3.59 \(\int \sqrt{3-x+2 x^2} (2+3 x+5 x^2)^3 \, dx\)

Optimal. Leaf size=166 \[ \frac{125}{16} \left (2 x^2-x+3\right )^{3/2} x^5+\frac{8825}{448} \left (2 x^2-x+3\right )^{3/2} x^4+\frac{247435 \left (2 x^2-x+3\right )^{3/2} x^3}{10752}+\frac{531681 \left (2 x^2-x+3\right )^{3/2} x^2}{71680}-\frac{9627393 \left (2 x^2-x+3\right )^{3/2} x}{1146880}-\frac{22548119 \left (2 x^2-x+3\right )^{3/2}}{4587520}-\frac{6766097 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}-\frac{155620231 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}} \]

[Out]

(-6766097*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2097152 - (22548119*(3 - x + 2*x^2)^(3/2))/4587520 - (9627393*x*(3 -
x + 2*x^2)^(3/2))/1146880 + (531681*x^2*(3 - x + 2*x^2)^(3/2))/71680 + (247435*x^3*(3 - x + 2*x^2)^(3/2))/1075
2 + (8825*x^4*(3 - x + 2*x^2)^(3/2))/448 + (125*x^5*(3 - x + 2*x^2)^(3/2))/16 - (155620231*ArcSinh[(1 - 4*x)/S
qrt[23]])/(4194304*Sqrt[2])

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Rubi [A]  time = 0.181374, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{125}{16} \left (2 x^2-x+3\right )^{3/2} x^5+\frac{8825}{448} \left (2 x^2-x+3\right )^{3/2} x^4+\frac{247435 \left (2 x^2-x+3\right )^{3/2} x^3}{10752}+\frac{531681 \left (2 x^2-x+3\right )^{3/2} x^2}{71680}-\frac{9627393 \left (2 x^2-x+3\right )^{3/2} x}{1146880}-\frac{22548119 \left (2 x^2-x+3\right )^{3/2}}{4587520}-\frac{6766097 (1-4 x) \sqrt{2 x^2-x+3}}{2097152}-\frac{155620231 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6766097*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2097152 - (22548119*(3 - x + 2*x^2)^(3/2))/4587520 - (9627393*x*(3 -
x + 2*x^2)^(3/2))/1146880 + (531681*x^2*(3 - x + 2*x^2)^(3/2))/71680 + (247435*x^3*(3 - x + 2*x^2)^(3/2))/1075
2 + (8825*x^4*(3 - x + 2*x^2)^(3/2))/448 + (125*x^5*(3 - x + 2*x^2)^(3/2))/16 - (155620231*ArcSinh[(1 - 4*x)/S
qrt[23]])/(4194304*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 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx &=\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{16} \int \sqrt{3-x+2 x^2} \left (128+576 x+1824 x^2+3312 x^3+2685 x^4+\frac{8825 x^5}{2}\right ) \, dx\\ &=\frac{8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{224} \int \sqrt{3-x+2 x^2} \left (1792+8064 x+25536 x^2-6582 x^3+\frac{247435 x^4}{4}\right ) \, dx\\ &=\frac{247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac{8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (21504+96768 x-\frac{1001187 x^2}{4}+\frac{1595043 x^3}{8}\right ) \, dx}{2688}\\ &=\frac{531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac{247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac{8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \left (215040-\frac{914409 x}{4}-\frac{28882179 x^2}{16}\right ) \sqrt{3-x+2 x^2} \, dx}{26880}\\ &=-\frac{9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac{531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac{247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac{8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \left (\frac{114171657}{16}-\frac{202933071 x}{32}\right ) \sqrt{3-x+2 x^2} \, dx}{215040}\\ &=-\frac{22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac{9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac{531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac{247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac{8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac{6766097 \int \sqrt{3-x+2 x^2} \, dx}{262144}\\ &=-\frac{6766097 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac{9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac{531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac{247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac{8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac{155620231 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{4194304}\\ &=-\frac{6766097 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac{9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac{531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac{247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac{8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac{\left (6766097 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4194304}\\ &=-\frac{6766097 (1-4 x) \sqrt{3-x+2 x^2}}{2097152}-\frac{22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac{9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac{531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac{247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac{8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}-\frac{155620231 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4194304 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.172468, size = 75, normalized size = 0.45 \[ \frac{4 \sqrt{2 x^2-x+3} \left (3440640000 x^7+6955008000 x^6+10958233600 x^5+11212171264 x^4+9872163456 x^3+4583812128 x^2-1621307916 x-3957369321\right )-16340124255 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{880803840} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-3957369321 - 1621307916*x + 4583812128*x^2 + 9872163456*x^3 + 11212171264*x^4 + 10958
233600*x^5 + 6955008000*x^6 + 3440640000*x^7) - 16340124255*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/880803840

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Maple [A]  time = 0.059, size = 132, normalized size = 0.8 \begin{align*}{\frac{125\,{x}^{5}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{8825\,{x}^{4}}{448} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{247435\,{x}^{3}}{10752} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{531681\,{x}^{2}}{71680} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{9627393\,x}{1146880} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-6766097+27064388\,x}{2097152}\sqrt{2\,{x}^{2}-x+3}}+{\frac{155620231\,\sqrt{2}}{8388608}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{22548119}{4587520} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2+3*x+2)^3*(2*x^2-x+3)^(1/2),x)

[Out]

125/16*x^5*(2*x^2-x+3)^(3/2)+8825/448*x^4*(2*x^2-x+3)^(3/2)+247435/10752*x^3*(2*x^2-x+3)^(3/2)+531681/71680*x^
2*(2*x^2-x+3)^(3/2)-9627393/1146880*x*(2*x^2-x+3)^(3/2)+6766097/2097152*(-1+4*x)*(2*x^2-x+3)^(1/2)[phone]/8
388608*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-22548119/4587520*(2*x^2-x+3)^(3/2)

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Maxima [A]  time = 1.47232, size = 193, normalized size = 1.16 \begin{align*} \frac{125}{16} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} + \frac{8825}{448} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + \frac{247435}{10752} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{531681}{71680} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{9627393}{1146880} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{22548119}{4587520} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{6766097}{524288} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{155620231}{8388608} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{6766097}{2097152} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x+2)^3*(2*x^2-x+3)^(1/2),x, algorithm="maxima")

[Out]

125/16*(2*x^2 - x + 3)^(3/2)*x^5 + 8825/448*(2*x^2 - x + 3)^(3/2)*x^4 + 247435/10752*(2*x^2 - x + 3)^(3/2)*x^3
+ 531681/71680*(2*x^2 - x + 3)^(3/2)*x^2 - 9627393/1146880*(2*x^2 - x + 3)^(3/2)*x - 22548119/4587520*(2*x^2
- x + 3)^(3/2) + 6766097/524288*sqrt(2*x^2 - x + 3)*x + 155620231/8388608*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x -
1)) - 6766097/2097152*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.52335, size = 352, normalized size = 2.12 \begin{align*} \frac{1}{220200960} \,{\left (3440640000 \, x^{7} + 6955008000 \, x^{6} + 10958233600 \, x^{5} + 11212171264 \, x^{4} + 9872163456 \, x^{3} + 4583812128 \, x^{2} - 1621307916 \, x - 3957369321\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{155620231}{16777216} \, \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((5*x^2+3*x+2)^3*(2*x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

1/220200960*(3440640000*x^7 + 6955008000*x^6 + 10958233600*x^5 + 11212171264*x^4 + 9872163456*x^3 + 4583812128
*x^2 - 1621307916*x - 3957369321)*sqrt(2*x^2 - x + 3) + 155620231/16777216*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 \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**2+3*x+2)**3*(2*x**2-x+3)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.32715, size = 112, normalized size = 0.67 \begin{align*} \frac{1}{220200960} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \,{\left (120 \,{\left (140 \, x + 283\right )} x + 53507\right )} x + 5474693\right )} x + 77126277\right )} x + 143244129\right )} x - 405326979\right )} x - 3957369321\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{155620231}{8388608} \, \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((5*x^2+3*x+2)^3*(2*x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

1/220200960*(4*(8*(4*(16*(100*(120*(140*x + 283)*x + 53507)*x + 5474693)*x + 77126277)*x + 143244129)*x - 4053
26979)*x - 3957369321)*sqrt(2*x^2 - x + 3) - 155620231/8388608*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2
- x + 3)) + 1)