### 3.363 $$\int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x)^3 (3-x+2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=135 $-\frac{4679797-2148263 x}{592344576 \sqrt{2 x^2-x+3}}-\frac{45979 \sqrt{2 x^2-x+3}}{26873856 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{373248 (2 x+5)^2}+\frac{65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}+\frac{774079 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{322486272 \sqrt{2}}$

[Out]

(65991 - 8779*x)/(12877056*(3 - x + 2*x^2)^(3/2)) - (4679797 - 2148263*x)/(592344576*Sqrt[3 - x + 2*x^2]) - (3
667*Sqrt[3 - x + 2*x^2])/(373248*(5 + 2*x)^2) - (45979*Sqrt[3 - x + 2*x^2])/(26873856*(5 + 2*x)) + (774079*Arc
Tanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(322486272*Sqrt[2])

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Rubi [A]  time = 0.220896, antiderivative size = 135, 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 = {1646, 1650, 806, 724, 206} $-\frac{4679797-2148263 x}{592344576 \sqrt{2 x^2-x+3}}-\frac{45979 \sqrt{2 x^2-x+3}}{26873856 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{373248 (2 x+5)^2}+\frac{65991-8779 x}{12877056 \left (2 x^2-x+3\right )^{3/2}}+\frac{774079 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{322486272 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(65991 - 8779*x)/(12877056*(3 - x + 2*x^2)^(3/2)) - (4679797 - 2148263*x)/(592344576*Sqrt[3 - x + 2*x^2]) - (3
667*Sqrt[3 - x + 2*x^2])/(373248*(5 + 2*x)^2) - (45979*Sqrt[3 - x + 2*x^2])/(26873856*(5 + 2*x)) + (774079*Arc
Tanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(322486272*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 1650

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

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)^3 \left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{11115283}{746496}+\frac{3198845 x}{62208}+\frac{605005 x^2}{6912}-\frac{8779 x^3}{23328}}{(5+2 x)^3 \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{-\frac{171639869}{2985984}-\frac{142392517 x}{746496}-\frac{16570925 x^2}{746496}}{(5+2 x)^3 \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{373248 (5+2 x)^2}-\frac{\int \frac{\frac{34040621}{10368}+\frac{28209983 x}{10368}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{57132}\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{373248 (5+2 x)^2}-\frac{45979 \sqrt{3-x+2 x^2}}{26873856 (5+2 x)}-\frac{774079 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{53747712}\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{373248 (5+2 x)^2}-\frac{45979 \sqrt{3-x+2 x^2}}{26873856 (5+2 x)}+\frac{774079 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{26873856}\\ &=\frac{65991-8779 x}{12877056 \left (3-x+2 x^2\right )^{3/2}}-\frac{4679797-2148263 x}{592344576 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{373248 (5+2 x)^2}-\frac{45979 \sqrt{3-x+2 x^2}}{26873856 (5+2 x)}+\frac{774079 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{322486272 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.306919, size = 97, normalized size = 0.72 $\frac{\frac{12 \sqrt{2} \left (217883368 x^5+107028732 x^4-1503926130 x^3-5919924791 x^2+2280511668 x-8953831359\right )}{529 (2 x+5)^2 \left (2 x^2-x+3\right )^{3/2}}+774079 \log \left (12 \sqrt{4 x^2-2 x+6}-22 x+17\right )-774079 \log (2 x+5)}{322486272 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((12*Sqrt[2]*(-8953831359 + 2280511668*x - 5919924791*x^2 - 1503926130*x^3 + 107028732*x^4 + 217883368*x^5))/(
529*(5 + 2*x)^2*(3 - x + 2*x^2)^(3/2)) - 774079*Log[5 + 2*x] + 774079*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]
])/(322486272*Sqrt[2])

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Maple [A]  time = 0.062, size = 200, normalized size = 1.5 \begin{align*} -{\frac{5}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{-149+596\,x}{1104} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{-149+596\,x}{1587}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{3667}{4608} \left ( x+{\frac{5}{2}} \right ) ^{-2} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{115369}{165888} \left ( x+{\frac{5}{2}} \right ) ^{-1} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}-{\frac{774079}{17915904} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-57937675+231750700\,x}{412065792} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-5366174813+21464699252\,x}{56865079296}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{774079}{107495424}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{774079\,\sqrt{2}}{644972544}{\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 ) } \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)^3/(2*x^2-x+3)^(5/2),x)

[Out]

-5/48/(2*x^2-x+3)^(3/2)-149/1104*(-1+4*x)/(2*x^2-x+3)^(3/2)-149/1587*(-1+4*x)/(2*x^2-x+3)^(1/2)-3667/4608/(x+5
/2)^2/(2*(x+5/2)^2-11*x-19/2)^(3/2)+115369/165888/(x+5/2)/(2*(x+5/2)^2-11*x-19/2)^(3/2)-774079/17915904/(2*(x+
5/2)^2-11*x-19/2)^(3/2)+57937675/412065792*(-1+4*x)/(2*(x+5/2)^2-11*x-19/2)^(3/2)+5366174813/56865079296*(-1+4
*x)/(2*(x+5/2)^2-11*x-19/2)^(1/2)-774079/107495424/(2*(x+5/2)^2-11*x-19/2)^(1/2)+774079/644972544*2^(1/2)*arct
anh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))

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Maxima [A]  time = 1.61319, size = 240, normalized size = 1.78 \begin{align*} -\frac{774079}{644972544} \, \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{27235421 \, x}{14216269824 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{36393601}{4738756608 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{2323723 \, x}{103016448 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{3667}{1152 \,{\left (4 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 20 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 25 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{115369}{82944 \,{\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{5254255}{34338816 \,{\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)^3/(2*x^2-x+3)^(5/2),x, algorithm="maxima")

[Out]

-774079/644972544*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 27235421/1421
6269824*x/sqrt(2*x^2 - x + 3) - 36393601/4738756608/sqrt(2*x^2 - x + 3) + 2323723/103016448*x/(2*x^2 - x + 3)^
(3/2) - 3667/1152/(4*(2*x^2 - x + 3)^(3/2)*x^2 + 20*(2*x^2 - x + 3)^(3/2)*x + 25*(2*x^2 - x + 3)^(3/2)) + 1153
69/82944/(2*(2*x^2 - x + 3)^(3/2)*x + 5*(2*x^2 - x + 3)^(3/2)) - 5254255/34338816/(2*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.40361, size = 502, normalized size = 3.72 \begin{align*} \frac{409487791 \, \sqrt{2}{\left (16 \, x^{6} + 64 \, x^{5} + 72 \, x^{4} + 136 \, x^{3} + 241 \, x^{2} + 30 \, x + 225\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 (217883368 \, x^{5} + 107028732 \, x^{4} - 1503926130 \, x^{3} - 5919924791 \, x^{2} + 2280511668 \, x - 8953831359\right )} \sqrt{2 \, x^{2} - x + 3}}{682380951552 \,{\left (16 \, x^{6} + 64 \, x^{5} + 72 \, x^{4} + 136 \, x^{3} + 241 \, x^{2} + 30 \, x + 225\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)^3/(2*x^2-x+3)^(5/2),x, algorithm="fricas")

[Out]

1/682380951552*(409487791*sqrt(2)*(16*x^6 + 64*x^5 + 72*x^4 + 136*x^3 + 241*x^2 + 30*x + 225)*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*(217883368*x^5 + 1070287
32*x^4 - 1503926130*x^3 - 5919924791*x^2 + 2280511668*x - 8953831359)*sqrt(2*x^2 - x + 3))/(16*x^6 + 64*x^5 +
72*x^4 + 136*x^3 + 241*x^2 + 30*x + 225)

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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 )^{3} \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)**3/(2*x**2-x+3)**(5/2),x)

[Out]

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

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Giac [B]  time = 1.21714, size = 308, normalized size = 2.28 \begin{align*} \frac{774079}{644972544} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{774079}{644972544} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{\sqrt{2}{\left (44558 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} - 10136238 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 16812201 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 10182217\right )}}{53747712 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} + \frac{{\left ({\left (4296526 \, x - 11507857\right )} x + 10720752\right )} x - 11003805}{592344576 \,{\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)^3/(2*x^2-x+3)^(5/2),x, algorithm="giac")

[Out]

774079/644972544*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 774079/644972544*sqrt(2)*l
og(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 1/53747712*sqrt(2)*(44558*sqrt(2)*(sqrt(2)*x - sq
rt(2*x^2 - x + 3))^3 - 10136238*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 16812201*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2
- x + 3)) - 10182217)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) -
11)^2 + 1/592344576*(((4296526*x - 11507857)*x + 10720752)*x - 11003805)/(2*x^2 - x + 3)^(3/2)