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

Optimal. Leaf size=94 \[ -\frac {7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}-\frac {5 (16-421 x) (2 x+3)^2}{54 \sqrt {3 x^2+2}}-\frac {50}{81} (93 x+299) \sqrt {3 x^2+2}+\frac {1600 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{27 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 94, 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, 780, 215} \begin {gather*} -\frac {7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}-\frac {5 (16-421 x) (2 x+3)^2}{54 \sqrt {3 x^2+2}}-\frac {50}{81} (93 x+299) \sqrt {3 x^2+2}+\frac {1600 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{27 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^4)/(18*(2 + 3*x^2)^(3/2)) - (5*(16 - 421*x)*(3 + 2*x)^2)/(54*Sqrt[2 + 3*x^2]) - (50*(2
99 + 93*x)*Sqrt[2 + 3*x^2])/81 + (1600*ArcSinh[Sqrt[3/2]*x])/(27*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])

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx &=-\frac {7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}+\frac {1}{18} \int \frac {(370-220 x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}-\frac {5 (16-421 x) (3+2 x)^2}{54 \sqrt {2+3 x^2}}+\frac {1}{108} \int \frac {(-2000-18600 x) (3+2 x)}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}-\frac {5 (16-421 x) (3+2 x)^2}{54 \sqrt {2+3 x^2}}-\frac {50}{81} (299+93 x) \sqrt {2+3 x^2}+\frac {1600}{27} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=-\frac {7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}-\frac {5 (16-421 x) (3+2 x)^2}{54 \sqrt {2+3 x^2}}-\frac {50}{81} (299+93 x) \sqrt {2+3 x^2}+\frac {1600 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{27 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 68, normalized size = 0.72 \begin {gather*} -\frac {864 x^5+4320 x^4-183945 x^3+147600 x^2-3200 \sqrt {3} \left (3 x^2+2\right )^{3/2} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-79215 x+134126}{162 \left (3 x^2+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/162*(134126 - 79215*x + 147600*x^2 - 183945*x^3 + 4320*x^4 + 864*x^5 - 3200*Sqrt[3]*(2 + 3*x^2)^(3/2)*ArcSi
nh[Sqrt[3/2]*x])/(2 + 3*x^2)^(3/2)

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IntegrateAlgebraic [A]  time = 0.47, size = 71, normalized size = 0.76 \begin {gather*} \frac {-864 x^5-4320 x^4+183945 x^3-147600 x^2+79215 x-134126}{162 \left (3 x^2+2\right )^{3/2}}-\frac {1600 \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right )}{27 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-134126 + 79215*x - 147600*x^2 + 183945*x^3 - 4320*x^4 - 864*x^5)/(162*(2 + 3*x^2)^(3/2)) - (1600*Log[-(Sqrt[
3]*x) + Sqrt[2 + 3*x^2]])/(27*Sqrt[3])

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fricas [A]  time = 0.42, size = 93, normalized size = 0.99 \begin {gather*} \frac {1600 \, \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^{5} + 4320 \, x^{4} - 183945 \, x^{3} + 147600 \, x^{2} - 79215 \, x + 134126\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)^5/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/162*(1600*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (864*x^5 + 4320*x^4 - 1
83945*x^3 + 147600*x^2 - 79215*x + 134126)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)

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giac [A]  time = 0.25, size = 55, normalized size = 0.59 \begin {gather*} -\frac {1600}{81} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left ({\left ({\left (288 \, {\left (x + 5\right )} x - 61315\right )} x + 49200\right )} x - 26405\right )} x + 134126}{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)^5/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

-1600/81*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(3*(((288*(x + 5)*x - 61315)*x + 49200)*x - 26405)*
x + 134126)/(3*x^2 + 2)^(3/2)

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maple [A]  time = 0.07, size = 105, normalized size = 1.12 \begin {gather*} -\frac {16 x^{5}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {80 x^{4}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {1600 x^{3}}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {8200 x^{2}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {21505 x}{54 \sqrt {3 x^{2}+2}}-\frac {615 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {1600 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{81}-\frac {67063}{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)^5/(3*x^2+2)^(5/2),x)

[Out]

-16/3/(3*x^2+2)^(3/2)*x^5-1600/27/(3*x^2+2)^(3/2)*x^3+21505/54/(3*x^2+2)^(1/2)*x+1600/81*arcsinh(1/2*6^(1/2)*x
)*3^(1/2)-80/3/(3*x^2+2)^(3/2)*x^4-8200/9/(3*x^2+2)^(3/2)*x^2-67063/81/(3*x^2+2)^(3/2)-615/2/(3*x^2+2)^(3/2)*x

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maxima [A]  time = 1.31, size = 119, normalized size = 1.27 \begin {gather*} -\frac {16 \, x^{5}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {80 \, x^{4}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {1600}{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 {1600}{81} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {70915 \, x}{162 \, \sqrt {3 \, x^{2} + 2}} - \frac {8200 \, x^{2}}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {615 \, x}{2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {67063}{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)^5/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

-16/3*x^5/(3*x^2 + 2)^(3/2) - 80/3*x^4/(3*x^2 + 2)^(3/2) - 1600/81*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^
(3/2)) + 1600/81*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 70915/162*x/sqrt(3*x^2 + 2) - 8200/9*x^2/(3*x^2 + 2)^(3/2) -
615/2*x/(3*x^2 + 2)^(3/2) - 67063/81/(3*x^2 + 2)^(3/2)

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mupad [B]  time = 0.05, size = 217, normalized size = 2.31 \begin {gather*} \frac {1600\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{81}-\frac {\sqrt {3}\,\left (\frac {16\,x}{9}+\frac {80}{9}\right )\,\sqrt {x^2+\frac {2}{3}}}{3}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {43799}{144}+\frac {\sqrt {6}\,18823{}\mathrm {i}}{144}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {43799}{216}+\frac {\sqrt {6}\,18823{}\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 {43799}{144}+\frac {\sqrt {6}\,18823{}\mathrm {i}}{144}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {43799}{216}+\frac {\sqrt {6}\,18823{}\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 (-567360+\sqrt {6}\,290595{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{23328\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (567360+\sqrt {6}\,290595{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{23328\,\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)^5*(x - 5))/(3*x^2 + 2)^(5/2),x)

[Out]

(1600*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/81 - (3^(1/2)*((16*x)/9 + 80/9)*(x^2 + 2/3)^(1/2))/3 + (3^(1/2)*(x
^2 + 2/3)^(1/2)*(((6^(1/2)*18823i)/144 - 43799/144)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*18823i)/216 - 43
799/216)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*18823i)/144 + 43799/144)/
(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*18823i)/216 + 43799/216)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1
/2)*6^(1/2)*(6^(1/2)*290595i - 567360)*(x^2 + 2/3)^(1/2)*1i)/(23328*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(
6^(1/2)*290595i + 567360)*(x^2 + 2/3)^(1/2)*1i)/(23328*(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 {3807 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 {4590 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 {2520 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 \left (- \frac {480 x^{4}}{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 {80 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 \frac {32 x^{6}}{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 {1215}{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)**5/(3*x**2+2)**(5/2),x)

[Out]

-Integral(-3807*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(-45
90*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(-2520*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(-480*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(80*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(32*x**6/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt
(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-1215/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) +
4*sqrt(3*x**2 + 2)), x)

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