3.1382 \(\int (5-x) (3+2 x)^3 (2+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=132 \[ -\frac{1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}+\frac{91}{270} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac{(4977 x+15244) \left (3 x^2+2\right )^{7/2}}{1620}+\frac{3731}{180} x \left (3 x^2+2\right )^{5/2}+\frac{3731}{72} x \left (3 x^2+2\right )^{3/2}+\frac{3731}{24} x \sqrt{3 x^2+2}+\frac{3731 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{12 \sqrt{3}} \]

[Out]

(3731*x*Sqrt[2 + 3*x^2])/24 + (3731*x*(2 + 3*x^2)^(3/2))/72 + (3731*x*(2 + 3*x^2)^(5/2))/180 + (91*(3 + 2*x)^2
*(2 + 3*x^2)^(7/2))/270 - ((3 + 2*x)^3*(2 + 3*x^2)^(7/2))/30 + ((15244 + 4977*x)*(2 + 3*x^2)^(7/2))/1620 + (37
31*ArcSinh[Sqrt[3/2]*x])/(12*Sqrt[3])

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Rubi [A]  time = 0.0622385, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac{1}{30} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}+\frac{91}{270} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac{(4977 x+15244) \left (3 x^2+2\right )^{7/2}}{1620}+\frac{3731}{180} x \left (3 x^2+2\right )^{5/2}+\frac{3731}{72} x \left (3 x^2+2\right )^{3/2}+\frac{3731}{24} x \sqrt{3 x^2+2}+\frac{3731 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{12 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3731*x*Sqrt[2 + 3*x^2])/24 + (3731*x*(2 + 3*x^2)^(3/2))/72 + (3731*x*(2 + 3*x^2)^(5/2))/180 + (91*(3 + 2*x)^2
*(2 + 3*x^2)^(7/2))/270 - ((3 + 2*x)^3*(2 + 3*x^2)^(7/2))/30 + ((15244 + 4977*x)*(2 + 3*x^2)^(7/2))/1620 + (37
31*ArcSinh[Sqrt[3/2]*x])/(12*Sqrt[3])

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])

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)^3 \left (2+3 x^2\right )^{5/2} \, dx &=-\frac{1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac{1}{30} \int (3+2 x)^2 (462+273 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac{91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac{1}{810} \int (3+2 x) (35238+29862 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac{91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac{(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac{3731}{30} \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac{3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac{91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac{(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac{3731}{18} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{3731}{72} x \left (2+3 x^2\right )^{3/2}+\frac{3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac{91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac{(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac{3731}{12} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{3731}{24} x \sqrt{2+3 x^2}+\frac{3731}{72} x \left (2+3 x^2\right )^{3/2}+\frac{3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac{91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac{(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac{3731}{12} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{3731}{24} x \sqrt{2+3 x^2}+\frac{3731}{72} x \left (2+3 x^2\right )^{3/2}+\frac{3731}{180} x \left (2+3 x^2\right )^{5/2}+\frac{91}{270} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}-\frac{1}{30} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}+\frac{(15244+4977 x) \left (2+3 x^2\right )^{7/2}}{1620}+\frac{3731 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{12 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0771452, size = 80, normalized size = 0.61 \[ \frac{335790 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\sqrt{3 x^2+2} \left (23328 x^9-12960 x^8-418446 x^7-1035720 x^6-1503522 x^5-2036880 x^4-1922805 x^3-1350240 x^2-1245915 x-299200\right )}{3240} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[2 + 3*x^2]*(-299200 - 1245915*x - 1350240*x^2 - 1922805*x^3 - 2036880*x^4 - 1503522*x^5 - 1035720*x^6
- 418446*x^7 - 12960*x^8 + 23328*x^9)) + 335790*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/3240

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Maple [A]  time = 0.006, size = 101, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{3}}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{319\,x}{60} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{3731\,x}{180} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{3731\,x}{72} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{3731\,x}{24}\sqrt{3\,{x}^{2}+2}}+{\frac{3731\,\sqrt{3}}{36}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{4\,{x}^{2}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{935}{81} \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*(3*x^2+2)^(5/2),x)

[Out]

-4/15*x^3*(3*x^2+2)^(7/2)+319/60*x*(3*x^2+2)^(7/2)+3731/180*x*(3*x^2+2)^(5/2)+3731/72*x*(3*x^2+2)^(3/2)+3731/2
4*x*(3*x^2+2)^(1/2)+3731/36*arcsinh(1/2*x*6^(1/2))*3^(1/2)+4/27*x^2*(3*x^2+2)^(7/2)+935/81*(3*x^2+2)^(7/2)

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Maxima [A]  time = 1.56745, size = 135, normalized size = 1.02 \begin{align*} -\frac{4}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{3} + \frac{4}{27} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{2} + \frac{319}{60} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x + \frac{935}{81} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} + \frac{3731}{180} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{3731}{72} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{3731}{24} \, \sqrt{3 \, x^{2} + 2} x + \frac{3731}{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*(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

-4/15*(3*x^2 + 2)^(7/2)*x^3 + 4/27*(3*x^2 + 2)^(7/2)*x^2 + 319/60*(3*x^2 + 2)^(7/2)*x + 935/81*(3*x^2 + 2)^(7/
2) + 3731/180*(3*x^2 + 2)^(5/2)*x + 3731/72*(3*x^2 + 2)^(3/2)*x + 3731/24*sqrt(3*x^2 + 2)*x + 3731/36*sqrt(3)*
arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.05964, size = 290, normalized size = 2.2 \begin{align*} -\frac{1}{3240} \,{\left (23328 \, x^{9} - 12960 \, x^{8} - 418446 \, x^{7} - 1035720 \, x^{6} - 1503522 \, x^{5} - 2036880 \, x^{4} - 1922805 \, x^{3} - 1350240 \, x^{2} - 1245915 \, x - 299200\right )} \sqrt{3 \, x^{2} + 2} + \frac{3731}{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*(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

-1/3240*(23328*x^9 - 12960*x^8 - 418446*x^7 - 1035720*x^6 - 1503522*x^5 - 2036880*x^4 - 1922805*x^3 - 1350240*
x^2 - 1245915*x - 299200)*sqrt(3*x^2 + 2) + 3731/72*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)

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Sympy [A]  time = 123.963, size = 180, normalized size = 1.36 \begin{align*} - \frac{36 x^{9} \sqrt{3 x^{2} + 2}}{5} + 4 x^{8} \sqrt{3 x^{2} + 2} + \frac{2583 x^{7} \sqrt{3 x^{2} + 2}}{20} + \frac{959 x^{6} \sqrt{3 x^{2} + 2}}{3} + \frac{9281 x^{5} \sqrt{3 x^{2} + 2}}{20} + \frac{1886 x^{4} \sqrt{3 x^{2} + 2}}{3} + \frac{14243 x^{3} \sqrt{3 x^{2} + 2}}{24} + \frac{11252 x^{2} \sqrt{3 x^{2} + 2}}{27} + \frac{9229 x \sqrt{3 x^{2} + 2}}{24} + \frac{7480 \sqrt{3 x^{2} + 2}}{81} + \frac{3731 \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*(3*x**2+2)**(5/2),x)

[Out]

-36*x**9*sqrt(3*x**2 + 2)/5 + 4*x**8*sqrt(3*x**2 + 2) + 2583*x**7*sqrt(3*x**2 + 2)/20 + 959*x**6*sqrt(3*x**2 +
2)/3 + 9281*x**5*sqrt(3*x**2 + 2)/20 + 1886*x**4*sqrt(3*x**2 + 2)/3 + 14243*x**3*sqrt(3*x**2 + 2)/24 + 11252*
x**2*sqrt(3*x**2 + 2)/27 + 9229*x*sqrt(3*x**2 + 2)/24 + 7480*sqrt(3*x**2 + 2)/81 + 3731*sqrt(3)*asinh(sqrt(6)*
x/2)/36

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Giac [A]  time = 1.21882, size = 103, normalized size = 0.78 \begin{align*} -\frac{1}{3240} \,{\left (3 \,{\left ({\left (9 \,{\left (2 \,{\left ({\left ({\left (3 \,{\left (16 \,{\left (9 \, x - 5\right )} x - 2583\right )} x - 19180\right )} x - 27843\right )} x - 37720\right )} x - 71215\right )} x - 450080\right )} x - 415305\right )} x - 299200\right )} \sqrt{3 \, x^{2} + 2} - \frac{3731}{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*(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

-1/3240*(3*((9*(2*(((3*(16*(9*x - 5)*x - 2583)*x - 19180)*x - 27843)*x - 37720)*x - 71215)*x - 450080)*x - 415
305)*x - 299200)*sqrt(3*x^2 + 2) - 3731/36*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))