12)*x - 455)*x + 48)*x - 385)*x + 32)*sqrt(3*x^2 + 2) - 25/6*sqrt(3)*log(-sqrt(3
)*x + sqrt(3*x^2 + 2))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 61, normalized size = 0.73 \[ \frac {25 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{12}+\frac {5 \left (3 x^{2}+2\right )^{\frac {5}{2}} x}{6}+\frac {25 \sqrt {3 x^{2}+2}\, x}{4}+\frac {25 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{6}-\frac {\left (3 x^{2}+2\right )^{\frac {7}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3*x^2+2)^(5/2),x)

[Out]

25/12*(3*x^2+2)^(3/2)*x+5/6*(3*x^2+2)^(5/2)*x-1/21*(3*x^2+2)^(7/2)+25/6*arcsinh(1/2*6^(1/2)*x)*3^(1/2)+25/4*(3
*x^2+2)^(1/2)*x

________________________________________________________________________________________

maxima [A]  time = 1.13, size = 60, normalized size = 0.72 \[ -\frac {1}{21} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {5}{6} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {25}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {25}{4} \, \sqrt {3 \, x^{2} + 2} x + \frac {25}{6} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

-1/21*(3*x^2 + 2)^(7/2) + 5/6*(3*x^2 + 2)^(5/2)*x + 25/12*(3*x^2 + 2)^(3/2)*x + 25/4*sqrt(3*x^2 + 2)*x + 25/6*
sqrt(3)*arcsinh(1/2*sqrt(6)*x)

________________________________________________________________________________________

mupad [B]  time = 1.74, size = 55, normalized size = 0.66 \[ \frac {25\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{6}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {27\,x^6}{7}-\frac {45\,x^5}{2}+\frac {54\,x^4}{7}-\frac {195\,x^3}{4}+\frac {36\,x^2}{7}-\frac {165\,x}{4}+\frac {8}{7}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^2 + 2)^(5/2)*(x - 5),x)

[Out]

(25*3^(1/2)*asinh((6^(1/2)*x)/2))/6 - (3^(1/2)*(x^2 + 2/3)^(1/2)*((36*x^2)/7 - (165*x)/4 - (195*x^3)/4 + (54*x
^4)/7 - (45*x^5)/2 + (27*x^6)/7 + 8/7))/3

________________________________________________________________________________________

sympy [A]  time = 20.99, size = 131, normalized size = 1.58 \[ - \frac {9 x^{6} \sqrt {3 x^{2} + 2}}{7} + \frac {15 x^{5} \sqrt {3 x^{2} + 2}}{2} - \frac {18 x^{4} \sqrt {3 x^{2} + 2}}{7} + \frac {65 x^{3} \sqrt {3 x^{2} + 2}}{4} - \frac {12 x^{2} \sqrt {3 x^{2} + 2}}{7} + \frac {55 x \sqrt {3 x^{2} + 2}}{4} - \frac {8 \sqrt {3 x^{2} + 2}}{21} + \frac {25 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3*x**2+2)**(5/2),x)

[Out]

-9*x**6*sqrt(3*x**2 + 2)/7 + 15*x**5*sqrt(3*x**2 + 2)/2 - 18*x**4*sqrt(3*x**2 + 2)/7 + 65*x**3*sqrt(3*x**2 + 2
)/4 - 12*x**2*sqrt(3*x**2 + 2)/7 + 55*x*sqrt(3*x**2 + 2)/4 - 8*sqrt(3*x**2 + 2)/21 + 25*sqrt(3)*asinh(sqrt(6)*
x/2)/6

________________________________________________________________________________________