3.13.46 \(\int \frac {(5-x) (3+2 x)^4}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=87 \[ -\frac {7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}-\frac {(318-1783 x) (2 x+3)}{54 \sqrt {3 x^2+2}}-\frac {2027}{81} \sqrt {3 x^2+2}-\frac {16 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {819, 641, 215} \begin {gather*} -\frac {7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}-\frac {(318-1783 x) (2 x+3)}{54 \sqrt {3 x^2+2}}-\frac {2027}{81} \sqrt {3 x^2+2}-\frac {16 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(3 + 2*x)^4)/(2 + 3*x^2)^(5/2),x]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^3)/(18*(2 + 3*x^2)^(3/2)) - ((318 - 1783*x)*(3 + 2*x))/(54*Sqrt[2 + 3*x^2]) - (2027*Sq
rt[2 + 3*x^2])/81 - (16*ArcSinh[Sqrt[3/2]*x])/(9*Sqrt[3])

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 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

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])

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac {7 (2-7 x) (3+2 x)^3}{18 \left (2+3 x^2\right )^{3/2}}+\frac {1}{18} \int \frac {(342-122 x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^3}{18 \left (2+3 x^2\right )^{3/2}}-\frac {(318-1783 x) (3+2 x)}{54 \sqrt {2+3 x^2}}+\frac {1}{108} \int \frac {-192-8108 x}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^3}{18 \left (2+3 x^2\right )^{3/2}}-\frac {(318-1783 x) (3+2 x)}{54 \sqrt {2+3 x^2}}-\frac {2027}{81} \sqrt {2+3 x^2}-\frac {16}{9} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^3}{18 \left (2+3 x^2\right )^{3/2}}-\frac {(318-1783 x) (3+2 x)}{54 \sqrt {2+3 x^2}}-\frac {2027}{81} \sqrt {2+3 x^2}-\frac {16 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 63, normalized size = 0.72 \begin {gather*} -\frac {864 x^4-57285 x^3+16560 x^2+96 \sqrt {3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-33381 x+25342}{162 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(3 + 2*x)^4)/(2 + 3*x^2)^(5/2),x]

[Out]

-1/162*(25342 - 33381*x + 16560*x^2 - 57285*x^3 + 864*x^4 + 96*Sqrt[3]*(2 + 3*x^2)^(3/2)*ArcSinh[Sqrt[3/2]*x])
/(2 + 3*x^2)^(3/2)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.45, size = 66, normalized size = 0.76 \begin {gather*} \frac {16 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{9 \sqrt {3}}+\frac {-864 x^4+57285 x^3-16560 x^2+33381 x-25342}{162 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((5 - x)*(3 + 2*x)^4)/(2 + 3*x^2)^(5/2),x]

[Out]

(-25342 + 33381*x - 16560*x^2 + 57285*x^3 - 864*x^4)/(162*(2 + 3*x^2)^(3/2)) + (16*Log[-(Sqrt[3]*x) + Sqrt[2 +
3*x^2]])/(9*Sqrt[3])

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 87, normalized size = 1.00 \begin {gather*} \frac {48 \, \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 (864 \, x^{4} - 57285 \, x^{3} + 16560 \, x^{2} - 33381 \, x + 25342\right )} \sqrt {3 \, x^{2} + 2}}{162 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/162*(48*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (864*x^4 - 57285*x^3 + 165
60*x^2 - 33381*x + 25342)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 52, normalized size = 0.60 \begin {gather*} \frac {16}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {9 \, {\left ({\left ({\left (96 \, x - 6365\right )} x + 1840\right )} x - 3709\right )} x + 25342}{162 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

16/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(9*(((96*x - 6365)*x + 1840)*x - 3709)*x + 25342)/(3*x
^2 + 2)^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 91, normalized size = 1.05 \begin {gather*} -\frac {16 x^{4}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {16 x^{3}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {920 x^{2}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {2111 x}{18 \sqrt {3 x^{2}+2}}-\frac {57 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {16 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{27}-\frac {12671}{81 \left (3 x^{2}+2\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(2*x+3)^4/(3*x^2+2)^(5/2),x)

[Out]

-16/3/(3*x^2+2)^(3/2)*x^4-920/9/(3*x^2+2)^(3/2)*x^2-12671/81/(3*x^2+2)^(3/2)+16/9/(3*x^2+2)^(3/2)*x^3+2111/18/
(3*x^2+2)^(1/2)*x-16/27*arcsinh(1/2*6^(1/2)*x)*3^(1/2)-57/2/(3*x^2+2)^(3/2)*x

________________________________________________________________________________________

maxima [A]  time = 1.35, size = 105, normalized size = 1.21 \begin {gather*} -\frac {16 \, x^{4}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {16}{27} \, 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 {16}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {6269 \, x}{54 \, \sqrt {3 \, x^{2} + 2}} - \frac {920 \, x^{2}}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {57 \, x}{2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {12671}{81 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

-16/3*x^4/(3*x^2 + 2)^(3/2) + 16/27*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) - 16/27*sqrt(3)*arcsinh(
1/2*sqrt(6)*x) + 6269/54*x/sqrt(3*x^2 + 2) - 920/9*x^2/(3*x^2 + 2)^(3/2) - 57/2*x/(3*x^2 + 2)^(3/2) - 12671/81
/(3*x^2 + 2)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 1.72, size = 212, normalized size = 2.44 \begin {gather*} -\frac {16\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{27}-\frac {16\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {1603}{48}+\frac {\sqrt {6}\,7343{}\mathrm {i}}{144}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {1603}{72}+\frac {\sqrt {6}\,7343{}\mathrm {i}}{216}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {1603}{48}+\frac {\sqrt {6}\,7343{}\mathrm {i}}{144}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {1603}{72}+\frac {\sqrt {6}\,7343{}\mathrm {i}}{216}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-20544+\sqrt {6}\,27063{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{7776\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (20544+\sqrt {6}\,27063{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{7776\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x + 3)^4*(x - 5))/(3*x^2 + 2)^(5/2),x)

[Out]

(3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*7343i)/144 - 1603/48)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*7343i)/2
16 - 1603/72)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (16*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/27 - (16*3^(1/2)
*(x^2 + 2/3)^(1/2))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*7343i)/144 + 1603/48)/(x + (6^(1/2)*1i)/3) + (6
^(1/2)*((6^(1/2)*7343i)/216 + 1603/72)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*27063i
- 20544)*(x^2 + 2/3)^(1/2)*1i)/(7776*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*27063i + 20544)*(x^2 +
2/3)^(1/2)*1i)/(7776*(x + (6^(1/2)*1i)/3))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {999 x}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {16 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \frac {16 x^{5}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {405}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**4/(3*x**2+2)**(5/2),x)

[Out]

-Integral(-999*x/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-864
*x**2/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-264*x**3/(9*x*
*4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(16*x**4/(9*x**4*sqrt(3*x**
2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(16*x**5/(9*x**4*sqrt(3*x**2 + 2) + 12*x
**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-405/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2
+ 2) + 4*sqrt(3*x**2 + 2)), x)

________________________________________________________________________________________