3.1381 \(\int (5-x) (3+2 x)^4 (2+3 x^2)^{5/2} \, dx\)

Optimal. Leaf size=154 \[ -\frac{1}{33} \left (3 x^2+2\right )^{7/2} (2 x+3)^4+\frac{49}{165} \left (3 x^2+2\right )^{7/2} (2 x+3)^3+\frac{6433 \left (3 x^2+2\right )^{7/2} (2 x+3)^2}{4455}+\frac{2 (62244 x+181243) \left (3 x^2+2\right )^{7/2}}{13365}+\frac{4991}{90} x \left (3 x^2+2\right )^{5/2}+\frac{4991}{36} x \left (3 x^2+2\right )^{3/2}+\frac{4991}{12} x \sqrt{3 x^2+2}+\frac{4991 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(4991*x*Sqrt[2 + 3*x^2])/12 + (4991*x*(2 + 3*x^2)^(3/2))/36 + (4991*x*(2 + 3*x^2)^(5/2))/90 + (6433*(3 + 2*x)^
2*(2 + 3*x^2)^(7/2))/4455 + (49*(3 + 2*x)^3*(2 + 3*x^2)^(7/2))/165 - ((3 + 2*x)^4*(2 + 3*x^2)^(7/2))/33 + (2*(
181243 + 62244*x)*(2 + 3*x^2)^(7/2))/13365 + (4991*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.0824702, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac{1}{33} \left (3 x^2+2\right )^{7/2} (2 x+3)^4+\frac{49}{165} \left (3 x^2+2\right )^{7/2} (2 x+3)^3+\frac{6433 \left (3 x^2+2\right )^{7/2} (2 x+3)^2}{4455}+\frac{2 (62244 x+181243) \left (3 x^2+2\right )^{7/2}}{13365}+\frac{4991}{90} x \left (3 x^2+2\right )^{5/2}+\frac{4991}{36} x \left (3 x^2+2\right )^{3/2}+\frac{4991}{12} x \sqrt{3 x^2+2}+\frac{4991 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4991*x*Sqrt[2 + 3*x^2])/12 + (4991*x*(2 + 3*x^2)^(3/2))/36 + (4991*x*(2 + 3*x^2)^(5/2))/90 + (6433*(3 + 2*x)^
2*(2 + 3*x^2)^(7/2))/4455 + (49*(3 + 2*x)^3*(2 + 3*x^2)^(7/2))/165 - ((3 + 2*x)^4*(2 + 3*x^2)^(7/2))/33 + (2*(
181243 + 62244*x)*(2 + 3*x^2)^(7/2))/13365 + (4991*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx &=-\frac{1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac{1}{33} \int (3+2 x)^3 (511+294 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac{49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac{1}{990} \int (3+2 x)^2 (42462+38598 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac{6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac{49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac{\int (3+2 x) (3130638+2987712 x) \left (2+3 x^2\right )^{5/2} \, dx}{26730}\\ &=\frac{6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac{49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac{2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac{4991}{15} \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac{4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac{6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac{49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac{2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac{4991}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{4991}{36} x \left (2+3 x^2\right )^{3/2}+\frac{4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac{6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac{49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac{2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac{4991}{6} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{4991}{12} x \sqrt{2+3 x^2}+\frac{4991}{36} x \left (2+3 x^2\right )^{3/2}+\frac{4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac{6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac{49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac{2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac{4991}{6} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{4991}{12} x \sqrt{2+3 x^2}+\frac{4991}{36} x \left (2+3 x^2\right )^{3/2}+\frac{4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac{6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac{49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac{1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac{2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac{4991 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0932964, size = 85, normalized size = 0.55 \[ \frac{\sqrt{3 x^2+2} \left (-699840 x^{10}-769824 x^9+12921120 x^8+50615928 x^7+93646260 x^6+129966606 x^5+150762600 x^4+127123425 x^3+92160240 x^2+64370295 x+19537120\right )}{53460}+\frac{4991 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(19537120 + 64370295*x + 92160240*x^2 + 127123425*x^3 + 150762600*x^4 + 129966606*x^5 + 93646
260*x^6 + 50615928*x^7 + 12921120*x^8 - 769824*x^9 - 699840*x^10))/53460 + (4991*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt
[3])

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Maple [A]  time = 0.014, size = 115, normalized size = 0.8 \begin{align*} -{\frac{16\,{x}^{4}}{33} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{8840\,{x}^{2}}{891} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{122107}{2673} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{x}^{3}}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{542\,x}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{4991\,x}{90} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{4991\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{4991\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{4991\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16/33*x^4*(3*x^2+2)^(7/2)+8840/891*x^2*(3*x^2+2)^(7/2)+122107/2673*(3*x^2+2)^(7/2)-8/15*x^3*(3*x^2+2)^(7/2)+5
42/15*x*(3*x^2+2)^(7/2)+4991/90*x*(3*x^2+2)^(5/2)+4991/36*x*(3*x^2+2)^(3/2)+4991/12*x*(3*x^2+2)^(1/2)+4991/18*
arcsinh(1/2*x*6^(1/2))*3^(1/2)

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Maxima [A]  time = 1.54148, size = 154, normalized size = 1. \begin{align*} -\frac{16}{33} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{4} - \frac{8}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{3} + \frac{8840}{891} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{2} + \frac{542}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x + \frac{122107}{2673} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} + \frac{4991}{90} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{4991}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{4991}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{4991}{18} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \end{align*}

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]

-16/33*(3*x^2 + 2)^(7/2)*x^4 - 8/15*(3*x^2 + 2)^(7/2)*x^3 + 8840/891*(3*x^2 + 2)^(7/2)*x^2 + 542/15*(3*x^2 + 2
)^(7/2)*x + 122107/2673*(3*x^2 + 2)^(7/2) + 4991/90*(3*x^2 + 2)^(5/2)*x + 4991/36*(3*x^2 + 2)^(3/2)*x + 4991/1
2*sqrt(3*x^2 + 2)*x + 4991/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 2.23001, size = 333, normalized size = 2.16 \begin{align*} -\frac{1}{53460} \,{\left (699840 \, x^{10} + 769824 \, x^{9} - 12921120 \, x^{8} - 50615928 \, x^{7} - 93646260 \, x^{6} - 129966606 \, x^{5} - 150762600 \, x^{4} - 127123425 \, x^{3} - 92160240 \, x^{2} - 64370295 \, x - 19537120\right )} \sqrt{3 \, x^{2} + 2} + \frac{4991}{36} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end{align*}

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]

-1/53460*(699840*x^10 + 769824*x^9 - 12921120*x^8 - 50615928*x^7 - 93646260*x^6 - 129966606*x^5 - 150762600*x^
4 - 127123425*x^3 - 92160240*x^2 - 64370295*x - 19537120)*sqrt(3*x^2 + 2) + 4991/36*sqrt(3)*log(-sqrt(3)*sqrt(
3*x^2 + 2)*x - 3*x^2 - 1)

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

[Out]

Timed out

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Giac [A]  time = 1.15371, size = 111, normalized size = 0.72 \begin{align*} -\frac{1}{53460} \,{\left (3 \,{\left ({\left (9 \,{\left (2 \,{\left ({\left (2 \,{\left (6 \,{\left (4 \,{\left (27 \,{\left (10 \, x + 11\right )} x - 4985\right )} x - 78111\right )} x - 867095\right )} x - 2406789\right )} x - 2791900\right )} x - 4708275\right )} x - 30720080\right )} x - 21456765\right )} x - 19537120\right )} \sqrt{3 \, x^{2} + 2} - \frac{4991}{18} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \end{align*}

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]

-1/53460*(3*((9*(2*((2*(6*(4*(27*(10*x + 11)*x - 4985)*x - 78111)*x - 867095)*x - 2406789)*x - 2791900)*x - 47
08275)*x - 30720080)*x - 21456765)*x - 19537120)*sqrt(3*x^2 + 2) - 4991/18*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2
+ 2))