### 3.362 $$\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^2 (3-x+2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=110 $-\frac{1255878-62021 x}{24681024 \sqrt{2 x^2-x+3}}-\frac{3667 \sqrt{2 x^2-x+3}}{186624 (2 x+5)}+\frac{2203 x+9897}{357696 \left (2 x^2-x+3\right )^{3/2}}-\frac{2821 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{2239488 \sqrt{2}}$

[Out]

(9897 + 2203*x)/(357696*(3 - x + 2*x^2)^(3/2)) - (1255878 - 62021*x)/(24681024*Sqrt[3 - x + 2*x^2]) - (3667*Sq
rt[3 - x + 2*x^2])/(186624*(5 + 2*x)) - (2821*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(2239488*
Sqrt[2])

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Rubi [A]  time = 0.152912, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 40, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {1646, 806, 724, 206} $-\frac{1255878-62021 x}{24681024 \sqrt{2 x^2-x+3}}-\frac{3667 \sqrt{2 x^2-x+3}}{186624 (2 x+5)}+\frac{2203 x+9897}{357696 \left (2 x^2-x+3\right )^{3/2}}-\frac{2821 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{2239488 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(9897 + 2203*x)/(357696*(3 - x + 2*x^2)^(3/2)) - (1255878 - 62021*x)/(24681024*Sqrt[3 - x + 2*x^2]) - (3667*Sq
rt[3 - x + 2*x^2])/(186624*(5 + 2*x)) - (2821*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(2239488*
Sqrt[2])

Rule 1646

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

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^2 \left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{119353}{20736}+\frac{481765 x}{10368}+\frac{113983 x^2}{1296}}{(5+2 x)^2 \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}-\frac{1255878-62021 x}{24681024 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{\frac{10109719}{124416}-\frac{4961491 x}{62208}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}-\frac{1255878-62021 x}{24681024 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{186624 (5+2 x)}+\frac{2821 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{373248}\\ &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}-\frac{1255878-62021 x}{24681024 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{186624 (5+2 x)}-\frac{2821 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{186624}\\ &=\frac{9897+2203 x}{357696 \left (3-x+2 x^2\right )^{3/2}}-\frac{1255878-62021 x}{24681024 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{186624 (5+2 x)}-\frac{2821 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{2239488 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.406562, size = 92, normalized size = 0.84 $\frac{-\frac{12 \sqrt{2} \left (6767036 x^4+10350004 x^3+63941915 x^2-18840090 x+79153407\right )}{529 (2 x+5) \left (2 x^2-x+3\right )^{3/2}}-2821 \log \left (12 \sqrt{4 x^2-2 x+6}-22 x+17\right )+2821 \log (2 x+5)}{2239488 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-12*Sqrt[2]*(79153407 - 18840090*x + 63941915*x^2 + 10350004*x^3 + 6767036*x^4))/(529*(5 + 2*x)*(3 - x + 2*x
^2)^(3/2)) + 2821*Log[5 + 2*x] - 2821*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(2239488*Sqrt[2])

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Maple [B]  time = 0.058, size = 194, normalized size = 1.8 \begin{align*} -{\frac{5\,x}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{203}{192} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-3173+12692\,x}{4416} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{-3173+12692\,x}{6348}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{2821}{124416} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-2081161+8324644\,x}{2861568} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}-{\frac{-199077743+796310972\,x}{394896384}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{2821}{746496}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{2821\,\sqrt{2}}{4478976}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }-{\frac{3667}{1152} \left ( x+{\frac{5}{2}} \right ) ^{-1} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/16*x/(2*x^2-x+3)^(3/2)+203/192/(2*x^2-x+3)^(3/2)+3173/4416*(-1+4*x)/(2*x^2-x+3)^(3/2)+3173/6348*(-1+4*x)/(2
*x^2-x+3)^(1/2)+2821/124416/(2*(x+5/2)^2-11*x-19/2)^(3/2)-2081161/2861568*(-1+4*x)/(2*(x+5/2)^2-11*x-19/2)^(3/
2)-199077743/394896384*(-1+4*x)/(2*(x+5/2)^2-11*x-19/2)^(1/2)+2821/746496/(2*(x+5/2)^2-11*x-19/2)^(1/2)-2821/4
478976*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))-3667/1152/(x+5/2)/(2*(x+5/2)^2-
11*x-19/2)^(3/2)

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Maxima [A]  time = 1.55799, size = 171, normalized size = 1.55 \begin{align*} \frac{2821}{4478976} \, \sqrt{2} \operatorname{arsinh}\left (\frac{22 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 5 \right |}} - \frac{17 \, \sqrt{23}}{23 \,{\left | 2 \, x + 5 \right |}}\right ) - \frac{1691759 \, x}{98724096 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{265339}{32908032 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{248617 \, x}{715392 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3667}{576 \,{\left (2 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 5 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{259621}{238464 \,{\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*x^4-x^3+3*x^2+x+2)/(5+2*x)^2/(2*x^2-x+3)^(5/2),x, algorithm="maxima")

[Out]

2821/4478976*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) - 1691759/98724096*x
/sqrt(2*x^2 - x + 3) + 265339/32908032/sqrt(2*x^2 - x + 3) - 248617/715392*x/(2*x^2 - x + 3)^(3/2) - 3667/576/
(2*(2*x^2 - x + 3)^(3/2)*x + 5*(2*x^2 - x + 3)^(3/2)) + 259621/238464/(2*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.31711, size = 425, normalized size = 3.86 \begin{align*} \frac{1492309 \, \sqrt{2}{\left (8 \, x^{5} + 12 \, x^{4} + 6 \, x^{3} + 53 \, x^{2} - 12 \, x + 45\right )} \log \left (-\frac{24 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) - 48 \,{\left (6767036 \, x^{4} + 10350004 \, x^{3} + 63941915 \, x^{2} - 18840090 \, x + 79153407\right )} \sqrt{2 \, x^{2} - x + 3}}{4738756608 \,{\left (8 \, x^{5} + 12 \, x^{4} + 6 \, x^{3} + 53 \, x^{2} - 12 \, x + 45\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4738756608*(1492309*sqrt(2)*(8*x^5 + 12*x^4 + 6*x^3 + 53*x^2 - 12*x + 45)*log(-(24*sqrt(2)*sqrt(2*x^2 - x +
3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25)) - 48*(6767036*x^4 + 10350004*x^3 + 63941915*x^
2 - 18840090*x + 79153407)*sqrt(2*x^2 - x + 3))/(8*x^5 + 12*x^4 + 6*x^3 + 53*x^2 - 12*x + 45)

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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