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

Optimal. Leaf size=85 \[ \frac{917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}-\frac{146729 x+335337}{1371168 \sqrt{2 x^2-x+3}}-\frac{3667 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{31104 \sqrt{2}} \]

[Out]

(1191 + 917*x)/(9936*(3 - x + 2*x^2)^(3/2)) - (335337 + 146729*x)/(1371168*Sqrt[3 - x + 2*x^2]) - (3667*ArcTan
h[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(31104*Sqrt[2])

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Rubi [A]  time = 0.128261, antiderivative size = 85, 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, 12, 724, 206} \[ \frac{917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}-\frac{146729 x+335337}{1371168 \sqrt{2 x^2-x+3}}-\frac{3667 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{31104 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1191 + 917*x)/(9936*(3 - x + 2*x^2)^(3/2)) - (335337 + 146729*x)/(1371168*Sqrt[3 - x + 2*x^2]) - (3667*ArcTan
h[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(31104*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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) \left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac{1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{-\frac{1877}{576}+\frac{695 x}{18}+\frac{345 x^2}{4}}{(5+2 x) \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac{1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac{335337+146729 x}{1371168 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{1939843}{6912 (5+2 x) \sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=\frac{1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac{335337+146729 x}{1371168 \sqrt{3-x+2 x^2}}+\frac{3667 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{5184}\\ &=\frac{1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac{335337+146729 x}{1371168 \sqrt{3-x+2 x^2}}-\frac{3667 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{2592}\\ &=\frac{1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac{335337+146729 x}{1371168 \sqrt{3-x+2 x^2}}-\frac{3667 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{31104 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.466597, size = 80, normalized size = 0.94 \[ \frac{-\frac{12 \sqrt{2} \left (293458 x^3+523945 x^2-21696 x+841653\right )}{529 \left (2 x^2-x+3\right )^{3/2}}-3667 \log \left (12 \sqrt{4 x^2-2 x+6}-22 x+17\right )+3667 \log (2 x+5)}{31104 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*Sqrt[2]*(841653 - 21696*x + 523945*x^2 + 293458*x^3))/(529*(3 - x + 2*x^2)^(3/2)) + 3667*Log[5 + 2*x] -
3667*Log[17 - 22*x + 12*Sqrt[6 - 2*x + 4*x^2]])/(31104*Sqrt[2])

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Maple [B]  time = 0.056, size = 190, normalized size = 2.2 \begin{align*} -{\frac{5\,{x}^{2}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{59\,x}{32} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{1597}{384} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{-3817+15268\,x}{2944} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{-3817+15268\,x}{4232}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{3667}{1728} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-40337+161348\,x}{39744} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{-{\frac{3}{2}}}}+{\frac{-4800103+19200412\,x}{5484672}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{3667}{10368}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{3667\,\sqrt{2}}{62208}{\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)/(2*x^2-x+3)^(5/2),x)

[Out]

-5/4*x^2/(2*x^2-x+3)^(3/2)+59/32*x/(2*x^2-x+3)^(3/2)-1597/384/(2*x^2-x+3)^(3/2)-3817/2944*(-1+4*x)/(2*x^2-x+3)
^(3/2)-3817/4232*(-1+4*x)/(2*x^2-x+3)^(1/2)+3667/1728/(2*(x+5/2)^2-11*x-19/2)^(3/2)+40337/39744*(-1+4*x)/(2*(x
+5/2)^2-11*x-19/2)^(3/2)+4800103/5484672*(-1+4*x)/(2*(x+5/2)^2-11*x-19/2)^(1/2)+3667/10368/(2*(x+5/2)^2-11*x-1
9/2)^(1/2)-3667/62208*2^(1/2)*arctanh(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 = 1.65677, size = 149, normalized size = 1.75 \begin{align*} \frac{3667}{62208} \, \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{146729 \, x}{1371168 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{5 \, x^{2}}{4 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{173881}{457056 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{7127 \, x}{9936 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{5813}{3312 \,{\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*x^2-x+3)^(5/2),x, algorithm="maxima")

[Out]

3667/62208*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) - 146729/1371168*x/sqr
t(2*x^2 - x + 3) - 5/4*x^2/(2*x^2 - x + 3)^(3/2) + 173881/457056/sqrt(2*x^2 - x + 3) + 7127/9936*x/(2*x^2 - x
+ 3)^(3/2) - 5813/3312/(2*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.31877, size = 362, normalized size = 4.26 \begin{align*} \frac{1939843 \, \sqrt{2}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\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 (293458 \, x^{3} + 523945 \, x^{2} - 21696 \, x + 841653\right )} \sqrt{2 \, x^{2} - x + 3}}{65816064 \,{\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*x^4-x^3+3*x^2+x+2)/(5+2*x)/(2*x^2-x+3)^(5/2),x, algorithm="fricas")

[Out]

1/65816064*(1939843*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*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*(293458*x^3 + 523945*x^2 - 21696*x + 841653)*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{5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\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*x**4-x**3+3*x**2+x+2)/(5+2*x)/(2*x**2-x+3)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.17316, size = 124, normalized size = 1.46 \begin{align*} -\frac{3667}{62208} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{3667}{62208} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{{\left ({\left (293458 \, x + 523945\right )} x - 21696\right )} x + 841653}{1371168 \,{\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*x^2-x+3)^(5/2),x, algorithm="giac")

[Out]

-3667/62208*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 3667/62208*sqrt(2)*log(abs(-2*s
qrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/1371168*(((293458*x + 523945)*x - 21696)*x + 841653)/(2*x^
2 - x + 3)^(3/2)