\(\int (5-x) (3+2 x) (2+3 x^2)^{5/2} \, dx\) [1384]

Optimal result

Integrand size = 22, antiderivative size = 88 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=\frac {455}{24} x \sqrt {2+3 x^2}+\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \]

[Out]

455/72*x*(3*x^2+2)^(3/2)+91/36*x*(3*x^2+2)^(5/2)+1/12*(4-x)*(3*x^2+2)^(7/2)+455/36*arcsinh(1/2*x*6^(1/2))*3^(1
/2)+455/24*x*(3*x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {794, 201, 221} \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=\frac {455 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}}+\frac {1}{12} (4-x) \left (3 x^2+2\right )^{7/2}+\frac {91}{36} x \left (3 x^2+2\right )^{5/2}+\frac {455}{72} x \left (3 x^2+2\right )^{3/2}+\frac {455}{24} x \sqrt {3 x^2+2} \]

[In]

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

[Out]

(455*x*Sqrt[2 + 3*x^2])/24 + (455*x*(2 + 3*x^2)^(3/2))/72 + (91*x*(2 + 3*x^2)^(5/2))/36 + ((4 - x)*(2 + 3*x^2)
^(7/2))/12 + (455*ArcSinh[Sqrt[3/2]*x])/(12*Sqrt[3])

Rule 201

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 221

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {91}{6} \int \left (2+3 x^2\right )^{5/2} \, dx \\ & = \frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{18} \int \left (2+3 x^2\right )^{3/2} \, dx \\ & = \frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{12} \int \sqrt {2+3 x^2} \, dx \\ & = \frac {455}{24} x \sqrt {2+3 x^2}+\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455}{12} \int \frac {1}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {455}{24} x \sqrt {2+3 x^2}+\frac {455}{72} x \left (2+3 x^2\right )^{3/2}+\frac {91}{36} x \left (2+3 x^2\right )^{5/2}+\frac {1}{12} (4-x) \left (2+3 x^2\right )^{7/2}+\frac {455 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{12 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.92 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{24} \sqrt {2+3 x^2} \left (-64-985 x-288 x^2-1111 x^3-432 x^4-438 x^5-216 x^6+54 x^7\right )-\frac {455 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{12 \sqrt {3}} \]

[In]

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

[Out]

-1/24*(Sqrt[2 + 3*x^2]*(-64 - 985*x - 288*x^2 - 1111*x^3 - 432*x^4 - 438*x^5 - 216*x^6 + 54*x^7)) - (455*Log[-
(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(12*Sqrt[3])

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {\left (54 x^{7}-216 x^{6}-438 x^{5}-432 x^{4}-1111 x^{3}-288 x^{2}-985 x -64\right ) \sqrt {3 x^{2}+2}}{24}+\frac {455 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{36}\)	\(60\)
default	\(\frac {91 x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{36}+\frac {455 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{72}+\frac {455 x \sqrt {3 x^{2}+2}}{24}+\frac {455 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{36}+\frac {\left (3 x^{2}+2\right )^{\frac {7}{2}}}{3}-\frac {x \left (3 x^{2}+2\right )^{\frac {7}{2}}}{12}\)	\(73\)
trager	\(\left (-\frac {9}{4} x^{7}+9 x^{6}+\frac {73}{4} x^{5}+18 x^{4}+\frac {1111}{24} x^{3}+12 x^{2}+\frac {985}{24} x +\frac {8}{3}\right ) \sqrt {3 x^{2}+2}-\frac {455 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{36}\)	\(77\)
meijerg	\(-\frac {75 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (\frac {3}{8} x^{4}+\frac {13}{16} x^{2}+\frac {11}{16}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{3}\right )}{2 \sqrt {\pi }}-\frac {35 \sqrt {2}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {8 \sqrt {\pi }\, \left (\frac {27}{4} x^{6}+\frac {27}{2} x^{4}+9 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{105}\right )}{2 \sqrt {\pi }}+\frac {10 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (162 x^{6}+306 x^{4}+177 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{720}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{24}\right )}{3 \sqrt {\pi }}\)	\(163\)

[In]

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

[Out]

-1/24*(54*x^7-216*x^6-438*x^5-432*x^4-1111*x^3-288*x^2-985*x-64)*(3*x^2+2)^(1/2)+455/36*arcsinh(1/2*x*6^(1/2))
*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{24} \, {\left (54 \, x^{7} - 216 \, x^{6} - 438 \, x^{5} - 432 \, x^{4} - 1111 \, x^{3} - 288 \, x^{2} - 985 \, x - 64\right )} \sqrt {3 \, x^{2} + 2} + \frac {455}{72} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]

[In]

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

[Out]

-1/24*(54*x^7 - 216*x^6 - 438*x^5 - 432*x^4 - 1111*x^3 - 288*x^2 - 985*x - 64)*sqrt(3*x^2 + 2) + 455/72*sqrt(3
)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

Sympy [A] (verification not implemented)

Time = 2.87 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.62 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=- \frac {9 x^{7} \sqrt {3 x^{2} + 2}}{4} + 9 x^{6} \sqrt {3 x^{2} + 2} + \frac {73 x^{5} \sqrt {3 x^{2} + 2}}{4} + 18 x^{4} \sqrt {3 x^{2} + 2} + \frac {1111 x^{3} \sqrt {3 x^{2} + 2}}{24} + 12 x^{2} \sqrt {3 x^{2} + 2} + \frac {985 x \sqrt {3 x^{2} + 2}}{24} + \frac {8 \sqrt {3 x^{2} + 2}}{3} + \frac {455 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{36} \]

[In]

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

[Out]

-9*x**7*sqrt(3*x**2 + 2)/4 + 9*x**6*sqrt(3*x**2 + 2) + 73*x**5*sqrt(3*x**2 + 2)/4 + 18*x**4*sqrt(3*x**2 + 2) +
1111*x**3*sqrt(3*x**2 + 2)/24 + 12*x**2*sqrt(3*x**2 + 2) + 985*x*sqrt(3*x**2 + 2)/24 + 8*sqrt(3*x**2 + 2)/3 +
455*sqrt(3)*asinh(sqrt(6)*x/2)/36

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {1}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {91}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {455}{72} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {455}{24} \, \sqrt {3 \, x^{2} + 2} x + \frac {455}{36} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]

[In]

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

[Out]

-1/12*(3*x^2 + 2)^(7/2)*x + 1/3*(3*x^2 + 2)^(7/2) + 91/36*(3*x^2 + 2)^(5/2)*x + 455/72*(3*x^2 + 2)^(3/2)*x + 4
55/24*sqrt(3*x^2 + 2)*x + 455/36*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{24} \, {\left ({\left ({\left ({\left (6 \, {\left ({\left (9 \, {\left (x - 4\right )} x - 73\right )} x - 72\right )} x - 1111\right )} x - 288\right )} x - 985\right )} x - 64\right )} \sqrt {3 \, x^{2} + 2} - \frac {455}{36} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]

[In]

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

[Out]

-1/24*((((6*((9*(x - 4)*x - 73)*x - 72)*x - 1111)*x - 288)*x - 985)*x - 64)*sqrt(3*x^2 + 2) - 455/36*sqrt(3)*l
og(-sqrt(3)*x + sqrt(3*x^2 + 2))

Mupad [B] (verification not implemented)

Time = 10.58 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx=\frac {455\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{36}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {27\,x^7}{4}+27\,x^6+\frac {219\,x^5}{4}+54\,x^4+\frac {1111\,x^3}{8}+36\,x^2+\frac {985\,x}{8}+8\right )}{3} \]

[In]

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

[Out]

(455*3^(1/2)*asinh((6^(1/2)*x)/2))/36 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((985*x)/8 + 36*x^2 + (1111*x^3)/8 + 54*x^4
+ (219*x^5)/4 + 27*x^6 - (27*x^7)/4 + 8))/3