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

Optimal. Leaf size=137 \[ \frac{369609-175877 x}{154524672 \sqrt{2 x^2-x+3}}+\frac{430799 \sqrt{2 x^2-x+3}}{107495424 (2 x+5)}+\frac{152885 \sqrt{2 x^2-x+3}}{4478976 (2 x+5)^2}-\frac{3667 \sqrt{2 x^2-x+3}}{31104 (2 x+5)^3}-\frac{3505819 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1289945088 \sqrt{2}} \]

[Out]

(369609 - 175877*x)/(154524672*Sqrt[3 - x + 2*x^2]) - (3667*Sqrt[3 - x + 2*x^2])/(31104*(5 + 2*x)^3) + (152885
*Sqrt[3 - x + 2*x^2])/(4478976*(5 + 2*x)^2) + (430799*Sqrt[3 - x + 2*x^2])/(107495424*(5 + 2*x)) - (3505819*Ar
cTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(1289945088*Sqrt[2])

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Rubi [A]  time = 0.203512, antiderivative size = 137, 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} \[ \frac{369609-175877 x}{154524672 \sqrt{2 x^2-x+3}}+\frac{430799 \sqrt{2 x^2-x+3}}{107495424 (2 x+5)}+\frac{152885 \sqrt{2 x^2-x+3}}{4478976 (2 x+5)^2}-\frac{3667 \sqrt{2 x^2-x+3}}{31104 (2 x+5)^3}-\frac{3505819 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1289945088 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(369609 - 175877*x)/(154524672*Sqrt[3 - x + 2*x^2]) - (3667*Sqrt[3 - x + 2*x^2])/(31104*(5 + 2*x)^3) + (152885
*Sqrt[3 - x + 2*x^2])/(4478976*(5 + 2*x)^2) + (430799*Sqrt[3 - x + 2*x^2])/(107495424*(5 + 2*x)) - (3505819*Ar
cTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(1289945088*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)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx &=\frac{369609-175877 x}{154524672 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{\frac{348877271}{26873856}+\frac{119871055 x}{4478976}+\frac{73960295 x^2}{2239488}+\frac{1302559 x^3}{3359232}}{(5+2 x)^4 \sqrt{3-x+2 x^2}} \, dx\\ &=\frac{369609-175877 x}{154524672 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{31104 (5+2 x)^3}-\frac{\int \frac{\frac{79609325}{124416}-\frac{71248733 x}{31104}-\frac{1302559 x^2}{31104}}{(5+2 x)^3 \sqrt{3-x+2 x^2}} \, dx}{2484}\\ &=\frac{369609-175877 x}{154524672 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{31104 (5+2 x)^3}+\frac{152885 \sqrt{3-x+2 x^2}}{4478976 (5+2 x)^2}+\frac{\int \frac{\frac{29340847}{1728}+\frac{1481453 x}{54}}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx}{357696}\\ &=\frac{369609-175877 x}{154524672 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{31104 (5+2 x)^3}+\frac{152885 \sqrt{3-x+2 x^2}}{4478976 (5+2 x)^2}+\frac{430799 \sqrt{3-x+2 x^2}}{107495424 (5+2 x)}+\frac{3505819 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{214990848}\\ &=\frac{369609-175877 x}{154524672 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{31104 (5+2 x)^3}+\frac{152885 \sqrt{3-x+2 x^2}}{4478976 (5+2 x)^2}+\frac{430799 \sqrt{3-x+2 x^2}}{107495424 (5+2 x)}-\frac{3505819 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{107495424}\\ &=\frac{369609-175877 x}{154524672 \sqrt{3-x+2 x^2}}-\frac{3667 \sqrt{3-x+2 x^2}}{31104 (5+2 x)^3}+\frac{152885 \sqrt{3-x+2 x^2}}{4478976 (5+2 x)^2}+\frac{430799 \sqrt{3-x+2 x^2}}{107495424 (5+2 x)}-\frac{3505819 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{1289945088 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.164399, size = 95, normalized size = 0.69 \[ \frac{24 \left (56754760 x^4+572739684 x^3+441046842 x^2+1257975811 x+1873786587\right )-80633837 (2 x+5)^3 \sqrt{4 x^2-2 x+6} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )}{59337474048 (2 x+5)^3 \sqrt{2 x^2-x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*(1873786587 + 1257975811*x + 441046842*x^2 + 572739684*x^3 + 56754760*x^4) - 80633837*(5 + 2*x)^3*Sqrt[6 -
2*x + 4*x^2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/(59337474048*(5 + 2*x)^3*Sqrt[3 - x + 2*x^2])

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Maple [A]  time = 0.063, size = 151, normalized size = 1.1 \begin{align*}{\frac{-5+20\,x}{184}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{314233}{995328} \left ( x+{\frac{5}{2}} \right ) ^{-2}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{3127169}{35831808} \left ( x+{\frac{5}{2}} \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{3505819}{429981696}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{-261644215+1046576860\,x}{9889579008}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{3505819\,\sqrt{2}}{2579890176}{\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}{13824} \left ( x+{\frac{5}{2}} \right ) ^{-3}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{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)^4/(2*x^2-x+3)^(3/2),x)

[Out]

5/184*(-1+4*x)/(2*x^2-x+3)^(1/2)+314233/995328/(x+5/2)^2/(2*(x+5/2)^2-11*x-19/2)^(1/2)-3127169/35831808/(x+5/2
)/(2*(x+5/2)^2-11*x-19/2)^(1/2)+3505819/429981696/(2*(x+5/2)^2-11*x-19/2)^(1/2)-261644215/9889579008*(-1+4*x)/
(2*(x+5/2)^2-11*x-19/2)^(1/2)-3505819/2579890176*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19
/2)^(1/2))-3667/13824/(x+5/2)^3/(2*(x+5/2)^2-11*x-19/2)^(1/2)

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Maxima [A]  time = 1.60093, size = 293, normalized size = 2.14 \begin{align*} \frac{3505819}{2579890176} \, \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{7094345 \, x}{2472394752 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{6128291}{824131584 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{3667}{1728 \,{\left (8 \, \sqrt{2 \, x^{2} - x + 3} x^{3} + 60 \, \sqrt{2 \, x^{2} - x + 3} x^{2} + 150 \, \sqrt{2 \, x^{2} - x + 3} x + 125 \, \sqrt{2 \, x^{2} - x + 3}\right )}} + \frac{314233}{248832 \,{\left (4 \, \sqrt{2 \, x^{2} - x + 3} x^{2} + 20 \, \sqrt{2 \, x^{2} - x + 3} x + 25 \, \sqrt{2 \, x^{2} - x + 3}\right )}} - \frac{3127169}{17915904 \,{\left (2 \, \sqrt{2 \, x^{2} - x + 3} x + 5 \, \sqrt{2 \, x^{2} - x + 3}\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)^4/(2*x^2-x+3)^(3/2),x, algorithm="maxima")

[Out]

3505819/2579890176*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 7094345/2472
394752*x/sqrt(2*x^2 - x + 3) + 6128291/824131584/sqrt(2*x^2 - x + 3) - 3667/1728/(8*sqrt(2*x^2 - x + 3)*x^3 +
60*sqrt(2*x^2 - x + 3)*x^2 + 150*sqrt(2*x^2 - x + 3)*x + 125*sqrt(2*x^2 - x + 3)) + 314233/248832/(4*sqrt(2*x^
2 - x + 3)*x^2 + 20*sqrt(2*x^2 - x + 3)*x + 25*sqrt(2*x^2 - x + 3)) - 3127169/17915904/(2*sqrt(2*x^2 - x + 3)*
x + 5*sqrt(2*x^2 - x + 3))

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Fricas [A]  time = 1.33425, size = 458, normalized size = 3.34 \begin{align*} \frac{80633837 \, \sqrt{2}{\left (16 \, x^{5} + 112 \, x^{4} + 264 \, x^{3} + 280 \, x^{2} + 325 \, x + 375\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 (56754760 \, x^{4} + 572739684 \, x^{3} + 441046842 \, x^{2} + 1257975811 \, x + 1873786587\right )} \sqrt{2 \, x^{2} - x + 3}}{118674948096 \,{\left (16 \, x^{5} + 112 \, x^{4} + 264 \, x^{3} + 280 \, x^{2} + 325 \, x + 375\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)^4/(2*x^2-x+3)^(3/2),x, algorithm="fricas")

[Out]

1/118674948096*(80633837*sqrt(2)*(16*x^5 + 112*x^4 + 264*x^3 + 280*x^2 + 325*x + 375)*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*(56754760*x^4 + 572739684*x^3 +
441046842*x^2 + 1257975811*x + 1873786587)*sqrt(2*x^2 - x + 3))/(16*x^5 + 112*x^4 + 264*x^3 + 280*x^2 + 325*x
+ 375)

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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 )^{4} \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**4-x**3+3*x**2+x+2)/(5+2*x)**4/(2*x**2-x+3)**(3/2),x)

[Out]

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

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Giac [B]  time = 1.18457, size = 366, normalized size = 2.67 \begin{align*} -\frac{3505819}{2579890176} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{3505819}{2579890176} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) - \frac{175877 \, x - 369609}{154524672 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{\sqrt{2}{\left (10398764 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{5} - 303070900 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{4} - 529738052 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 3644644652 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 2612608649 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1052284471\right )}}{214990848 \,{\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 )}^{3}} \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)^4/(2*x^2-x+3)^(3/2),x, algorithm="giac")

[Out]

-3505819/2579890176*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 3505819/2579890176*sqrt
(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/154524672*(175877*x - 369609)/sqrt(2*x^2 -
x + 3) - 1/214990848*sqrt(2)*(10398764*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 - 303070900*(sqrt(2)*x - s
qrt(2*x^2 - x + 3))^4 - 529738052*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 3644644652*(sqrt(2)*x - sqrt(2
*x^2 - x + 3))^2 - 2612608649*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1052284471)/(2*(sqrt(2)*x - sqrt(2*x
^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^3