3.1384 \(\int (5-x) (3+2 x) (2+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=88 \[ \frac{1}{12} (4-x) \left (3 x^2+2\right )^{7/2}+\frac{91}{36} x \left (3 x^2+2\right )^{5/2}+\frac{455}{72} x \left (3 x^2+2\right )^{3/2}+\frac{455}{24} x \sqrt{3 x^2+2}+\frac{455 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{12 \sqrt{3}} \]

[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])

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Rubi [A]  time = 0.0250976, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {780, 195, 215} \[ \frac{1}{12} (4-x) \left (3 x^2+2\right )^{7/2}+\frac{91}{36} x \left (3 x^2+2\right )^{5/2}+\frac{455}{72} x \left (3 x^2+2\right )^{3/2}+\frac{455}{24} x \sqrt{3 x^2+2}+\frac{455 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{12 \sqrt{3}} \]

Antiderivative was successfully verified.

[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 780

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]

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]

Rubi steps

\begin{align*} \int (5-x) (3+2 x) \left (2+3 x^2\right )^{5/2} \, dx &=\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]  time = 0.0604959, size = 70, normalized size = 0.8 \[ \frac{1}{72} \left (910 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-3 \sqrt{3 x^2+2} \left (54 x^7-216 x^6-438 x^5-432 x^4-1111 x^3-288 x^2-985 x-64\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*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) + 910*Sqrt[3]*Ar
cSinh[Sqrt[3/2]*x])/72

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Maple [A]  time = 0.005, size = 73, normalized size = 0.8 \begin{align*} -{\frac{x}{12} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{91\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{455\,x}{72} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{455\,x}{24}\sqrt{3\,{x}^{2}+2}}+{\frac{455\,\sqrt{3}}{36}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{1}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49092, size = 97, normalized size = 1.1 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.74597, size = 212, normalized size = 2.41 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 53.2683, size = 143, normalized size = 1.62 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.15884, size = 85, normalized size = 0.97 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))