3.1580 \(\int \frac{b+2 c x}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=328 \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

[Out]

((2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((8*c^2*d^2 + 5*b^2*e^2 - 4*c*
e*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d
^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt[a + b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - ((
b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[
c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(16*(c*d^2 - b*d*e + a*e^2)^(7/2))

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Rubi [A]  time = 0.448662, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {834, 806, 724, 206} \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}+\frac{\sqrt{a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

((2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((8*c^2*d^2 + 5*b^2*e^2 - 4*c*
e*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d
^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt[a + b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - ((
b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[
c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(16*(c*d^2 - b*d*e + a*e^2)^(7/2))

Rule 834

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

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{b+2 c x}{(d+e x)^4 \sqrt{a+b x+c x^2}} \, dx &=\frac{(2 c d-b e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac{\int \frac{\frac{1}{2} \left (-4 b c d+5 b^2 e-12 a c e\right )-2 c (2 c d-b e) x}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{(2 c d-b e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{\int \frac{\frac{1}{4} \left (-28 b^2 c d e+80 a c^2 d e+15 b^3 e^2+4 b c \left (2 c d^2-13 a e^2\right )\right )+\frac{1}{2} c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx}{6 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{(2 c d-b e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (\left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{(2 c d-b e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac{\left (\left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x}{\sqrt{a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac{(2 c d-b e) \sqrt{a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac{\left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt{a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac{(2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt{a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac{\left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.633233, size = 323, normalized size = 0.98 \[ \frac{\frac{\sqrt{a+x (b+c x)} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2}+\frac{\sqrt{a+x (b+c x)} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{4 (d+e x) \left (e (a e-b d)+c d^2\right )}+\frac{3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{2 \sqrt{a+x (b+c x)} (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{(d+e x)^3}}{6 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]

[Out]

((2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^3 + ((8*c^2*d^2 + 5*b^2*e^2 - 4*
c*e*(2*b*d + 3*a*e))*Sqrt[a + x*(b + c*x)])/(2*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(
2*b*d + 13*a*e))*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (3*(b^2 - 4*a*c)*e*(16*c^2*
d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) +
a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(6*(c*d^2 + e*(-(b*d) + a*e))^2)

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Maple [B]  time = 0.017, size = 2659, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x)

[Out]

-10/e/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e
)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)
)*b*c^3*d^3+2/3/e^3/(a*e^2-b*d*e+c*d^2)/(x+d/e)^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2)*c*d+5/3/e^2/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2)*c^2*d^2+5/e/(a*e^2-b*d*e+c*d^2)^3/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)*c^3*d^3+5/e^2/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c
*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2))/(x+d/e))*c^4*d^4-15/2/(a*e^2-b*d*e+c*d^2)^3/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)*b*c^2*d^2+15/2/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^
2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2))/(x+d/e))*b^2*c^2*d^2+5/12/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-6*c^3/e^2/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-
b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a
*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*d^2-13/3*c^2/e/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*d+5/16*e^2/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2
*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^4-1/3/e^2/(a*e^2-b*d*e+c*d^2)/(x+d/e)^3*((x+d/e)^2*c+(b*e-2*c
*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b-5/8*e^2/(a*e^2-b*d*e+c*d^2)^3/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/
e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3-c/e^2/(a*e^2-b*d*e+c*d^2)/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x
+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+13/6*c/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b-3/2*c/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e
+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-
b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^2+c^2/e^2/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2
-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(
a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))+6*c^2/e/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a
*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b*d-5/3/e/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)
/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d+15/4*e/(a*e^2-b*d*e+c*d^2)^3/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/
e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*c*d-5/2*e/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*l
n((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/
e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^3*c*d

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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((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 41.8555, size = 6263, normalized size = 19.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/96*(3*(16*(b^2*c^2 - 4*a*c^3)*d^5*e - 16*(b^3*c - 4*a*b*c^2)*d^4*e^2 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^
3*e^3 + (16*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 16*(b^3*c - 4*a*b*c^2)*d*e^5 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*e^6
)*x^3 + 3*(16*(b^2*c^2 - 4*a*c^3)*d^3*e^3 - 16*(b^3*c - 4*a*b*c^2)*d^2*e^4 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)
*d*e^5)*x^2 + 3*(16*(b^2*c^2 - 4*a*c^3)*d^4*e^2 - 16*(b^3*c - 4*a*b*c^2)*d^3*e^3 + (5*b^4 - 24*a*b^2*c + 16*a^
2*c^2)*d^2*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8
*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d
- b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(48*c^4*d^7 - 14
4*b*c^3*d^6*e - 8*a^3*b*e^7 + 32*(6*b^2*c^2 - a*c^3)*d^5*e^2 - (129*b^3*c + 4*a*b*c^2)*d^4*e^3 + 11*(3*b^4 + 1
0*a*b^2*c - 8*a^2*c^2)*d^3*e^4 - (59*a*b^3 - 12*a^2*b*c)*d^2*e^5 + 2*(17*a^2*b^2 - 4*a^3*c)*d*e^6 + (16*c^4*d^
5*e^2 - 40*b*c^3*d^4*e^3 + 2*(31*b^2*c^2 - 44*a*c^3)*d^3*e^4 - (53*b^3*c - 132*a*b*c^2)*d^2*e^5 + (15*b^4 - 14
*a*b^2*c - 104*a^2*c^2)*d*e^6 - (15*a*b^3 - 52*a^2*b*c)*e^7)*x^2 + 2*(24*c^4*d^6*e - 64*b*c^3*d^5*e^2 + 7*(13*
b^2*c^2 - 12*a*c^3)*d^4*e^3 - (71*b^3*c - 124*a*b*c^2)*d^3*e^4 + 20*(b^4 - 6*a^2*c^2)*d^2*e^5 - (25*a*b^3 - 68
*a^2*b*c)*d*e^6 + (5*a^2*b^2 - 12*a^3*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^11 - 4*b*c^3*d^10*e - 4*a^3*b*d
^4*e^7 + a^4*d^3*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^3 + (b^4 + 12*a*b^2*c + 6
*a^2*c^2)*d^7*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^6 + (c^4*d^8*e^3 - 4*b*c^3*d
^7*e^4 - 4*a^3*b*d*e^10 + a^4*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^6 + (b^4 +
12*a*b^2*c + 6*a^2*c^2)*d^4*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^9)*x^3 + 3*(c^
4*d^9*e^2 - 4*b*c^3*d^8*e^3 - 4*a^3*b*d^2*e^9 + a^4*d*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^4 - 4*(b^3*c + 3*a*
b*c^2)*d^6*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^7 + 2*(3*a^2*b^2 + 2*a^3
*c)*d^3*e^8)*x^2 + 3*(c^4*d^10*e - 4*b*c^3*d^9*e^2 - 4*a^3*b*d^3*e^8 + a^4*d^2*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d
^8*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^
6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^7)*x), -1/48*(3*(16*(b^2*c^2 - 4*a*c^3)*d^5*e - 16*(b^3*c - 4*a*b*c^2)*d^4*e
^2 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^3*e^3 + (16*(b^2*c^2 - 4*a*c^3)*d^2*e^4 - 16*(b^3*c - 4*a*b*c^2)*d*e^
5 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*e^6)*x^3 + 3*(16*(b^2*c^2 - 4*a*c^3)*d^3*e^3 - 16*(b^3*c - 4*a*b*c^2)*d^
2*e^4 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d*e^5)*x^2 + 3*(16*(b^2*c^2 - 4*a*c^3)*d^4*e^2 - 16*(b^3*c - 4*a*b*c
^2)*d^3*e^3 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^2*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d
^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*
d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 2*(48*c^4*d^7 - 144*b*c^3*d^6*e - 8*a^3*b*e
^7 + 32*(6*b^2*c^2 - a*c^3)*d^5*e^2 - (129*b^3*c + 4*a*b*c^2)*d^4*e^3 + 11*(3*b^4 + 10*a*b^2*c - 8*a^2*c^2)*d^
3*e^4 - (59*a*b^3 - 12*a^2*b*c)*d^2*e^5 + 2*(17*a^2*b^2 - 4*a^3*c)*d*e^6 + (16*c^4*d^5*e^2 - 40*b*c^3*d^4*e^3
+ 2*(31*b^2*c^2 - 44*a*c^3)*d^3*e^4 - (53*b^3*c - 132*a*b*c^2)*d^2*e^5 + (15*b^4 - 14*a*b^2*c - 104*a^2*c^2)*d
*e^6 - (15*a*b^3 - 52*a^2*b*c)*e^7)*x^2 + 2*(24*c^4*d^6*e - 64*b*c^3*d^5*e^2 + 7*(13*b^2*c^2 - 12*a*c^3)*d^4*e
^3 - (71*b^3*c - 124*a*b*c^2)*d^3*e^4 + 20*(b^4 - 6*a^2*c^2)*d^2*e^5 - (25*a*b^3 - 68*a^2*b*c)*d*e^6 + (5*a^2*
b^2 - 12*a^3*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^11 - 4*b*c^3*d^10*e - 4*a^3*b*d^4*e^7 + a^4*d^3*e^8 + 2*
(3*b^2*c^2 + 2*a*c^3)*d^9*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^4 - 4*(a*
b^3 + 3*a^2*b*c)*d^6*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^6 + (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 - 4*a^3*b*d*e^10 +
a^4*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d
^4*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^9)*x^3 + 3*(c^4*d^9*e^2 - 4*b*c^3*d^8*e
^3 - 4*a^3*b*d^2*e^9 + a^4*d*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^5 + (b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*d^5*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^8)*x^2 + 3*(c^4
*d^10*e - 4*b*c^3*d^9*e^2 - 4*a^3*b*d^3*e^8 + a^4*d^2*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^3 - 4*(b^3*c + 3*a*b
*c^2)*d^7*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^6*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^6 + 2*(3*a^2*b^2 + 2*a^3*
c)*d^4*e^7)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b + 2 c x}{\left (d + e x\right )^{4} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((b + 2*c*x)/((d + e*x)**4*sqrt(a + b*x + c*x**2)), x)

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Giac [B]  time = 2.52354, size = 3599, normalized size = 10.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

-1/8*(16*b^2*c^2*d^2*e - 64*a*c^3*d^2*e - 16*b^3*c*d*e^2 + 64*a*b*c^2*d*e^2 + 5*b^4*e^3 - 24*a*b^2*c*e^3 + 16*
a^2*c^2*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d
^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a
^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/24*(128*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^3*c^5*d^6 - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^4*d^5*e + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*b*c^(9/2)*d^6 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(7/2)*d^5*e - 384*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))^2*a*c^(9/2)*d^5*e + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c^4*d^6 + 736*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*b^2*c^3*d^4*e^2 - 1024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^4*d^4*e^2 - 192*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))*b^3*c^3*d^5*e - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^4*d^5*e + 16*b^3*c^(
7/2)*d^6 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(5/2)*d^3*e^3 - 960*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^4*a*c^(7/2)*d^3*e^3 + 864*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*c^(5/2)*d^4*e^2 - 576*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*a*b*c^(7/2)*d^4*e^2 - 24*b^4*c^(5/2)*d^5*e - 96*a*b^2*c^(7/2)*d^5*e + 48*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^5*b^2*c^2*d^2*e^4 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^3*d^2*e^4 - 352*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c^2*d^3*e^3 + 128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^3*d^3*e^3 +
324*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*c^2*d^4*e^2 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c^3*d^4
*e^2 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^4*d^4*e^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^3
*c^(3/2)*d^2*e^4 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c^(5/2)*d^2*e^4 - 546*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*b^4*c^(3/2)*d^3*e^3 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(5/2)*d^3*e^3 + 1632*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(7/2)*d^3*e^3 + 38*b^5*c^(3/2)*d^4*e^2 + 64*a*b^3*c^(5/2)*d^4*e^2 +
96*a^2*b*c^(7/2)*d^4*e^2 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c*d*e^5 + 192*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^5*a*b*c^2*d*e^5 - 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^4*c*d^2*e^4 + 240*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^3*a*b^2*c^2*d^2*e^4 + 1248*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*c^3*d^2*e^4 - 186*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))*b^5*c*d^3*e^3 - 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^3*c^2*d^3*e^3 + 1248
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c^3*d^3*e^3 + 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*sqrt(c)*
d*e^5 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(3/2)*d*e^5 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^4*a^2*c^(5/2)*d*e^5 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^5*sqrt(c)*d^2*e^4 + 384*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))^2*a*b^3*c^(3/2)*d^2*e^4 - 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b*c^(5/2)*d^2*e^4 -
15*b^6*sqrt(c)*d^3*e^3 - 158*a*b^4*c^(3/2)*d^3*e^3 + 240*a^2*b^2*c^(5/2)*d^3*e^3 - 32*a^3*c^(7/2)*d^3*e^3 + 15
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*e^6 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c*e^6 + 48*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^2*e^6 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*d*e^5 - 64*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*c*d*e^5 - 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^2*d*e^5 + 3
3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^6*d^2*e^4 + 402*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^4*c*d^2*e^4 -
432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^2*c^2*d^2*e^4 - 1056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*c^3
*d^2*e^4 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*sqrt(c)*d*e^5 - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^2*a^3*c^(5/2)*d*e^5 + 78*a*b^5*sqrt(c)*d^2*e^4 + 48*a^2*b^3*c^(3/2)*d^2*e^4 - 480*a^3*b*c^(5/2)*d^2*e^4 -
40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^4*e^6 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^2*c*e^6
- 66*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*d*e^5 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*c*d*e^5
+ 672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b*c^2*d*e^5 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b*c
^(3/2)*e^6 - 111*a^2*b^4*sqrt(c)*d*e^5 + 200*a^3*b^2*c^(3/2)*d*e^5 + 208*a^4*c^(5/2)*d*e^5 + 33*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*a^2*b^4*e^6 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^2*c*e^6 - 48*(sqrt(c)*x - sqr
t(c*x^2 + b*x + a))*a^4*c^2*e^6 + 48*a^3*b^3*sqrt(c)*e^6 - 128*a^4*b*c^(3/2)*e^6)/((c^3*d^6*e - 3*b*c^2*d^5*e^
2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*
a^2*b*d*e^6 + a^3*e^7)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c
)*d + b*d - a*e)^3)