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

Optimal. Leaf size=89 \[ -\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}-\frac {151}{27} \sqrt {3 x^2+2} (2 x+3)^2-\frac {10}{81} (207 x+185) \sqrt {3 x^2+2}+\frac {880 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {819, 833, 780, 215} \begin {gather*} -\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}-\frac {151}{27} \sqrt {3 x^2+2} (2 x+3)^2-\frac {10}{81} (207 x+185) \sqrt {3 x^2+2}+\frac {880 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^3)/(6*Sqrt[2 + 3*x^2]) - (151*(3 + 2*x)^2*Sqrt[2 + 3*x^2])/27 - (10*(185 + 207*x)*Sqrt
[2 + 3*x^2])/81 + (880*ArcSinh[Sqrt[3/2]*x])/(3*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 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 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])

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx &=-\frac {7 (2-7 x) (3+2 x)^3}{6 \sqrt {2+3 x^2}}+\frac {1}{6} \int \frac {(72-302 x) (3+2 x)^2}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^3}{6 \sqrt {2+3 x^2}}-\frac {151}{27} (3+2 x)^2 \sqrt {2+3 x^2}+\frac {1}{54} \int \frac {(4360-4140 x) (3+2 x)}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^3}{6 \sqrt {2+3 x^2}}-\frac {151}{27} (3+2 x)^2 \sqrt {2+3 x^2}-\frac {10}{81} (185+207 x) \sqrt {2+3 x^2}+\frac {880}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^3}{6 \sqrt {2+3 x^2}}-\frac {151}{27} (3+2 x)^2 \sqrt {2+3 x^2}-\frac {10}{81} (185+207 x) \sqrt {2+3 x^2}+\frac {880 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 58, normalized size = 0.65 \begin {gather*} -\frac {288 x^4+432 x^3-15024 x^2-15840 \sqrt {9 x^2+6} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )+14715 x+33914}{162 \sqrt {3 x^2+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/162*(33914 + 14715*x - 15024*x^2 + 432*x^3 + 288*x^4 - 15840*Sqrt[6 + 9*x^2]*ArcSinh[Sqrt[3/2]*x])/Sqrt[2 +
3*x^2]

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IntegrateAlgebraic [A]  time = 0.38, size = 66, normalized size = 0.74 \begin {gather*} \frac {-288 x^4-432 x^3+15024 x^2-14715 x-33914}{162 \sqrt {3 x^2+2}}-\frac {880 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-33914 - 14715*x + 15024*x^2 - 432*x^3 - 288*x^4)/(162*Sqrt[2 + 3*x^2]) - (880*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*
x^2]])/(3*Sqrt[3])

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fricas [A]  time = 0.41, size = 78, normalized size = 0.88 \begin {gather*} \frac {7920 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (288 \, x^{4} + 432 \, x^{3} - 15024 \, x^{2} + 14715 \, x + 33914\right )} \sqrt {3 \, x^{2} + 2}}{162 \, {\left (3 \, x^{2} + 2\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)^(3/2),x, algorithm="fricas")

[Out]

1/162*(7920*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (288*x^4 + 432*x^3 - 15024*x^2 +
14715*x + 33914)*sqrt(3*x^2 + 2))/(3*x^2 + 2)

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giac [A]  time = 0.19, size = 54, normalized size = 0.61 \begin {gather*} -\frac {880}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left (16 \, {\left (3 \, {\left (2 \, x + 3\right )} x - 313\right )} x + 4905\right )} x + 33914}{162 \, \sqrt {3 \, x^{2} + 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)^(3/2),x, algorithm="giac")

[Out]

-880/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(3*(16*(3*(2*x + 3)*x - 313)*x + 4905)*x + 33914)/sqr
t(3*x^2 + 2)

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maple [A]  time = 0.06, size = 79, normalized size = 0.89 \begin {gather*} -\frac {16 x^{4}}{9 \sqrt {3 x^{2}+2}}-\frac {8 x^{3}}{3 \sqrt {3 x^{2}+2}}+\frac {2504 x^{2}}{27 \sqrt {3 x^{2}+2}}-\frac {545 x}{6 \sqrt {3 x^{2}+2}}+\frac {880 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}-\frac {16957}{81 \sqrt {3 x^{2}+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)^(3/2),x)

[Out]

-16/9*x^4/(3*x^2+2)^(1/2)+2504/27*x^2/(3*x^2+2)^(1/2)-16957/81/(3*x^2+2)^(1/2)-8/3*x^3/(3*x^2+2)^(1/2)-545/6*x
/(3*x^2+2)^(1/2)+880/9*arcsinh(1/2*6^(1/2)*x)*3^(1/2)

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maxima [A]  time = 1.41, size = 78, normalized size = 0.88 \begin {gather*} -\frac {16 \, x^{4}}{9 \, \sqrt {3 \, x^{2} + 2}} - \frac {8 \, x^{3}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {2504 \, x^{2}}{27 \, \sqrt {3 \, x^{2} + 2}} + \frac {880}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {545 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} - \frac {16957}{81 \, \sqrt {3 \, x^{2} + 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)^(3/2),x, algorithm="maxima")

[Out]

-16/9*x^4/sqrt(3*x^2 + 2) - 8/3*x^3/sqrt(3*x^2 + 2) + 2504/27*x^2/sqrt(3*x^2 + 2) + 880/9*sqrt(3)*arcsinh(1/2*
sqrt(6)*x) - 545/6*x/sqrt(3*x^2 + 2) - 16957/81/sqrt(3*x^2 + 2)

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mupad [B]  time = 0.06, size = 110, normalized size = 1.24 \begin {gather*} \frac {880\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {16\,x^2}{9}+\frac {8\,x}{3}-\frac {2536}{27}\right )}{3}+\frac {\sqrt {3}\,\sqrt {6}\,\left (-44058+\sqrt {6}\,4809{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\left (44058+\sqrt {6}\,4809{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\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)^(3/2),x)

[Out]

(880*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/9 - (3^(1/2)*(x^2 + 2/3)^(1/2)*((8*x)/3 + (16*x^2)/9 - 2536/27))/3
+ (3^(1/2)*6^(1/2)*(6^(1/2)*4809i - 44058)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x + (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2
)*(6^(1/2)*4809i + 44058)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x - (6^(1/2)*1i)/3))

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

[Out]

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

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