### 3.358 $$\int \frac{(5+2 x)^2 (2+x+3 x^2-x^3+5 x^4)}{(3-x+2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=105 $\frac{4 (18982-20383 x)}{1587 \sqrt{2 x^2-x+3}}+\frac{5}{4} x \sqrt{2 x^2-x+3}+\frac{247}{16} \sqrt{2 x^2-x+3}-\frac{4 (346-533 x)}{69 \left (2 x^2-x+3\right )^{3/2}}-\frac{1471 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}}$

[Out]

(-4*(346 - 533*x))/(69*(3 - x + 2*x^2)^(3/2)) + (4*(18982 - 20383*x))/(1587*Sqrt[3 - x + 2*x^2]) + (247*Sqrt[3
- x + 2*x^2])/16 + (5*x*Sqrt[3 - x + 2*x^2])/4 - (1471*ArcSinh[(1 - 4*x)/Sqrt[23]])/(32*Sqrt[2])

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Rubi [A]  time = 0.130773, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {1660, 1661, 640, 619, 215} $\frac{4 (18982-20383 x)}{1587 \sqrt{2 x^2-x+3}}+\frac{5}{4} x \sqrt{2 x^2-x+3}+\frac{247}{16} \sqrt{2 x^2-x+3}-\frac{4 (346-533 x)}{69 \left (2 x^2-x+3\right )^{3/2}}-\frac{1471 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-4*(346 - 533*x))/(69*(3 - x + 2*x^2)^(3/2)) + (4*(18982 - 20383*x))/(1587*Sqrt[3 - x + 2*x^2]) + (247*Sqrt[3
- x + 2*x^2])/16 + (5*x*Sqrt[3 - x + 2*x^2])/4 - (1471*ArcSinh[(1 - 4*x)/Sqrt[23]])/(32*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{(5+2 x)^2 \left (2+x+3 x^2-x^3+5 x^4\right )}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{-145-\frac{1725 x}{2}+2415 x^2+\frac{3657 x^3}{2}+345 x^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{\frac{33327}{2}+\frac{46023 x}{4}+\frac{7935 x^2}{4}}{\sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{5}{4} x \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{242811}{4}+\frac{391989 x}{8}}{\sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{247}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}+\frac{1471}{32} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{247}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}+\frac{1471 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{32 \sqrt{46}}\\ &=-\frac{4 (346-533 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{4 (18982-20383 x)}{1587 \sqrt{3-x+2 x^2}}+\frac{247}{16} \sqrt{3-x+2 x^2}+\frac{5}{4} x \sqrt{3-x+2 x^2}-\frac{1471 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.711421, size = 65, normalized size = 0.62 $\frac{126960 x^5+1440996 x^4-3764360 x^3+8639625 x^2-6410082 x+6663133}{25392 \left (2 x^2-x+3\right )^{3/2}}-\frac{1471 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(6663133 - 6410082*x + 8639625*x^2 - 3764360*x^3 + 1440996*x^4 + 126960*x^5)/(25392*(3 - x + 2*x^2)^(3/2)) - (
1471*ArcSinh[(1 - 4*x)/Sqrt[23]])/(32*Sqrt[2])

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Maple [B]  time = 0.059, size = 180, normalized size = 1.7 \begin{align*} 5\,{\frac{{x}^{5}}{ \left ( 2\,{x}^{2}-x+3 \right ) ^{3/2}}}-{\frac{1471\,{x}^{3}}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{19073\,{x}^{2}}{64} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{1471\,\sqrt{2}}{64}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{-162931+651724\,x}{50784}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{-753223+3012892\,x}{141312} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{32257\,x}{512} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{227\,{x}^{4}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{1471\,x}{32}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{1471}{128}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{577397}{2048} \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+2*x)^2*(5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x)

[Out]

5*x^5/(2*x^2-x+3)^(3/2)-1471/48*x^3/(2*x^2-x+3)^(3/2)+19073/64*x^2/(2*x^2-x+3)^(3/2)+1471/64*2^(1/2)*arcsinh(4
/23*23^(1/2)*(x-1/4))-162931/50784*(-1+4*x)/(2*x^2-x+3)^(1/2)-753223/141312*(-1+4*x)/(2*x^2-x+3)^(3/2)-32257/5
12*x/(2*x^2-x+3)^(3/2)+227/4*x^4/(2*x^2-x+3)^(3/2)-1471/32*x/(2*x^2-x+3)^(1/2)-1471/128/(2*x^2-x+3)^(1/2)+5773
97/2048/(2*x^2-x+3)^(3/2)

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Maxima [B]  time = 1.55226, size = 296, normalized size = 2.82 \begin{align*} \frac{5 \, x^{5}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{227 \, x^{4}}{4 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1471}{50784} \, x{\left (\frac{284 \, x}{\sqrt{2 \, x^{2} - x + 3}} - \frac{3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{71}{\sqrt{2 \, x^{2} - x + 3}} + \frac{805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}\right )} + \frac{1471}{64} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{104441}{25392} \, \sqrt{2 \, x^{2} - x + 3} - \frac{383581 \, x}{12696 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{321 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{15965}{4232 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{4147 \, x}{46 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{42883}{138 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)^2*(5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="maxima")

[Out]

5*x^5/(2*x^2 - x + 3)^(3/2) + 227/4*x^4/(2*x^2 - x + 3)^(3/2) + 1471/50784*x*(284*x/sqrt(2*x^2 - x + 3) - 3174
*x^2/(2*x^2 - x + 3)^(3/2) - 71/sqrt(2*x^2 - x + 3) + 805*x/(2*x^2 - x + 3)^(3/2) - 3243/(2*x^2 - x + 3)^(3/2)
) + 1471/64*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 104441/25392*sqrt(2*x^2 - x + 3) - 383581/12696*x/sqrt(
2*x^2 - x + 3) + 321*x^2/(2*x^2 - x + 3)^(3/2) - 15965/4232/sqrt(2*x^2 - x + 3) - 4147/46*x/(2*x^2 - x + 3)^(3
/2) + 42883/138/(2*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.33287, size = 359, normalized size = 3.42 \begin{align*} \frac{2334477 \, \sqrt{2}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\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 (126960 \, x^{5} + 1440996 \, x^{4} - 3764360 \, x^{3} + 8639625 \, x^{2} - 6410082 \, x + 6663133\right )} \sqrt{2 \, x^{2} - x + 3}}{203136 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)^2*(5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="fricas")

[Out]

1/203136*(2334477*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32
*x^2 + 16*x - 25) + 8*(126960*x^5 + 1440996*x^4 - 3764360*x^3 + 8639625*x^2 - 6410082*x + 6663133)*sqrt(2*x^2
- x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (2 x + 5\right )^{2} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x^{2} - x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16211, size = 96, normalized size = 0.91 \begin{align*} -\frac{1471}{64} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left ({\left (4 \,{\left (1587 \,{\left (20 \, x + 227\right )} x - 941090\right )} x + 8639625\right )} x - 6410082\right )} x + 6663133}{25392 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+2*x)^2*(5*x^4-x^3+3*x^2+x+2)/(2*x^2-x+3)^(5/2),x, algorithm="giac")

[Out]

-1471/64*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/25392*(((4*(1587*(20*x + 227)*x - 9
41090)*x + 8639625)*x - 6410082)*x + 6663133)/(2*x^2 - x + 3)^(3/2)