3.1626 \(\int \frac{(b+2 c x) \sqrt{d+e x}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=463 \[ \frac{\sqrt{c} e \left (-2 c e \left (-d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e \left (-2 c e \left (d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{e \sqrt{d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

-Sqrt[d + e*x]/(2*(a + b*x + c*x^2)^2) - (e*Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(4*(b
^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) + (Sqrt[c]*e*(8*c^2*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*e^2
- 2*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - 6*a*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)) - (Sqrt[c]*e*(8*c^2*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*a*e))*A
rcTanh[(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))

________________________________________________________________________________________

Rubi [A]  time = 1.8903, antiderivative size = 463, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {768, 740, 826, 1166, 208} \[ \frac{\sqrt{c} e \left (-2 c e \left (-d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt{b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e \left (-2 c e \left (d \sqrt{b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt{b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} \left (b^2-4 a c\right )^{3/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{e \sqrt{d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x}}{2 \left (a+b x+c x^2\right )^2} \]

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]/(2*(a + b*x + c*x^2)^2) - (e*Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(4*(b
^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) + (Sqrt[c]*e*(8*c^2*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*e^2
- 2*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - 6*a*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)) - (Sqrt[c]*e*(8*c^2*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*a*e))*A
rcTanh[(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))

Rule 768

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

Rule 740

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

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

Mathematica [B]  time = 6.39562, size = 1080, normalized size = 2.33 \[ -\frac{\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 ) (d+e x)^{3/2}}{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{\left (\frac{3}{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+b e) \left (-e b^2+c d b+2 a c e\right ) e+c \left (\frac{3}{2} c \left (b^2-4 a c\right ) e^2 (b d-2 a e)-\frac{1}{2} \left (b^2-4 a c\right ) e (2 c d-b e) (c d+b e)\right ) x\right ) (d+e x)^{3/2}}{\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 )}-\frac{\frac{1}{2} \left (b^2-4 a c\right ) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \sqrt{d+e x} e^2+\frac{4 \left (\frac{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e} \left (-\frac{1}{8} c^2 \left (b^2-4 a c\right ) (2 c d-b e) \left (c d^2-e (b d-a e)\right ) e^2-\frac{\frac{1}{8} c^2 \left (b^2-4 a c\right ) (2 c d-b e) (b e-2 c d) \left (c d^2-e (b d-a e)\right ) e^2+2 c \left (\frac{1}{8} c^2 \left (b^2-4 a c\right ) d e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )-\frac{1}{8} c \left (b^2-4 a c\right ) e^2 \left (c d^2-b e d+a e^2\right ) \left (4 c^2 d^2-3 b c e d-b^2 e^2+6 a c e^2\right )\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{\frac{1}{8} c^2 \left (b^2-4 a c\right ) (2 c d-b e) (b e-2 c d) \left (c d^2-e (b d-a e)\right ) e^2+2 c \left (\frac{1}{8} c^2 \left (b^2-4 a c\right ) d e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )-\frac{1}{8} c \left (b^2-4 a c\right ) e^2 \left (c d^2-b e d+a e^2\right ) \left (4 c^2 d^2-3 b c e d-b^2 e^2+6 a c e^2\right )\right )}{\sqrt{b^2-4 a c} e}-\frac{1}{8} c^2 \left (b^2-4 a c\right ) e^2 (2 c d-b e) \left (c d^2-e (b d-a e)\right )\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 )}{c}}{\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]

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

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Maple [B]  time = 0.05, size = 3056, normalized size = 6.6 \begin{align*} \text{output 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)^(1/2)/(c*x^2+b*x+a)^3,x)

[Out]

e^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2/(-e^2*(4*a*c-b^2))^(1/2)*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*d+e^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2/(-e^2*(4*a*c-b^2))^
(1/2)*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*d-1/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*c^2*d^2+1/2
*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)
^(5/2)*a-1/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^
2)*(e*x+d)^(5/2)*b^2-3/2*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*a*c-1/4*e^3/(c*e^2*x^2+b*e^
2*x+a*e^2)^2*c^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(7/2)*b+1/2*e^2/(c*
e^2*x^2+b*e^2*x+a*e^2)^2*c^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(7/2)*d
-3/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*
x+d)^(5/2)*d^2+3/2*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*
c*d^2)*(e*x+d)^(3/2)*c^3*d^3-e^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^3/(-e^2*(
4*a*c-b^2))^(1/2)*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^2-3/2*e^4/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b
^2*c*d^2)*c^2/(-e^2*(4*a*c-b^2))^(1/2)*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))*a+1/8*e^4/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e
+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c/(-e^2*(4*a*c-b^2))^(1/2)*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^2-e^2/(4*a^2*c*e^2-a*b
^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^3/(-e^2*(4*a*c-b^2))^(1/2)*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^2-
3/2*e^4/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2/(-e^2*(4*a*c-b^2))^(1/2)*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))*a+1/8*e^4/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c/(-e^2*(4*a*c-b^
2))^(1/2)*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^2-1/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4
*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(3/2)*b^3+1/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*
b^2+3/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*
(e*x+d)^(5/2)*b*d+1/4*e^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2*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+1/8*e^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^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-1/4*e^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*c^2*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-1/8*e^3/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^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-1/2*
e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(3/2
)*a*c^2*d+1/4*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2
)*(e*x+d)^(3/2)*c*a*b-9/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d
*e-b^2*c*d^2)*(e*x+d)^(3/2)*b*c^2*d^2+1/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a*c-b^2)*(e*x+d)^(1/2)*b*c*d+5/4*
e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)*(e*x+d)^(3/2
)*b^2*c*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

[Out]

Timed out

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

[Out]

Timed out