3.1418 \(\int \frac{(5-x) (3+2 x)^6}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}+\frac{(2427 x+158) (2 x+3)^3}{54 \sqrt{3 x^2+2}}-\frac{2639}{81} \sqrt{3 x^2+2} (2 x+3)^2-\frac{70}{243} (801 x+2167) \sqrt{3 x^2+2}+\frac{20720 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^5)/(18*(2 + 3*x^2)^(3/2)) + ((3 + 2*x)^3*(158 + 2427*x))/(54*Sqrt[2 + 3*x^2]) - (2639*
(3 + 2*x)^2*Sqrt[2 + 3*x^2])/81 - (70*(2167 + 801*x)*Sqrt[2 + 3*x^2])/243 + (20720*ArcSinh[Sqrt[3/2]*x])/(27*S
qrt[3])

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Rubi [A]  time = 0.060576, antiderivative size = 116, normalized size of antiderivative = 1., 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 = {819, 833, 780, 215} \[ -\frac{7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}+\frac{(2427 x+158) (2 x+3)^3}{54 \sqrt{3 x^2+2}}-\frac{2639}{81} \sqrt{3 x^2+2} (2 x+3)^2-\frac{70}{243} (801 x+2167) \sqrt{3 x^2+2}+\frac{20720 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(3 + 2*x)^6)/(2 + 3*x^2)^(5/2),x]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^5)/(18*(2 + 3*x^2)^(3/2)) + ((3 + 2*x)^3*(158 + 2427*x))/(54*Sqrt[2 + 3*x^2]) - (2639*
(3 + 2*x)^2*Sqrt[2 + 3*x^2])/81 - (70*(2167 + 801*x)*Sqrt[2 + 3*x^2])/243 + (20720*ArcSinh[Sqrt[3/2]*x])/(27*S
qrt[3])

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

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 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 \frac{(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac{7 (2-7 x) (3+2 x)^5}{18 \left (2+3 x^2\right )^{3/2}}+\frac{1}{18} \int \frac{(398-318 x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^5}{18 \left (2+3 x^2\right )^{3/2}}+\frac{(3+2 x)^3 (158+2427 x)}{54 \sqrt{2+3 x^2}}+\frac{1}{108} \int \frac{(-5712-31668 x) (3+2 x)^2}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^5}{18 \left (2+3 x^2\right )^{3/2}}+\frac{(3+2 x)^3 (158+2427 x)}{54 \sqrt{2+3 x^2}}-\frac{2639}{81} (3+2 x)^2 \sqrt{2+3 x^2}+\frac{1}{972} \int \frac{(99120-672840 x) (3+2 x)}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^5}{18 \left (2+3 x^2\right )^{3/2}}+\frac{(3+2 x)^3 (158+2427 x)}{54 \sqrt{2+3 x^2}}-\frac{2639}{81} (3+2 x)^2 \sqrt{2+3 x^2}-\frac{70}{243} (2167+801 x) \sqrt{2+3 x^2}+\frac{20720}{27} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=-\frac{7 (2-7 x) (3+2 x)^5}{18 \left (2+3 x^2\right )^{3/2}}+\frac{(3+2 x)^3 (158+2427 x)}{54 \sqrt{2+3 x^2}}-\frac{2639}{81} (3+2 x)^2 \sqrt{2+3 x^2}-\frac{70}{243} (2167+801 x) \sqrt{2+3 x^2}+\frac{20720 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.079336, size = 73, normalized size = 0.63 \[ -\frac{3456 x^6+20736 x^5-130464 x^4-1125999 x^3+2363976 x^2-124320 \sqrt{3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-139815 x+1798610}{486 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(3 + 2*x)^6)/(2 + 3*x^2)^(5/2),x]

[Out]

-(1798610 - 139815*x + 2363976*x^2 - 1125999*x^3 - 130464*x^4 + 20736*x^5 + 3456*x^6 - 124320*Sqrt[3]*(2 + 3*x
^2)^(3/2)*ArcSinh[Sqrt[3/2]*x])/(486*(2 + 3*x^2)^(3/2))

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Maple [A]  time = 0.02, size = 119, normalized size = 1. \begin{align*} -{\frac{64\,{x}^{6}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{2416\,{x}^{4}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{131332\,{x}^{2}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{899305}{243} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{128\,{x}^{5}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{20720\,{x}^{3}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{55517\,x}{54}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}+{\frac{20720\,\sqrt{3}}{81}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{3537\,x}{2} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^6/(3*x^2+2)^(5/2),x)

[Out]

-64/9*x^6/(3*x^2+2)^(3/2)+2416/9*x^4/(3*x^2+2)^(3/2)-131332/27*x^2/(3*x^2+2)^(3/2)-899305/243/(3*x^2+2)^(3/2)-
128/3*x^5/(3*x^2+2)^(3/2)-20720/27*x^3/(3*x^2+2)^(3/2)+55517/54*x/(3*x^2+2)^(1/2)+20720/81*arcsinh(1/2*x*6^(1/
2))*3^(1/2)-3537/2*x/(3*x^2+2)^(3/2)

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Maxima [A]  time = 1.49757, size = 180, normalized size = 1.55 \begin{align*} -\frac{64 \, x^{6}}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{128 \, x^{5}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{2416 \, x^{4}}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{20720}{81} \, x{\left (\frac{9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\right )} + \frac{20720}{81} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{249431 \, x}{162 \, \sqrt{3 \, x^{2} + 2}} - \frac{131332 \, x^{2}}{27 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{3537 \, x}{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{899305}{243 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^6/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

-64/9*x^6/(3*x^2 + 2)^(3/2) - 128/3*x^5/(3*x^2 + 2)^(3/2) + 2416/9*x^4/(3*x^2 + 2)^(3/2) - 20720/81*x*(9*x^2/(
3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) + 20720/81*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 249431/162*x/sqrt(3*x^2 +
2) - 131332/27*x^2/(3*x^2 + 2)^(3/2) - 3537/2*x/(3*x^2 + 2)^(3/2) - 899305/243/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 1.61609, size = 286, normalized size = 2.47 \begin{align*} \frac{62160 \, \sqrt{3}{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) -{\left (3456 \, x^{6} + 20736 \, x^{5} - 130464 \, x^{4} - 1125999 \, x^{3} + 2363976 \, x^{2} - 139815 \, x + 1798610\right )} \sqrt{3 \, x^{2} + 2}}{486 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^6/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/486*(62160*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (3456*x^6 + 20736*x^5
- 130464*x^4 - 1125999*x^3 + 2363976*x^2 - 139815*x + 1798610)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**6/(3*x**2+2)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.20622, size = 81, normalized size = 0.7 \begin{align*} -\frac{20720}{81} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{9 \,{\left ({\left ({\left (96 \,{\left (4 \,{\left (x + 6\right )} x - 151\right )} x - 125111\right )} x + 262664\right )} x - 15535\right )} x + 1798610}{486 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^6/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

-20720/81*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/486*(9*(((96*(4*(x + 6)*x - 151)*x - 125111)*x + 26266
4)*x - 15535)*x + 1798610)/(3*x^2 + 2)^(3/2)