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

Optimal. Leaf size=94 \[ -\frac {1}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}+\frac {2}{315} (160 x+611) \left (3 x^2+2\right )^{5/2}+\frac {397}{36} x \left (3 x^2+2\right )^{3/2}+\frac {397}{12} x \sqrt {3 x^2+2}+\frac {397 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \begin {gather*} -\frac {1}{21} (2 x+3)^2 \left (3 x^2+2\right )^{5/2}+\frac {2}{315} (160 x+611) \left (3 x^2+2\right )^{5/2}+\frac {397}{36} x \left (3 x^2+2\right )^{3/2}+\frac {397}{12} x \sqrt {3 x^2+2}+\frac {397 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(397*x*Sqrt[2 + 3*x^2])/12 + (397*x*(2 + 3*x^2)^(3/2))/36 - ((3 + 2*x)^2*(2 + 3*x^2)^(5/2))/21 + (2*(611 + 160
*x)*(2 + 3*x^2)^(5/2))/315 + (397*ArcSinh[Sqrt[3/2]*x])/(6*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 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 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rubi steps

\begin {align*} \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{3/2} \, dx &=-\frac {1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac {1}{21} \int (3+2 x) (323+192 x) \left (2+3 x^2\right )^{3/2} \, dx\\ &=-\frac {1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac {2}{315} (611+160 x) \left (2+3 x^2\right )^{5/2}+\frac {397}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {397}{36} x \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac {2}{315} (611+160 x) \left (2+3 x^2\right )^{5/2}+\frac {397}{6} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {397}{12} x \sqrt {2+3 x^2}+\frac {397}{36} x \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac {2}{315} (611+160 x) \left (2+3 x^2\right )^{5/2}+\frac {397}{6} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {397}{12} x \sqrt {2+3 x^2}+\frac {397}{36} x \left (2+3 x^2\right )^{3/2}-\frac {1}{21} (3+2 x)^2 \left (2+3 x^2\right )^{5/2}+\frac {2}{315} (611+160 x) \left (2+3 x^2\right )^{5/2}+\frac {397 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 65, normalized size = 0.69 \begin {gather*} \frac {27790 \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\sqrt {3 x^2+2} \left (2160 x^6-5040 x^5-36252 x^4-48405 x^3-51216 x^2-71715 x-17392\right )}{1260} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-17392 - 71715*x - 51216*x^2 - 48405*x^3 - 36252*x^4 - 5040*x^5 + 2160*x^6)) + 27790*Sqrt[
3]*ArcSinh[Sqrt[3/2]*x])/1260

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IntegrateAlgebraic [A]  time = 0.31, size = 76, normalized size = 0.81 \begin {gather*} \frac {\sqrt {3 x^2+2} \left (-2160 x^6+5040 x^5+36252 x^4+48405 x^3+51216 x^2+71715 x+17392\right )}{1260}-\frac {397 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{6 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(17392 + 71715*x + 51216*x^2 + 48405*x^3 + 36252*x^4 + 5040*x^5 - 2160*x^6))/1260 - (397*Log[
-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(6*Sqrt[3])

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fricas [A]  time = 0.42, size = 70, normalized size = 0.74 \begin {gather*} -\frac {1}{1260} \, {\left (2160 \, x^{6} - 5040 \, x^{5} - 36252 \, x^{4} - 48405 \, x^{3} - 51216 \, x^{2} - 71715 \, x - 17392\right )} \sqrt {3 \, x^{2} + 2} + \frac {397}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1260*(2160*x^6 - 5040*x^5 - 36252*x^4 - 48405*x^3 - 51216*x^2 - 71715*x - 17392)*sqrt(3*x^2 + 2) + 397/36*s
qrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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giac [A]  time = 0.17, size = 62, normalized size = 0.66 \begin {gather*} -\frac {1}{1260} \, {\left (3 \, {\left ({\left ({\left (12 \, {\left (20 \, {\left (3 \, x - 7\right )} x - 1007\right )} x - 16135\right )} x - 17072\right )} x - 23905\right )} x - 17392\right )} \sqrt {3 \, x^{2} + 2} - \frac {397}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1260*(3*(((12*(20*(3*x - 7)*x - 1007)*x - 16135)*x - 17072)*x - 23905)*x - 17392)*sqrt(3*x^2 + 2) - 397/18*
sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))

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maple [A]  time = 0.06, size = 75, normalized size = 0.80 \begin {gather*} -\frac {4 \left (3 x^{2}+2\right )^{\frac {5}{2}} x^{2}}{21}+\frac {4 \left (3 x^{2}+2\right )^{\frac {5}{2}} x}{9}+\frac {397 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{36}+\frac {397 \sqrt {3 x^{2}+2}\, x}{12}+\frac {397 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{18}+\frac {1087 \left (3 x^{2}+2\right )^{\frac {5}{2}}}{315} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/21*(3*x^2+2)^(5/2)*x^2+1087/315*(3*x^2+2)^(5/2)+4/9*(3*x^2+2)^(5/2)*x+397/36*(3*x^2+2)^(3/2)*x+397/12*(3*x^
2+2)^(1/2)*x+397/18*arcsinh(1/2*6^(1/2)*x)*3^(1/2)

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maxima [A]  time = 1.33, size = 74, normalized size = 0.79 \begin {gather*} -\frac {4}{21} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x^{2} + \frac {4}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {1087}{315} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {397}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {397}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {397}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/21*(3*x^2 + 2)^(5/2)*x^2 + 4/9*(3*x^2 + 2)^(5/2)*x + 1087/315*(3*x^2 + 2)^(5/2) + 397/36*(3*x^2 + 2)^(3/2)*
x + 397/12*sqrt(3*x^2 + 2)*x + 397/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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mupad [B]  time = 0.04, size = 55, normalized size = 0.59 \begin {gather*} \frac {397\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {36\,x^6}{7}+12\,x^5+\frac {3021\,x^4}{35}+\frac {461\,x^3}{4}+\frac {4268\,x^2}{35}+\frac {683\,x}{4}+\frac {4348}{105}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(397*3^(1/2)*asinh((6^(1/2)*x)/2))/18 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((683*x)/4 + (4268*x^2)/35 + (461*x^3)/4 +
(3021*x^4)/35 + 12*x^5 - (36*x^6)/7 + 4348/105))/3

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sympy [A]  time = 8.16, size = 129, normalized size = 1.37 \begin {gather*} - \frac {12 x^{6} \sqrt {3 x^{2} + 2}}{7} + 4 x^{5} \sqrt {3 x^{2} + 2} + \frac {1007 x^{4} \sqrt {3 x^{2} + 2}}{35} + \frac {461 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {4268 x^{2} \sqrt {3 x^{2} + 2}}{105} + \frac {683 x \sqrt {3 x^{2} + 2}}{12} + \frac {4348 \sqrt {3 x^{2} + 2}}{315} + \frac {397 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12*x**6*sqrt(3*x**2 + 2)/7 + 4*x**5*sqrt(3*x**2 + 2) + 1007*x**4*sqrt(3*x**2 + 2)/35 + 461*x**3*sqrt(3*x**2 +
2)/12 + 4268*x**2*sqrt(3*x**2 + 2)/105 + 683*x*sqrt(3*x**2 + 2)/12 + 4348*sqrt(3*x**2 + 2)/315 + 397*sqrt(3)*
asinh(sqrt(6)*x/2)/18

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