3.1620 \(\int \frac {(b+2 c x) \sqrt {d+e x}}{(a+b x+c x^2)^2} \, dx\)

Optimal. Leaf size=223 \[ -\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x}}{a+b x+c x^2} \]

[Out]

-(e*x+d)^(1/2)/(c*x^2+b*x+a)-e*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*2
^(1/2)*c^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)+e*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/
2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))*2^(1/2)*c^(1/2)/(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))
^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.32, antiderivative size = 223, normalized size of antiderivative = 1.00, 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 = {768, 707, 1093, 208} \[ -\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x}}{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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}, 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]

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 1093

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.75, size = 221, normalized size = 0.99 \[ \frac {-\frac {\sqrt {b^2-4 a c} \sqrt {d+e x}}{a+x (b+c x)}-\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {e \sqrt {b^2-4 a c}-b e+2 c d}}\right )}{\sqrt {e \left (\sqrt {b^2-4 a c}-b\right )+2 c d}}+\frac {\sqrt {2} \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}{\sqrt {b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.09, size = 2750, normalized size = 12.33 \[ \text {result too large to display} \]

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)^2,x, algorithm="fricas")

[Out]

-1/2*(sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a
*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4)
)*((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b
*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(2*sqrt(e*x + d)*c*e^4 + sqrt(1/2)*((b^2 - 4*a*c)*e^4 - sqrt(e^6/((b^2*c^
2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)
*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*(2*(b^2*c^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c
- 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3))*sqrt((2*c*d*e^2 - b*e^3 + sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 -
2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 -
4*a^3*c)*e^4))*((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))/((b^2*c - 4*a*c^2)*d^2
- (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))) - sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 + sqrt(
e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3
- 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c
)*e^2))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(2*sqrt(e*x + d)*c*e^4 - sqr
t(1/2)*((b^2 - 4*a*c)*e^4 - sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c
- 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*(2*(b^2*c^2 - 4*a*c^3)*d^3 - 3
*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3))*sqrt((2*c*d*e^2 -
b*e^3 + sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e
^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a
*b^2 - 4*a^2*c)*e^2))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))) + sqrt(1/2)*(c*x
^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4
- 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*((b^2*c - 4*a*c^2)
*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4
*a^2*c)*e^2))*log(2*sqrt(e*x + d)*c*e^4 + sqrt(1/2)*((b^2 - 4*a*c)*e^4 + sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2
*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 -
4*a^3*c)*e^4))*(2*(b^2*c^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2
- (a*b^3 - 4*a^2*b*c)*e^3))*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2
)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*((b^
2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*
e + (a*b^2 - 4*a^2*c)*e^2))) - sqrt(1/2)*(c*x^2 + b*x + a)*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/((b^2*c^2 - 4*a*
c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 +
(a^2*b^2 - 4*a^3*c)*e^4))*((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))/((b^2*c - 4*
a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))*log(2*sqrt(e*x + d)*c*e^4 - sqrt(1/2)*((b^2 - 4*a*c
)*e^4 + sqrt(e^6/((b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^
2 - 2*(a*b^3 - 4*a^2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*(2*(b^2*c^2 - 4*a*c^3)*d^3 - 3*(b^3*c - 4*a*b*c^2)
*d^2*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d*e^2 - (a*b^3 - 4*a^2*b*c)*e^3))*sqrt((2*c*d*e^2 - b*e^3 - sqrt(e^6/((
b^2*c^2 - 4*a*c^3)*d^4 - 2*(b^3*c - 4*a*b*c^2)*d^3*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^2 - 2*(a*b^3 - 4*a^
2*b*c)*d*e^3 + (a^2*b^2 - 4*a^3*c)*e^4))*((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2)
)/((b^2*c - 4*a*c^2)*d^2 - (b^3 - 4*a*b*c)*d*e + (a*b^2 - 4*a^2*c)*e^2))) + 2*sqrt(e*x + d))/(c*x^2 + b*x + a)

________________________________________________________________________________________

giac [A]  time = 1.08, size = 351, normalized size = 1.57 \[ -\frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} c \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right ) e}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c d + {\left (b^{2} - 4 \, a c - \sqrt {b^{2} - 4 \, a c} b\right )} e\right )} {\left | c \right |}} + \frac {\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} c \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right ) e}{{\left (2 \, \sqrt {b^{2} - 4 \, a c} c d - {\left (b^{2} - 4 \, a c + \sqrt {b^{2} - 4 \, a c} b\right )} e\right )} {\left | c \right |}} - \frac {\sqrt {x e + d} e^{2}}{{\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e + a e^{2}} \]

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)^2,x, algorithm="giac")

[Out]

-sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*c*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*c*d - b*e + sqrt
((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))*e/((2*sqrt(b^2 - 4*a*c)*c*d + (b^2 - 4*a*c - sqrt(b^2 - 4
*a*c)*b)*e)*abs(c)) + sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*c*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt
(-(2*c*d - b*e - sqrt((2*c*d - b*e)^2 - 4*(c*d^2 - b*d*e + a*e^2)*c))/c))*e/((2*sqrt(b^2 - 4*a*c)*c*d - (b^2 -
4*a*c + sqrt(b^2 - 4*a*c)*b)*e)*abs(c)) - sqrt(x*e + d)*e^2/((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e +
d)*b*e - b*d*e + a*e^2)

________________________________________________________________________________________

maple [A]  time = 0.09, size = 232, normalized size = 1.04 \[ -\frac {\sqrt {2}\, c \,e^{2} \arctanh \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (-b e +2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {\sqrt {2}\, c \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, \sqrt {2}\, c}{\sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}\right )}{\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\, \sqrt {\left (b e -2 c d +\sqrt {-\left (4 a c -b^{2}\right ) e^{2}}\right ) c}}-\frac {\sqrt {e x +d}\, e^{2}}{c \,e^{2} x^{2}+b \,e^{2} x +a \,e^{2}} \]

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)^2,x)

[Out]

-e^2*(e*x+d)^(1/2)/(c*e^2*x^2+b*e^2*x+a*e^2)-e^2*c/(-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)
*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*2^(1/2)/((-b*e+2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)-e^2*c/(
-(4*a*c-b^2)*e^2)^(1/2)*2^(1/2)/((b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*2^(1/2)/((
b*e-2*c*d+(-(4*a*c-b^2)*e^2)^(1/2))*c)^(1/2)*c)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, c x + b\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{2}}\,{d x} \]

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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 2.93, size = 4814, normalized size = 21.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((((4*b^2*c^2*e^4 - 16*a*c^3*e^4 + (d + e*x)^(1/2)*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3
+ 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*
b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 -
32*a*b*c^3*e^3 + 64*a*c^4*d*e^2))*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*
b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b
^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 4*c^3*e^4*(d + e*x)^(1/2))*(-(b^3*e^3 + e^3*(-(4*a*c -
b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a
^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*1i + ((16
*a*c^3*e^4 - 4*b^2*c^2*e^4 + (d + e*x)^(1/2)*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2
*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^
2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3
*e^3 + 64*a*c^4*d*e^2))*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^
2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 -
16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 4*c^3*e^4*(d + e*x)^(1/2))*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1
/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2
- b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*1i)/(((4*b^2*c^2*e^
4 - 16*a*c^3*e^4 + (d + e*x)^(1/2)*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2
*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*
b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*
a*c^4*d*e^2))*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b
^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*
c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 4*c^3*e^4*(d + e*x)^(1/2))*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*
b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e
- 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) - ((16*a*c^3*e^4 - 4*b^2*c^2*
e^4 + (d + e*x)^(1/2)*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)
/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 1
6*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2))
*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*
c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a
*b^3*c*d*e)))^(1/2) + 4*c^3*e^4*(d + e*x)^(1/2))*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c*e^3 + 8*a
*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^
2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)))*(-(b^3*e^3 + e^3*(-(4*a*c - b^2)^3)^(1/2
) - 4*a*b*c*e^3 + 8*a*c^2*d*e^2 - 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 -
b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*2i + atan((((4*b^2*c^
2*e^4 - 16*a*c^3*e^4 + (d + e*x)^(1/2)*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2
+ 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a
^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 +
64*a*c^4*d*e^2))*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2)/(8*(a
*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*
b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 4*c^3*e^4*(d + e*x)^(1/2))*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a
*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*
e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*1i + ((16*a*c^3*e^4 - 4*b^2*
c^2*e^4 + (d + e*x)^(1/2)*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e
^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2
- 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^
2))*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^
4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8
*a*b^3*c*d*e)))^(1/2) + 4*c^3*e^4*(d + e*x)^(1/2))*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*
a*c^2*d*e^2 + 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c
^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*1i)/(((4*b^2*c^2*e^4 - 16*a*c^3*e^4 + (d
+ e*x)^(1/2)*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2)/(8*(a*b^4
*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^
2*d*e + 8*a*b^3*c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2))*((e^3*(-(
4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*
a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)
))^(1/2) + 4*c^3*e^4*(d + e*x)^(1/2))*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 +
2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^
2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) - ((16*a*c^3*e^4 - 4*b^2*c^2*e^4 + (d + e*x)^(1/2)*((e
^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2
+ 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*
c*d*e)))^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2))*((e^3*(-(4*a*c - b^2)^3)^
(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16
*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2) + 4*c^3
*e^4*(d + e*x)^(1/2))*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e^2 + 2*b^2*c*d*e^2)/
(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 - 8*a^2*b^2*c*e^2 - 16
*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)))*((e^3*(-(4*a*c - b^2)^3)^(1/2) - b^3*e^3 + 4*a*b*c*e^3 - 8*a*c^2*d*e
^2 + 2*b^2*c*d*e^2)/(8*(a*b^4*e^2 + b^4*c*d^2 + 16*a^2*c^3*d^2 + 16*a^3*c^2*e^2 - b^5*d*e - 8*a*b^2*c^2*d^2 -
8*a^2*b^2*c*e^2 - 16*a^2*b*c^2*d*e + 8*a*b^3*c*d*e)))^(1/2)*2i - (e^2*(d + e*x)^(1/2))/((b*e - 2*c*d)*(d + e*x
) + c*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**2,x)

[Out]

Timed out

________________________________________________________________________________________