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

Optimal. Leaf size=149 \[ \frac{5}{64} (2 x+5) \left (2 x^2-x+3\right )^{3/2}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}-\frac{541}{384} \left (2 x^2-x+3\right )^{3/2}-\frac{(1996953-333380 x) \sqrt{2 x^2-x+3}}{18432}+\frac{239201 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{384 \sqrt{2}}-\frac{2551847 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}} \]

[Out]

-((1996953 - 333380*x)*Sqrt[3 - x + 2*x^2])/18432 - (541*(3 - x + 2*x^2)^(3/2))/384 - (3667*(3 - x + 2*x^2)^(3
/2))/(576*(5 + 2*x)) + (5*(5 + 2*x)*(3 - x + 2*x^2)^(3/2))/64 - (2551847*ArcSinh[(1 - 4*x)/Sqrt[23]])/(4096*Sq
rt[2]) + (239201*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(384*Sqrt[2])

________________________________________________________________________________________

Rubi [A]  time = 0.236645, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1650, 1653, 814, 843, 619, 215, 724, 206} \[ \frac{5}{64} (2 x+5) \left (2 x^2-x+3\right )^{3/2}-\frac{3667 \left (2 x^2-x+3\right )^{3/2}}{576 (2 x+5)}-\frac{541}{384} \left (2 x^2-x+3\right )^{3/2}-\frac{(1996953-333380 x) \sqrt{2 x^2-x+3}}{18432}+\frac{239201 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{384 \sqrt{2}}-\frac{2551847 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1996953 - 333380*x)*Sqrt[3 - x + 2*x^2])/18432 - (541*(3 - x + 2*x^2)^(3/2))/384 - (3667*(3 - x + 2*x^2)^(3
/2))/(576*(5 + 2*x)) + (5*(5 + 2*x)*(3 - x + 2*x^2)^(3/2))/64 - (2551847*ArcSinh[(1 - 4*x)/Sqrt[23]])/(4096*Sq
rt[2]) + (239201*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(384*Sqrt[2])

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 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

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]

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{\sqrt{3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{576 (5+2 x)}-\frac{1}{72} \int \frac{\sqrt{3-x+2 x^2} \left (\frac{19341}{16}-\frac{6313 x}{2}+486 x^2-180 x^3\right )}{5+2 x} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{576 (5+2 x)}+\frac{5}{64} (5+2 x) \left (3-x+2 x^2\right )^{3/2}-\frac{\int \frac{\sqrt{3-x+2 x^2} \left (74664-158096 x+77904 x^2\right )}{5+2 x} \, dx}{4608}\\ &=-\frac{541}{384} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{576 (5+2 x)}+\frac{5}{64} (5+2 x) \left (3-x+2 x^2\right )^{3/2}-\frac{\int \frac{(2960496-8001120 x) \sqrt{3-x+2 x^2}}{5+2 x} \, dx}{110592}\\ &=-\frac{(1996953-333380 x) \sqrt{3-x+2 x^2}}{18432}-\frac{541}{384} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{576 (5+2 x)}+\frac{5}{64} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac{\int \frac{-2202879456+4409591616 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{3538944}\\ &=-\frac{(1996953-333380 x) \sqrt{3-x+2 x^2}}{18432}-\frac{541}{384} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{576 (5+2 x)}+\frac{5}{64} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac{2551847 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{4096}-\frac{239201}{64} \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{(1996953-333380 x) \sqrt{3-x+2 x^2}}{18432}-\frac{541}{384} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{576 (5+2 x)}+\frac{5}{64} (5+2 x) \left (3-x+2 x^2\right )^{3/2}+\frac{239201}{32} \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )+\frac{2551847 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4096 \sqrt{46}}\\ &=-\frac{(1996953-333380 x) \sqrt{3-x+2 x^2}}{18432}-\frac{541}{384} \left (3-x+2 x^2\right )^{3/2}-\frac{3667 \left (3-x+2 x^2\right )^{3/2}}{576 (5+2 x)}+\frac{5}{64} (5+2 x) \left (3-x+2 x^2\right )^{3/2}-\frac{2551847 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4096 \sqrt{2}}+\frac{239201 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{384 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.163372, size = 98, normalized size = 0.66 \[ \frac{\frac{4 \sqrt{2 x^2-x+3} \left (3840 x^4-17344 x^3+94936 x^2-728410 x-3539439\right )}{2 x+5}+7654432 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )-7655541 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{24576} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*Sqrt[3 - x + 2*x^2]*(-3539439 - 728410*x + 94936*x^2 - 17344*x^3 + 3840*x^4))/(5 + 2*x) - 7655541*Sqrt[2]*
ArcSinh[(1 - 4*x)/Sqrt[23]] + 7654432*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/24576

________________________________________________________________________________________

Maple [A]  time = 0.059, size = 152, normalized size = 1. \begin{align*}{\frac{5\,x}{32} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{391}{384} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-6001+24004\,x}{2048}\sqrt{2\,{x}^{2}-x+3}}+{\frac{2551847\,\sqrt{2}}{8192}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{239201}{2304}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{239201\,\sqrt{2}}{768}{\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 ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{-3667+14668\,x}{2304}\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)*(2*x^2-x+3)^(1/2)/(5+2*x)^2,x)

[Out]

5/32*x*(2*x^2-x+3)^(3/2)-391/384*(2*x^2-x+3)^(3/2)+6001/2048*(-1+4*x)*(2*x^2-x+3)^(1/2)+2551847/8192*2^(1/2)*a
rcsinh(4/23*23^(1/2)*(x-1/4))-239201/2304*(2*(x+5/2)^2-11*x-19/2)^(1/2)+239201/768*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)+3667/2304*(-1+4*x
)*(2*(x+5/2)^2-11*x-19/2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.53635, size = 178, normalized size = 1.19 \begin{align*} \frac{5}{32} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{391}{384} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{6001}{512} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{2551847}{8192} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) - \frac{239201}{768} \, \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{182769}{2048} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \, \sqrt{2 \, x^{2} - x + 3}}{32 \,{\left (2 \, x + 5\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)*(2*x^2-x+3)^(1/2)/(5+2*x)^2,x, algorithm="maxima")

[Out]

5/32*(2*x^2 - x + 3)^(3/2)*x - 391/384*(2*x^2 - x + 3)^(3/2) + 6001/512*sqrt(2*x^2 - x + 3)*x + 2551847/8192*s
qrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 239201/768*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 1
7/23*sqrt(23)/abs(2*x + 5)) - 182769/2048*sqrt(2*x^2 - x + 3) - 3667/32*sqrt(2*x^2 - x + 3)/(2*x + 5)

________________________________________________________________________________________

Fricas [A]  time = 1.41511, size = 431, normalized size = 2.89 \begin{align*} \frac{7655541 \, \sqrt{2}{\left (2 \, x + 5\right )} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 7654432 \, \sqrt{2}{\left (2 \, x + 5\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 ) + 8 \,{\left (3840 \, x^{4} - 17344 \, x^{3} + 94936 \, x^{2} - 728410 \, x - 3539439\right )} \sqrt{2 \, x^{2} - x + 3}}{49152 \,{\left (2 \, x + 5\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)*(2*x^2-x+3)^(1/2)/(5+2*x)^2,x, algorithm="fricas")

[Out]

1/49152*(7655541*sqrt(2)*(2*x + 5)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 765443
2*sqrt(2)*(2*x + 5)*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)) + 8*(3840*x^4 - 17344*x^3 + 94936*x^2 - 728410*x - 3539439)*sqrt(2*x^2 - x + 3))/(2*x + 5)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 x^{2} - x + 3} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{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)*(2*x**2-x+3)**(1/2)/(5+2*x)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.31159, size = 717, normalized size = 4.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4-x^3+3*x^2+x+2)*(2*x^2-x+3)^(1/2)/(5+2*x)^2,x, algorithm="giac")

[Out]

1/24576*sqrt(2)*(7654432*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11)*sgn(1/(2*x + 5))
+ 7655541*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) + 1))*sgn(1/(2*x + 5)) - 7655541*log
(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) - 1))*sgn(1/(2*x + 5)) - 1408128*sqrt(-11/(2*x + 5
) + 36/(2*x + 5)^2 + 1)*sgn(1/(2*x + 5)) + 2*(16367883*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5)
)^7*sgn(1/(2*x + 5)) - 34896384*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^6*sgn(1/(2*x + 5)) -
93395*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^5*sgn(1/(2*x + 5)) + 25574400*(sqrt(-11/(2*x +
5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^4*sgn(1/(2*x + 5)) + 19752365*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1
) + 6/(2*x + 5))^3*sgn(1/(2*x + 5)) - 31921920*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2*sgn(
1/(2*x + 5)) - 2445813*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))*sgn(1/(2*x + 5)) + 7663104*sgn
(1/(2*x + 5)))/((sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2 - 1)^4)