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

Optimal. Leaf size=147 \[ \frac{25}{16} \left (2 x^2-x+3\right )^{5/2} x^3+\frac{1235}{448} \left (2 x^2-x+3\right )^{5/2} x^2+\frac{24499 \left (2 x^2-x+3\right )^{5/2} x}{10752}+\frac{73861 \left (2 x^2-x+3\right )^{5/2}}{215040}+\frac{24293 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{196608}+\frac{558739 (1-4 x) \sqrt{2 x^2-x+3}}{1048576}+\frac{12850997 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}} \]

[Out]

(558739*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1048576 + (24293*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/196608 + (73861*(3 -
x + 2*x^2)^(5/2))/215040 + (24499*x*(3 - x + 2*x^2)^(5/2))/10752 + (1235*x^2*(3 - x + 2*x^2)^(5/2))/448 + (25*
x^3*(3 - x + 2*x^2)^(5/2))/16 + (12850997*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2097152*Sqrt[2])

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Rubi [A]  time = 0.122099, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \[ \frac{25}{16} \left (2 x^2-x+3\right )^{5/2} x^3+\frac{1235}{448} \left (2 x^2-x+3\right )^{5/2} x^2+\frac{24499 \left (2 x^2-x+3\right )^{5/2} x}{10752}+\frac{73861 \left (2 x^2-x+3\right )^{5/2}}{215040}+\frac{24293 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{196608}+\frac{558739 (1-4 x) \sqrt{2 x^2-x+3}}{1048576}+\frac{12850997 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(558739*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1048576 + (24293*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/196608 + (73861*(3 -
x + 2*x^2)^(5/2))/215040 + (24499*x*(3 - x + 2*x^2)^(5/2))/10752 + (1235*x^2*(3 - x + 2*x^2)^(5/2))/448 + (25*
x^3*(3 - x + 2*x^2)^(5/2))/16 + (12850997*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2097152*Sqrt[2])

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

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 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx &=\frac{25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{16} \int \left (3-x+2 x^2\right )^{3/2} \left (64+192 x+239 x^2+\frac{1235 x^3}{2}\right ) \, dx\\ &=\frac{1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{224} \int \left (3-x+2 x^2\right )^{3/2} \left (896-1017 x+\frac{24499 x^2}{4}\right ) \, dx\\ &=\frac{24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac{1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (-\frac{30489}{4}+\frac{73861 x}{8}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{2688}\\ &=\frac{73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac{24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac{1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac{24293 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{12288}\\ &=\frac{24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac{73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac{24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac{1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac{558739 \int \sqrt{3-x+2 x^2} \, dx}{131072}\\ &=\frac{558739 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}+\frac{24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac{73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac{24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac{1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac{12850997 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{2097152}\\ &=\frac{558739 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}+\frac{24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac{73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac{24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac{1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac{\left (558739 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{2097152}\\ &=\frac{558739 (1-4 x) \sqrt{3-x+2 x^2}}{1048576}+\frac{24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac{73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac{24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac{1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac{12850997 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2097152 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.14373, size = 75, normalized size = 0.51 \[ \frac{4 \sqrt{2 x^2-x+3} \left (688128000 x^7+525926400 x^6+2025840640 x^5+2061273088 x^4+2728413312 x^3+1799647136 x^2+1619403428 x+439831323\right )[phone] \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{440401920} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(439831323 + 1619403428*x + 1799647136*x^2 + 2728413312*x^3 + 2061273088*x^4 + 20258406
40*x^5 + 525926400*x^6 + 688128000*x^7) + 1349354685*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/440401920

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Maple [A]  time = 0.055, size = 117, normalized size = 0.8 \begin{align*}{\frac{25\,{x}^{3}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{1235\,{x}^{2}}{448} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{24499\,x}{10752} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{-558739+2234956\,x}{1048576}\sqrt{2\,{x}^{2}-x+3}}-{\frac{12850997\,\sqrt{2}}{4194304}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{-24293+97172\,x}{196608} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{73861}{215040} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/16*x^3*(2*x^2-x+3)^(5/2)+1235/448*x^2*(2*x^2-x+3)^(5/2)+24499/10752*x*(2*x^2-x+3)^(5/2)-558739/1048576*(-1+
4*x)*(2*x^2-x+3)^(1/2)-12850997/4194304*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-24293/196608*(-1+4*x)*(2*x^2-x+
3)^(3/2)+73861/215040*(2*x^2-x+3)^(5/2)

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Maxima [A]  time = 1.49987, size = 186, normalized size = 1.27 \begin{align*} \frac{25}{16} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{1235}{448} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{24499}{10752} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{73861}{215040} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} - \frac{24293}{49152} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{24293}{196608} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{558739}{262144} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{12850997}{4194304} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{558739}{1048576} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/16*(2*x^2 - x + 3)^(5/2)*x^3 + 1235/448*(2*x^2 - x + 3)^(5/2)*x^2 + 24499/10752*(2*x^2 - x + 3)^(5/2)*x + 7
3861/215040*(2*x^2 - x + 3)^(5/2) - 24293/49152*(2*x^2 - x + 3)^(3/2)*x + 24293/196608*(2*x^2 - x + 3)^(3/2) -
558739/262144*sqrt(2*x^2 - x + 3)*x - 12850997/4194304*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 558739/1048
576*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.51727, size = 342, normalized size = 2.33 \begin{align*} \frac{1}{110100480} \,{\left (688128000 \, x^{7} + 525926400 \, x^{6} + 2025840640 \, x^{5} + 2061273088 \, x^{4} + 2728413312 \, x^{3} + 1799647136 \, x^{2} + 1619403428 \, x + 439831323\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{12850997}{8388608} \, \sqrt{2} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/110100480*(688128000*x^7 + 525926400*x^6 + 2025840640*x^5 + 2061273088*x^4 + 2728413312*x^3 + 1799647136*x^2
+ 1619403428*x + 439831323)*sqrt(2*x^2 - x + 3) + 12850997/8388608*sqrt(2)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*
(4*x - 1) - 32*x^2 + 16*x - 25)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((2*x**2 - x + 3)**(3/2)*(5*x**2 + 3*x + 2)**2, x)

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Giac [A]  time = 1.15408, size = 112, normalized size = 0.76 \begin{align*} \frac{1}{110100480} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (120 \,{\left (140 \, x + 107\right )} x + 49459\right )} x + 1006481\right )} x + 21315729\right )} x + 56238973\right )} x + 404850857\right )} x + 439831323\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{12850997}{4194304} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/110100480*(4*(8*(4*(16*(20*(120*(140*x + 107)*x + 49459)*x + 1006481)*x + 21315729)*x + 56238973)*x + 404850
857)*x + 439831323)*sqrt(2*x^2 - x + 3) + 12850997/4194304*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x
+ 3)) + 1)