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

Optimal. Leaf size=110 \[ -\frac {1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac {1}{81} (63 x+226) \left (3 x^2+2\right )^{7/2}+\frac {133}{18} x \left (3 x^2+2\right )^{5/2}+\frac {665}{36} x \left (3 x^2+2\right )^{3/2}+\frac {665}{12} x \sqrt {3 x^2+2}+\frac {665 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, 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}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac {1}{81} (63 x+226) \left (3 x^2+2\right )^{7/2}+\frac {133}{18} x \left (3 x^2+2\right )^{5/2}+\frac {665}{36} x \left (3 x^2+2\right )^{3/2}+\frac {665}{12} x \sqrt {3 x^2+2}+\frac {665 \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)^(5/2),x]

[Out]

(665*x*Sqrt[2 + 3*x^2])/12 + (665*x*(2 + 3*x^2)^(3/2))/36 + (133*x*(2 + 3*x^2)^(5/2))/18 - ((3 + 2*x)^2*(2 + 3
*x^2)^(7/2))/27 + ((226 + 63*x)*(2 + 3*x^2)^(7/2))/81 + (665*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 )^{5/2} \, dx &=-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{27} \int (3+2 x) (413+252 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {133}{3} \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {665}{36} x \left (2+3 x^2\right )^{3/2}+\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665}{6} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {665}{12} x \sqrt {2+3 x^2}+\frac {665}{36} x \left (2+3 x^2\right )^{3/2}+\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665}{6} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {665}{12} x \sqrt {2+3 x^2}+\frac {665}{36} x \left (2+3 x^2\right )^{3/2}+\frac {133}{18} x \left (2+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac {1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac {665 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 75, normalized size = 0.68 \begin {gather*} \frac {1}{324} \sqrt {3 x^2+2} \left (-1296 x^8+2916 x^7+18900 x^6+27378 x^5+41256 x^4+50571 x^3+28272 x^2+40365 x+6368\right )+\frac {665 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(6368 + 40365*x + 28272*x^2 + 50571*x^3 + 41256*x^4 + 27378*x^5 + 18900*x^6 + 2916*x^7 - 1296
*x^8))/324 + (665*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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IntegrateAlgebraic [A]  time = 0.38, size = 86, normalized size = 0.78 \begin {gather*} \frac {1}{324} \sqrt {3 x^2+2} \left (-1296 x^8+2916 x^7+18900 x^6+27378 x^5+41256 x^4+50571 x^3+28272 x^2+40365 x+6368\right )-\frac {665 \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)^(5/2),x]

[Out]

(Sqrt[2 + 3*x^2]*(6368 + 40365*x + 28272*x^2 + 50571*x^3 + 41256*x^4 + 27378*x^5 + 18900*x^6 + 2916*x^7 - 1296
*x^8))/324 - (665*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(6*Sqrt[3])

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fricas [A]  time = 0.43, size = 80, normalized size = 0.73 \begin {gather*} -\frac {1}{324} \, {\left (1296 \, x^{8} - 2916 \, x^{7} - 18900 \, x^{6} - 27378 \, x^{5} - 41256 \, x^{4} - 50571 \, x^{3} - 28272 \, x^{2} - 40365 \, x - 6368\right )} \sqrt {3 \, x^{2} + 2} + \frac {665}{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)^(5/2),x, algorithm="fricas")

[Out]

-1/324*(1296*x^8 - 2916*x^7 - 18900*x^6 - 27378*x^5 - 41256*x^4 - 50571*x^3 - 28272*x^2 - 40365*x - 6368)*sqrt
(3*x^2 + 2) + 665/36*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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giac [A]  time = 0.19, size = 72, normalized size = 0.65 \begin {gather*} -\frac {1}{324} \, {\left (3 \, {\left ({\left (9 \, {\left (2 \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 9\right )} x - 175\right )} x - 507\right )} x - 764\right )} x - 1873\right )} x - 9424\right )} x - 13455\right )} x - 6368\right )} \sqrt {3 \, x^{2} + 2} - \frac {665}{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)^(5/2),x, algorithm="giac")

[Out]

-1/324*(3*((9*(2*((2*(3*(4*x - 9)*x - 175)*x - 507)*x - 764)*x - 1873)*x - 9424)*x - 13455)*x - 6368)*sqrt(3*x
^2 + 2) - 665/18*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))

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maple [A]  time = 0.05, size = 87, normalized size = 0.79 \begin {gather*} -\frac {4 \left (3 x^{2}+2\right )^{\frac {7}{2}} x^{2}}{27}+\frac {\left (3 x^{2}+2\right )^{\frac {7}{2}} x}{3}+\frac {133 \left (3 x^{2}+2\right )^{\frac {5}{2}} x}{18}+\frac {665 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{36}+\frac {665 \sqrt {3 x^{2}+2}\, x}{12}+\frac {665 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{18}+\frac {199 \left (3 x^{2}+2\right )^{\frac {7}{2}}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/27*(3*x^2+2)^(7/2)*x^2+199/81*(3*x^2+2)^(7/2)+1/3*(3*x^2+2)^(7/2)*x+133/18*(3*x^2+2)^(5/2)*x+665/36*(3*x^2+
2)^(3/2)*x+665/12*(3*x^2+2)^(1/2)*x+665/18*arcsinh(1/2*6^(1/2)*x)*3^(1/2)

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maxima [A]  time = 1.32, size = 86, normalized size = 0.78 \begin {gather*} -\frac {4}{27} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{2} + \frac {1}{3} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {199}{81} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {133}{18} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {665}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {665}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {665}{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)^(5/2),x, algorithm="maxima")

[Out]

-4/27*(3*x^2 + 2)^(7/2)*x^2 + 1/3*(3*x^2 + 2)^(7/2)*x + 199/81*(3*x^2 + 2)^(7/2) + 133/18*(3*x^2 + 2)^(5/2)*x
+ 665/36*(3*x^2 + 2)^(3/2)*x + 665/12*sqrt(3*x^2 + 2)*x + 665/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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mupad [B]  time = 0.06, size = 65, normalized size = 0.59 \begin {gather*} \frac {665\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-12\,x^8+27\,x^7+175\,x^6+\frac {507\,x^5}{2}+382\,x^4+\frac {1873\,x^3}{4}+\frac {2356\,x^2}{9}+\frac {1495\,x}{4}+\frac {1592}{27}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(665*3^(1/2)*asinh((6^(1/2)*x)/2))/18 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((1495*x)/4 + (2356*x^2)/9 + (1873*x^3)/4 +
382*x^4 + (507*x^5)/2 + 175*x^6 + 27*x^7 - 12*x^8 + 1592/27))/3

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sympy [A]  time = 50.47, size = 162, normalized size = 1.47 \begin {gather*} - 4 x^{8} \sqrt {3 x^{2} + 2} + 9 x^{7} \sqrt {3 x^{2} + 2} + \frac {175 x^{6} \sqrt {3 x^{2} + 2}}{3} + \frac {169 x^{5} \sqrt {3 x^{2} + 2}}{2} + \frac {382 x^{4} \sqrt {3 x^{2} + 2}}{3} + \frac {1873 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {2356 x^{2} \sqrt {3 x^{2} + 2}}{27} + \frac {1495 x \sqrt {3 x^{2} + 2}}{12} + \frac {1592 \sqrt {3 x^{2} + 2}}{81} + \frac {665 \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)**(5/2),x)

[Out]

-4*x**8*sqrt(3*x**2 + 2) + 9*x**7*sqrt(3*x**2 + 2) + 175*x**6*sqrt(3*x**2 + 2)/3 + 169*x**5*sqrt(3*x**2 + 2)/2
+ 382*x**4*sqrt(3*x**2 + 2)/3 + 1873*x**3*sqrt(3*x**2 + 2)/12 + 2356*x**2*sqrt(3*x**2 + 2)/27 + 1495*x*sqrt(3
*x**2 + 2)/12 + 1592*sqrt(3*x**2 + 2)/81 + 665*sqrt(3)*asinh(sqrt(6)*x/2)/18

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