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

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

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Rubi [A]  time = 0.06, antiderivative size = 116, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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)^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*}

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Mathematica [A]  time = 0.08, size = 73, normalized size = 0.63 \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/486*(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])/(2 + 3*x^2)^(3/2)

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IntegrateAlgebraic [A]  time = 0.48, size = 76, normalized size = 0.66 \begin {gather*} \frac {-3456 x^6-20736 x^5+130464 x^4+1125999 x^3-2363976 x^2+139815 x-1798610}{486 \left (3 x^2+2\right )^{3/2}}-\frac {20720 \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)^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)/(486*(2 + 3*x^2)^(3/2))
- (20720*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(27*Sqrt[3])

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fricas [A]  time = 0.42, size = 98, normalized size = 0.84 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.18, size = 60, normalized size = 0.52 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.06, size = 119, normalized size = 1.03 \begin {gather*} -\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 x^{3}}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {131332 x^{2}}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {55517 x}{54 \sqrt {3 x^{2}+2}}-\frac {3537 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {20720 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{81}-\frac {899305}{243 \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)^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/(3*x^2+2)^(1/2)*x+20720/81*arcsinh(1/2*6^(1/2)
*x)*3^(1/2)-3537/2*x/(3*x^2+2)^(3/2)

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maxima [A]  time = 1.44, size = 133, normalized size = 1.15 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.71, size = 222, normalized size = 1.91 \begin {gather*} \frac {20720\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{81}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {64\,x^2}{27}+\frac {128\,x}{9}-\frac {7504}{81}\right )}{3}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {206689}{144}+\frac {\sqrt {6}\,81809{}\mathrm {i}}{432}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {206689}{216}+\frac {\sqrt {6}\,81809{}\mathrm {i}}{648}\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 {206689}{144}+\frac {\sqrt {6}\,81809{}\mathrm {i}}{432}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {206689}{216}+\frac {\sqrt {6}\,81809{}\mathrm {i}}{648}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-3390048+\sqrt {6}\,719421{}\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 (3390048+\sqrt {6}\,719421{}\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)^6*(x - 5))/(3*x^2 + 2)^(5/2),x)

[Out]

(20720*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/81 - (3^(1/2)*(x^2 + 2/3)^(1/2)*((128*x)/9 + (64*x^2)/27 - 7504/8
1))/3 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*81809i)/432 - 206689/144)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1
/2)*81809i)/648 - 206689/216)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*8180
9i)/432 + 206689/144)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*81809i)/648 + 206689/216)*1i)/(2*(x + (6^(1/2)
*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*719421i - 3390048)*(x^2 + 2/3)^(1/2)*1i)/(23328*(x - (6^(1/2)*1i)/
3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*719421i + 3390048)*(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 {13851 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 {21384 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 {16740 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 {6480 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 \left (- \frac {720 x^{5}}{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 {256 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 \frac {64 x^{7}}{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 {3645}{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)**6/(3*x**2+2)**(5/2),x)

[Out]

-Integral(-13851*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(-2
1384*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(-16740*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(-6480*x**4/(9*x**4*s
qrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-720*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(256*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(64*x**7/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**
2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-3645/(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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