3.281 \(\int \frac{x (d+e x+f x^2+g x^3)}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=245 \[ \frac{\sqrt{a+b x+c x^2} \left (2 c x \left (-36 a c g+35 b^2 g-40 b c f+48 c^2 e\right )-16 c^2 (8 a f+9 b e)+20 b c (11 a g+6 b f)-105 b^3 g+192 c^3 d\right )}{192 c^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-24 b^2 c (2 c e-5 a g)+32 b c^2 (2 c d-3 a f)+16 a c^2 (4 c e-3 a g)+40 b^3 c f-35 b^4 g\right )}{128 c^{9/2}}+\frac{x^2 \sqrt{a+b x+c x^2} (8 c f-7 b g)}{24 c^2}+\frac{g x^3 \sqrt{a+b x+c x^2}}{4 c} \]

[Out]

((8*c*f - 7*b*g)*x^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (g*x^3*Sqrt[a + b*x + c*x^2])/(4*c) + ((192*c^3*d - 16*
c^2*(9*b*e + 8*a*f) - 105*b^3*g + 20*b*c*(6*b*f + 11*a*g) + 2*c*(48*c^2*e - 40*b*c*f + 35*b^2*g - 36*a*c*g)*x)
*Sqrt[a + b*x + c*x^2])/(192*c^4) - ((40*b^3*c*f + 32*b*c^2*(2*c*d - 3*a*f) - 35*b^4*g - 24*b^2*c*(2*c*e - 5*a
*g) + 16*a*c^2*(4*c*e - 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(9/2))

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Rubi [A]  time = 0.437547, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {1653, 779, 621, 206} \[ \frac{\sqrt{a+b x+c x^2} \left (2 c x \left (-36 a c g+35 b^2 g-40 b c f+48 c^2 e\right )-16 c^2 (8 a f+9 b e)+20 b c (11 a g+6 b f)-105 b^3 g+192 c^3 d\right )}{192 c^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-24 b^2 c (2 c e-5 a g)+32 b c^2 (2 c d-3 a f)+16 a c^2 (4 c e-3 a g)+40 b^3 c f-35 b^4 g\right )}{128 c^{9/2}}+\frac{x^2 \sqrt{a+b x+c x^2} (8 c f-7 b g)}{24 c^2}+\frac{g x^3 \sqrt{a+b x+c x^2}}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x + f*x^2 + g*x^3))/Sqrt[a + b*x + c*x^2],x]

[Out]

((8*c*f - 7*b*g)*x^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (g*x^3*Sqrt[a + b*x + c*x^2])/(4*c) + ((192*c^3*d - 16*
c^2*(9*b*e + 8*a*f) - 105*b^3*g + 20*b*c*(6*b*f + 11*a*g) + 2*c*(48*c^2*e - 40*b*c*f + 35*b^2*g - 36*a*c*g)*x)
*Sqrt[a + b*x + c*x^2])/(192*c^4) - ((40*b^3*c*f + 32*b*c^2*(2*c*d - 3*a*f) - 35*b^4*g - 24*b^2*c*(2*c*e - 5*a
*g) + 16*a*c^2*(4*c*e - 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(9/2))

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 779

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

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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{x \left (d+e x+f x^2+g x^3\right )}{\sqrt{a+b x+c x^2}} \, dx &=\frac{g x^3 \sqrt{a+b x+c x^2}}{4 c}+\frac{\int \frac{x \left (4 c d+(4 c e-3 a g) x+\frac{1}{2} (8 c f-7 b g) x^2\right )}{\sqrt{a+b x+c x^2}} \, dx}{4 c}\\ &=\frac{(8 c f-7 b g) x^2 \sqrt{a+b x+c x^2}}{24 c^2}+\frac{g x^3 \sqrt{a+b x+c x^2}}{4 c}+\frac{\int \frac{x \left (12 c^2 d-8 a c f+7 a b g+\frac{1}{4} \left (48 c^2 e-40 b c f+35 b^2 g-36 a c g\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{12 c^2}\\ &=\frac{(8 c f-7 b g) x^2 \sqrt{a+b x+c x^2}}{24 c^2}+\frac{g x^3 \sqrt{a+b x+c x^2}}{4 c}+\frac{\left (192 c^3 d-16 c^2 (9 b e+8 a f)-105 b^3 g+20 b c (6 b f+11 a g)+2 c \left (48 c^2 e-40 b c f+35 b^2 g-36 a c g\right ) x\right ) \sqrt{a+b x+c x^2}}{192 c^4}-\frac{\left (40 b^3 c f+32 b c^2 (2 c d-3 a f)-35 b^4 g-24 b^2 c (2 c e-5 a g)+16 a c^2 (4 c e-3 a g)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c^4}\\ &=\frac{(8 c f-7 b g) x^2 \sqrt{a+b x+c x^2}}{24 c^2}+\frac{g x^3 \sqrt{a+b x+c x^2}}{4 c}+\frac{\left (192 c^3 d-16 c^2 (9 b e+8 a f)-105 b^3 g+20 b c (6 b f+11 a g)+2 c \left (48 c^2 e-40 b c f+35 b^2 g-36 a c g\right ) x\right ) \sqrt{a+b x+c x^2}}{192 c^4}-\frac{\left (40 b^3 c f+32 b c^2 (2 c d-3 a f)-35 b^4 g-24 b^2 c (2 c e-5 a g)+16 a c^2 (4 c e-3 a g)\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{64 c^4}\\ &=\frac{(8 c f-7 b g) x^2 \sqrt{a+b x+c x^2}}{24 c^2}+\frac{g x^3 \sqrt{a+b x+c x^2}}{4 c}+\frac{\left (192 c^3 d-16 c^2 (9 b e+8 a f)-105 b^3 g+20 b c (6 b f+11 a g)+2 c \left (48 c^2 e-40 b c f+35 b^2 g-36 a c g\right ) x\right ) \sqrt{a+b x+c x^2}}{192 c^4}-\frac{\left (40 b^3 c f+32 b c^2 (2 c d-3 a f)-35 b^4 g-24 b^2 c (2 c e-5 a g)+16 a c^2 (4 c e-3 a g)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.45855, size = 199, normalized size = 0.81 \[ \frac{\sqrt{a+x (b+c x)} \left (-8 c^2 \left (16 a f+9 a g x+18 b e+10 b f x+7 b g x^2\right )+10 b c (22 a g+12 b f+7 b g x)-105 b^3 g+16 c^3 \left (12 d+x \left (6 e+4 f x+3 g x^2\right )\right )\right )}{192 c^4}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (24 b^2 c (2 c e-5 a g)+32 b c^2 (3 a f-2 c d)+16 a c^2 (3 a g-4 c e)-40 b^3 c f+35 b^4 g\right )}{128 c^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(d + e*x + f*x^2 + g*x^3))/Sqrt[a + b*x + c*x^2],x]

[Out]

(Sqrt[a + x*(b + c*x)]*(-105*b^3*g + 10*b*c*(12*b*f + 22*a*g + 7*b*g*x) - 8*c^2*(18*b*e + 16*a*f + 10*b*f*x +
9*a*g*x + 7*b*g*x^2) + 16*c^3*(12*d + x*(6*e + 4*f*x + 3*g*x^2))))/(192*c^4) + ((-40*b^3*c*f + 32*b*c^2*(-2*c*
d + 3*a*f) + 35*b^4*g + 24*b^2*c*(2*c*e - 5*a*g) + 16*a*c^2*(-4*c*e + 3*a*g))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*S
qrt[a + x*(b + c*x)])])/(128*c^(9/2))

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Maple [B]  time = 0.055, size = 532, normalized size = 2.2 \begin{align*}{\frac{g{x}^{3}}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{7\,bg{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,{b}^{2}gx}{96\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,{b}^{3}g}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,{b}^{4}g}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}-{\frac{15\,{b}^{2}ga}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{55\,bga}{48\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,agx}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{a}^{2}g}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{f{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,bfx}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{2}f}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,f{b}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,abf}{4}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{2\,af}{3\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{ex}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,be}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}e}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{ae}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{d}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{bd}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x)

[Out]

1/4*g*x^3*(c*x^2+b*x+a)^(1/2)/c-7/24*g*b/c^2*x^2*(c*x^2+b*x+a)^(1/2)+35/96*g*b^2/c^3*x*(c*x^2+b*x+a)^(1/2)-35/
64*g*b^3/c^4*(c*x^2+b*x+a)^(1/2)+35/128*g*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/16*g*b^2/
c^(7/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+55/48*g*b/c^3*a*(c*x^2+b*x+a)^(1/2)-3/8*g*a/c^2*x*(c*x^2
+b*x+a)^(1/2)+3/8*g*a^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/3*f*x^2/c*(c*x^2+b*x+a)^(1/2)-5/
12*f*b/c^2*x*(c*x^2+b*x+a)^(1/2)+5/8*f*b^2/c^3*(c*x^2+b*x+a)^(1/2)-5/16*f*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))+3/4*f*b/c^(5/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/3*f*a/c^2*(c*x^2+b*x+a)^(1
/2)+1/2*e*x/c*(c*x^2+b*x+a)^(1/2)-3/4*e*b/c^2*(c*x^2+b*x+a)^(1/2)+3/8*e*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))-1/2*e*a/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+d/c*(c*x^2+b*x+a)^(1/2)-1/2*d*b/
c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62188, size = 1173, normalized size = 4.79 \begin{align*} \left [-\frac{3 \,{\left (64 \, b c^{3} d - 16 \,{\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} e + 8 \,{\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} f -{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} g\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \,{\left (48 \, c^{4} g x^{3} + 192 \, c^{4} d - 144 \, b c^{3} e + 8 \,{\left (8 \, c^{4} f - 7 \, b c^{3} g\right )} x^{2} + 8 \,{\left (15 \, b^{2} c^{2} - 16 \, a c^{3}\right )} f - 5 \,{\left (21 \, b^{3} c - 44 \, a b c^{2}\right )} g + 2 \,{\left (48 \, c^{4} e - 40 \, b c^{3} f +{\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} g\right )} x\right )} \sqrt{c x^{2} + b x + a}}{768 \, c^{5}}, \frac{3 \,{\left (64 \, b c^{3} d - 16 \,{\left (3 \, b^{2} c^{2} - 4 \, a c^{3}\right )} e + 8 \,{\left (5 \, b^{3} c - 12 \, a b c^{2}\right )} f -{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} g\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \,{\left (48 \, c^{4} g x^{3} + 192 \, c^{4} d - 144 \, b c^{3} e + 8 \,{\left (8 \, c^{4} f - 7 \, b c^{3} g\right )} x^{2} + 8 \,{\left (15 \, b^{2} c^{2} - 16 \, a c^{3}\right )} f - 5 \,{\left (21 \, b^{3} c - 44 \, a b c^{2}\right )} g + 2 \,{\left (48 \, c^{4} e - 40 \, b c^{3} f +{\left (35 \, b^{2} c^{2} - 36 \, a c^{3}\right )} g\right )} x\right )} \sqrt{c x^{2} + b x + a}}{384 \, c^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/768*(3*(64*b*c^3*d - 16*(3*b^2*c^2 - 4*a*c^3)*e + 8*(5*b^3*c - 12*a*b*c^2)*f - (35*b^4 - 120*a*b^2*c + 48*
a^2*c^2)*g)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*
(48*c^4*g*x^3 + 192*c^4*d - 144*b*c^3*e + 8*(8*c^4*f - 7*b*c^3*g)*x^2 + 8*(15*b^2*c^2 - 16*a*c^3)*f - 5*(21*b^
3*c - 44*a*b*c^2)*g + 2*(48*c^4*e - 40*b*c^3*f + (35*b^2*c^2 - 36*a*c^3)*g)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/3
84*(3*(64*b*c^3*d - 16*(3*b^2*c^2 - 4*a*c^3)*e + 8*(5*b^3*c - 12*a*b*c^2)*f - (35*b^4 - 120*a*b^2*c + 48*a^2*c
^2)*g)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(48*c^4*g*x
^3 + 192*c^4*d - 144*b*c^3*e + 8*(8*c^4*f - 7*b*c^3*g)*x^2 + 8*(15*b^2*c^2 - 16*a*c^3)*f - 5*(21*b^3*c - 44*a*
b*c^2)*g + 2*(48*c^4*e - 40*b*c^3*f + (35*b^2*c^2 - 36*a*c^3)*g)*x)*sqrt(c*x^2 + b*x + a))/c^5]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (d + e x + f x^{2} + g x^{3}\right )}{\sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(g*x**3+f*x**2+e*x+d)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(x*(d + e*x + f*x**2 + g*x**3)/sqrt(a + b*x + c*x**2), x)

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Giac [A]  time = 1.19921, size = 308, normalized size = 1.26 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (\frac{6 \, g x}{c} + \frac{8 \, c^{3} f - 7 \, b c^{2} g}{c^{4}}\right )} x - \frac{40 \, b c^{2} f - 35 \, b^{2} c g + 36 \, a c^{2} g - 48 \, c^{3} e}{c^{4}}\right )} x + \frac{192 \, c^{3} d + 120 \, b^{2} c f - 128 \, a c^{2} f - 105 \, b^{3} g + 220 \, a b c g - 144 \, b c^{2} e}{c^{4}}\right )} + \frac{{\left (64 \, b c^{3} d + 40 \, b^{3} c f - 96 \, a b c^{2} f - 35 \, b^{4} g + 120 \, a b^{2} c g - 48 \, a^{2} c^{2} g - 48 \, b^{2} c^{2} e + 64 \, a c^{3} e\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(g*x^3+f*x^2+e*x+d)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*g*x/c + (8*c^3*f - 7*b*c^2*g)/c^4)*x - (40*b*c^2*f - 35*b^2*c*g + 36*a*c^
2*g - 48*c^3*e)/c^4)*x + (192*c^3*d + 120*b^2*c*f - 128*a*c^2*f - 105*b^3*g + 220*a*b*c*g - 144*b*c^2*e)/c^4)
+ 1/128*(64*b*c^3*d + 40*b^3*c*f - 96*a*b*c^2*f - 35*b^4*g + 120*a*b^2*c*g - 48*a^2*c^2*g - 48*b^2*c^2*e + 64*
a*c^3*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)