3.1627 \(\int \frac{b+2 c x}{\sqrt{d+e x} (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=673 \[ \frac{e \sqrt{d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} e \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{c} e \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 5.2562, antiderivative size = 673, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {822, 826, 1166, 208} \[ \frac{e \sqrt{d+e x} \left (c x \left (-2 c e (5 a e+b d)+3 b^2 e^2+2 c^2 d^2\right )+(c d-3 b e) \left (2 a c e+b^2 (-e)+b c d\right )+5 a c e (2 c d-b e)\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{c} e \left (-2 c^2 d e \left (-d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (b d \sqrt{b^2-4 a c}+5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )^2}+\frac{\sqrt{c} e \left (-2 c^2 d e \left (d \sqrt{b^2-4 a c}-16 a e+6 b d\right )-2 c e^2 \left (-b d \sqrt{b^2-4 a c}-5 a e \sqrt{b^2-4 a c}+8 a b e+b^2 d\right )+3 b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+8 c^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )^2}-\frac{\sqrt{d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 822

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

Rule 826

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

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 6.20852, size = 1033, normalized size = 1.53 \[ -\frac{\sqrt{d+e x} \left (-2 a c (2 c d-b e)+b \left (-e b^2+c d b+2 a c e\right )+c (b (2 c d-b e)-2 c (b d-2 a e)) x\right )}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}-\frac{-\frac{\sqrt{d+e x} \left (\frac{5}{2} a c \left (b^2-4 a c\right ) (2 c d-b e) e^2+\frac{1}{2} \left (b^2-4 a c\right ) (c d-3 b e) \left (-e b^2+c d b+2 a c e\right ) e+c \left (\frac{5}{2} c \left (b^2-4 a c\right ) (b d-2 a e) e^2+\frac{1}{2} \left (b^2-4 a c\right ) (c d-3 b e) (2 c d-b e) e\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac{2 \left (\frac{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e} \left (\frac{1}{4} c \left (b^2-4 a c\right ) e^2 \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right )-\frac{2 c \left (\frac{1}{4} \left (b^2-4 a c\right ) e^2 \left (4 c^3 d^3-c^2 e (5 b d-16 a e) d+3 b^3 e^3-b c e^2 (2 b d+13 a e)\right )-\frac{1}{4} c \left (b^2-4 a c\right ) d e^2 \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right )\right )-\frac{1}{4} c \left (b^2-4 a c\right ) e^2 (b e-2 c d) \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right )}{\sqrt{b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e}}\right )}{\sqrt{2} \sqrt{c} \left (-2 c d+b e+\sqrt{b^2-4 a c} e\right )}+\frac{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e} \left (\frac{1}{4} c \left (b^2-4 a c\right ) \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right ) e^2+\frac{2 c \left (\frac{1}{4} \left (b^2-4 a c\right ) e^2 \left (4 c^3 d^3-c^2 e (5 b d-16 a e) d+3 b^3 e^3-b c e^2 (2 b d+13 a e)\right )-\frac{1}{4} c \left (b^2-4 a c\right ) d e^2 \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right )\right )-\frac{1}{4} c \left (b^2-4 a c\right ) e^2 (b e-2 c d) \left (2 c^2 d^2+3 b^2 e^2-2 c e (b d+5 a e)\right )}{\sqrt{b^2-4 a c} e}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}}\right )}{\sqrt{2} \sqrt{c} \left (-2 c d+b e-\sqrt{b^2-4 a c} e\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x]*(-2*a*c*(2*c*d - b*e) + b*(b*c*d - b^2*e + 2*a*c*e) + c*(-2*c*(b*d - 2*a*e) + b*(2*c*d - b*e))
*x))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (-((Sqrt[d + e*x]*((5*a*c*(b^2 - 4*a*c)*e
^2*(2*c*d - b*e))/2 + ((b^2 - 4*a*c)*e*(c*d - 3*b*e)*(b*c*d - b^2*e + 2*a*c*e))/2 + c*((5*c*(b^2 - 4*a*c)*e^2*
(b*d - 2*a*e))/2 + ((b^2 - 4*a*c)*e*(c*d - 3*b*e)*(2*c*d - b*e))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)
*(a + b*x + c*x^2))) - (2*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]*((c*(b^2 - 4*a*c)*e^2*(2*c^2*d^2 + 3*b^2*e
^2 - 2*c*e*(b*d + 5*a*e)))/4 - (-(c*(b^2 - 4*a*c)*e^2*(-2*c*d + b*e)*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a
*e)))/4 + 2*c*(-(c*(b^2 - 4*a*c)*d*e^2*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e)))/4 + ((b^2 - 4*a*c)*e^2*(
4*c^3*d^3 + 3*b^3*e^3 - c^2*d*e*(5*b*d - 16*a*e) - b*c*e^2*(2*b*d + 13*a*e)))/4))/(Sqrt[b^2 - 4*a*c]*e))*ArcTa
nh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e +
Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*((c*(b^2 - 4*a*c)*e^2*(2*c^2*d^2 + 3*b^2*e^2
- 2*c*e*(b*d + 5*a*e)))/4 + (-(c*(b^2 - 4*a*c)*e^2*(-2*c*d + b*e)*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e)
))/4 + 2*c*(-(c*(b^2 - 4*a*c)*d*e^2*(2*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(b*d + 5*a*e)))/4 + ((b^2 - 4*a*c)*e^2*(4*c
^3*d^3 + 3*b^3*e^3 - c^2*d*e*(5*b*d - 16*a*e) - b*c*e^2*(2*b*d + 13*a*e)))/4))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[
(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e - Sqr
t[b^2 - 4*a*c]*e))))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))

________________________________________________________________________________________

Maple [B]  time = 0.116, size = 2743, normalized size = 4.1 \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)/(c*x^2+b*x+a)^3/(e*x+d)^(1/2),x)

[Out]

e^3*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*
e^2-2*b*c*d*e+2*c^2*d^2-b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)+2*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)*b-2*e^2
*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2
-2*b*c*d*e+2*c^2*d^2-b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)+2*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)*d-5/2*e^2*
c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-
2*b*c*d*e+2*c^2*d^2-b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)+2*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)*(-4*a*c*e^2
+b^2*e^2)^(1/2)-e^3*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-
b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2)*b+2*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*
b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2)*d+7/2*e^2*c/(-e^
2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d+(-4*a*c*e^2+b^2*
e^2)^(1/2))*(e*x+d)^(1/2)*(-4*a*c*e^2+b^2*e^2)^(1/2)+4*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(8*a*c*e^2
-4*b^2*e^2+8*b*c*d*e-8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/2)/((-b
*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1
/2))*c)^(1/2))*b-8*e^2*c^4/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(8*a*c*e^2-4*b^2*e^2+8*b*c*d*e-8*c^2*d^2+4*b*e
*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c
)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d-10*e^2*c^3/(-e^2*(4
*a*c-b^2))^(1/2)/(4*a*c-b^2)/(8*a*c*e^2-4*b^2*e^2+8*b*c*d*e-8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*(-4
*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(
1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)-e^3*c^2/(-e^2*(4*a*c-b^2))^(1
/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+b*e*(
-4*a*c*e^2+b^2*e^2)^(1/2)-2*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)*b+2*e^2*c^3/(-e^2*(4*a*c-b^2))^(1/2)
/(4*a*c-b^2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+b*e*(-4*
a*c*e^2+b^2*e^2)^(1/2)-2*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)*d-5/2*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/
(4*a*c-b^2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-2*a*c*e^2+b^2*e^2-2*b*c*d*e+2*c^2*d^2+b*e*(-4*a
*c*e^2+b^2*e^2)^(1/2)-2*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*(e*x+d)^(3/2)*(-4*a*c*e^2+b^2*e^2)^(1/2)+e^3*c/(-e^2*(
4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2
)^(1/2))*(e*x+d)^(1/2)*b-2*e^2*c^2/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2)
)^(1/2))^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2)*d+7/2*e^2*c/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b
^2)/(e*x+1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))^2/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(e*x+d)^(1/2)*(-4
*a*c*e^2+b^2*e^2)^(1/2)-4*e^3*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d
^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(
1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b+8*e^2*c^4/(-e^
2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-8*a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*
d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*
2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d-10*e^2*c^3/(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-8*
a*c*e^2+4*b^2*e^2-8*b*c*d*e+8*c^2*d^2+4*b*e*(-4*a*c*e^2+b^2*e^2)^(1/2)-8*d*(-4*a*c*e^2+b^2*e^2)^(1/2)*c)*2^(1/
2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2)
)^(1/2))*c)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*c*x + b)/((c*x^2 + b*x + a)^3*sqrt(e*x + d)), x)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out