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

Optimal. Leaf size=124 \[ \frac{125}{16} \sqrt{2 x^2-x+3} x^3+\frac{1825}{64} \sqrt{2 x^2-x+3} x^2+\frac{15565}{512} \sqrt{2 x^2-x+3} x-\frac{181561 \sqrt{2 x^2-x+3}}{2048}-\frac{1331 (17-45 x)}{368 \sqrt{2 x^2-x+3}}+\frac{1168881 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}} \]

[Out]

(-1331*(17 - 45*x))/(368*Sqrt[3 - x + 2*x^2]) - (181561*Sqrt[3 - x + 2*x^2])/2048 + (15565*x*Sqrt[3 - x + 2*x^
2])/512 + (1825*x^2*Sqrt[3 - x + 2*x^2])/64 + (125*x^3*Sqrt[3 - x + 2*x^2])/16 + (1168881*ArcSinh[(1 - 4*x)/Sq
rt[23]])/(4096*Sqrt[2])

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Rubi [A]  time = 0.127095, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{125}{16} \sqrt{2 x^2-x+3} x^3+\frac{1825}{64} \sqrt{2 x^2-x+3} x^2+\frac{15565}{512} \sqrt{2 x^2-x+3} x-\frac{181561 \sqrt{2 x^2-x+3}}{2048}-\frac{1331 (17-45 x)}{368 \sqrt{2 x^2-x+3}}+\frac{1168881 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*(17 - 45*x))/(368*Sqrt[3 - x + 2*x^2]) - (181561*Sqrt[3 - x + 2*x^2])/2048 + (15565*x*Sqrt[3 - x + 2*x^
2])/512 + (1825*x^2*Sqrt[3 - x + 2*x^2])/64 + (125*x^3*Sqrt[3 - x + 2*x^2])/16 + (1168881*ArcSinh[(1 - 4*x)/Sq
rt[23]])/(4096*Sqrt[2])

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

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 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 \frac{\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{3/2}} \, dx &=-\frac{1331 (17-45 x)}{368 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{-\frac{110285}{64}-\frac{19067 x}{32}+\frac{22195 x^2}{16}+\frac{13225 x^3}{8}+\frac{2875 x^4}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{1331 (17-45 x)}{368 \sqrt{3-x+2 x^2}}+\frac{125}{16} x^3 \sqrt{3-x+2 x^2}+\frac{1}{92} \int \frac{-\frac{110285}{8}-\frac{19067 x}{4}+\frac{18515 x^2}{4}+\frac{125925 x^3}{8}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{1331 (17-45 x)}{368 \sqrt{3-x+2 x^2}}+\frac{1825}{64} x^2 \sqrt{3-x+2 x^2}+\frac{125}{16} x^3 \sqrt{3-x+2 x^2}+\frac{1}{552} \int \frac{-\frac{330855}{4}-\frac{492177 x}{4}+\frac{1073985 x^2}{16}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{1331 (17-45 x)}{368 \sqrt{3-x+2 x^2}}+\frac{15565}{512} x \sqrt{3-x+2 x^2}+\frac{1825}{64} x^2 \sqrt{3-x+2 x^2}+\frac{125}{16} x^3 \sqrt{3-x+2 x^2}+\frac{\int \frac{-\frac{8515635}{16}-\frac{12527709 x}{32}}{\sqrt{3-x+2 x^2}} \, dx}{2208}\\ &=-\frac{1331 (17-45 x)}{368 \sqrt{3-x+2 x^2}}-\frac{181561 \sqrt{3-x+2 x^2}}{2048}+\frac{15565}{512} x \sqrt{3-x+2 x^2}+\frac{1825}{64} x^2 \sqrt{3-x+2 x^2}+\frac{125}{16} x^3 \sqrt{3-x+2 x^2}-\frac{1168881 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{4096}\\ &=-\frac{1331 (17-45 x)}{368 \sqrt{3-x+2 x^2}}-\frac{181561 \sqrt{3-x+2 x^2}}{2048}+\frac{15565}{512} x \sqrt{3-x+2 x^2}+\frac{1825}{64} x^2 \sqrt{3-x+2 x^2}+\frac{125}{16} x^3 \sqrt{3-x+2 x^2}-\frac{1168881 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4096 \sqrt{46}}\\ &=-\frac{1331 (17-45 x)}{368 \sqrt{3-x+2 x^2}}-\frac{181561 \sqrt{3-x+2 x^2}}{2048}+\frac{15565}{512} x \sqrt{3-x+2 x^2}+\frac{1825}{64} x^2 \sqrt{3-x+2 x^2}+\frac{125}{16} x^3 \sqrt{3-x+2 x^2}+\frac{1168881 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.228601, size = 65, normalized size = 0.52 \[ \frac{\frac{4 \left (736000 x^5+2318400 x^4+2624760 x^3-5754186 x^2+16138403 x-15423965\right )}{\sqrt{2 x^2-x+3}}-26884263 \sqrt{2} \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{188416} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(-15423965 + 16138403*x - 5754186*x^2 + 2624760*x^3 + 2318400*x^4 + 736000*x^5))/Sqrt[3 - x + 2*x^2] - 268
84263*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]])/188416

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Maple [A]  time = 0.059, size = 132, normalized size = 1.1 \begin{align*}{\frac{125\,{x}^{5}}{8}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{1575\,{x}^{4}}{32}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{1168881\,\sqrt{2}}{8192}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-5392543+21570172\,x}{376832}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{1168881\,x}{4096}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{14265\,{x}^{3}}{256}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{125091\,{x}^{2}}{1024}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{5130399}{16384}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}} \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)^(3/2),x)

[Out]

125/8*x^5/(2*x^2-x+3)^(1/2)+1575/32*x^4/(2*x^2-x+3)^(1/2)-1168881/8192*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+
5392543/376832*(-1+4*x)/(2*x^2-x+3)^(1/2)+1168881/4096*x/(2*x^2-x+3)^(1/2)+14265/256*x^3/(2*x^2-x+3)^(1/2)-125
091/1024*x^2/(2*x^2-x+3)^(1/2)-5130399/16384/(2*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.52628, size = 154, normalized size = 1.24 \begin{align*} \frac{125 \, x^{5}}{8 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{1575 \, x^{4}}{32 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{14265 \, x^{3}}{256 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{125091 \, x^{2}}{1024 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{1168881}{8192} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{16138403 \, x}{47104 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{15423965}{47104 \, \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)^(3/2),x, algorithm="maxima")

[Out]

125/8*x^5/sqrt(2*x^2 - x + 3) + 1575/32*x^4/sqrt(2*x^2 - x + 3) + 14265/256*x^3/sqrt(2*x^2 - x + 3) - 125091/1
024*x^2/sqrt(2*x^2 - x + 3) - 1168881/8192*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 16138403/47104*x/sqrt(2*
x^2 - x + 3) - 15423965/47104/sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.32687, size = 311, normalized size = 2.51 \begin{align*} \frac{26884263 \, \sqrt{2}{\left (2 \, x^{2} - x + 3\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \,{\left (736000 \, x^{5} + 2318400 \, x^{4} + 2624760 \, x^{3} - 5754186 \, x^{2} + 16138403 \, x - 15423965\right )} \sqrt{2 \, x^{2} - x + 3}}{376832 \,{\left (2 \, x^{2} - x + 3\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)^(3/2),x, algorithm="fricas")

[Out]

1/376832*(26884263*sqrt(2)*(2*x^2 - x + 3)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) +
8*(736000*x^5 + 2318400*x^4 + 2624760*x^3 - 5754186*x^2 + 16138403*x - 15423965)*sqrt(2*x^2 - x + 3))/(2*x^2
- x + 3)

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

[Out]

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

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Giac [A]  time = 1.20565, size = 97, normalized size = 0.78 \begin{align*} \frac{1168881}{8192} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left (46 \,{\left (20 \,{\left (40 \,{\left (20 \, x + 63\right )} x + 2853\right )} x - 125091\right )} x + 16138403\right )} x - 15423965}{47104 \, \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)^(3/2),x, algorithm="giac")

[Out]

1168881/8192*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/47104*((46*(20*(40*(20*x + 63)*
x + 2853)*x - 125091)*x + 16138403)*x - 15423965)/sqrt(2*x^2 - x + 3)