3.1579 \(\int \frac {b+2 c x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx\)

Optimal. Leaf size=225 \[ \frac {\sqrt {a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {3 e \left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.27, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {834, 806, 724, 206} \[ \frac {\sqrt {a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {3 e \left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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 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 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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.32, size = 220, normalized size = 0.98 \[ \frac {\sqrt {a+x (b+c x)} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{4 (d+e x) \left (e (a e-b d)+c d^2\right )^2}-\frac {3 e \left (b^2-4 a c\right ) (b e-2 c d) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{8 \left (e (a e-b d)+c d^2\right )^{5/2}}+\frac {\sqrt {a+x (b+c x)} (2 c d-b e)}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.94, size = 1500, normalized size = 6.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*a*b*c)*d^2*e^2 + (2*(b^2*c - 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e
^4)*x^2 + 2*(2*(b^2*c - 4*a*c^2)*d^2*e^2 - (b^3 - 4*a*b*c)*d*e^3)*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*(8*c^3*d^5 - 18*b*c^2*d^4*e - 2*a^2*b*e^5 + (15*b^2*c + 4*a*c^2)*d^3*e^2 - (
5*b^3 + 8*a*b*c)*d^2*e^3 + (7*a*b^2 - 4*a^2*c)*d*e^4 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + (7*b^2*c - 4*a*c^2)*d^
2*e^3 - (3*b^3 - 4*a*b*c)*d*e^4 + (3*a*b^2 - 8*a^2*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^8 - 3*b*c^2*d^7*e
- 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*
e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^3 + 6*a*b*c)*d
^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^
2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x), -1/8*(3*(2*(b^2*c - 4*a*c^2)*d
^3*e - (b^3 - 4*a*b*c)*d^2*e^2 + (2*(b^2*c - 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*x^2 + 2*(2*(b^2*c - 4*a*c^2
)*d^2*e^2 - (b^3 - 4*a*b*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sq
rt(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*(8*c^3*d^5 - 18*b*c^2*d^4*e - 2*a^2*b*e^5 + (15*b^2*c + 4*a*c^2
)*d^3*e^2 - (5*b^3 + 8*a*b*c)*d^2*e^3 + (7*a*b^2 - 4*a^2*c)*d*e^4 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + (7*b^2*c
- 4*a*c^2)*d^2*e^3 - (3*b^3 - 4*a*b*c)*d*e^4 + (3*a*b^2 - 8*a^2*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))/(c^3*d^8 - 3
*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2
+ a^2*c)*d^4*e^4 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4*e^4 - (b^3
+ 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*
d*e^7 + 3*(b^2*c + a*c^2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x)]

________________________________________________________________________________________

giac [B]  time = 0.39, size = 1046, normalized size = 4.65 \[ -\frac {3 \, {\left (2 \, b^{2} c d e - 8 \, a c^{2} d e - b^{3} e^{2} + 4 \, a b c e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} + \frac {16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {7}{2}} d^{4} - 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b c^{\frac {5}{2}} d^{3} e + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{3} d^{4} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} c^{2} d^{3} e - 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c^{3} d^{3} e + 4 \, b^{2} c^{\frac {5}{2}} d^{4} + 34 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} c^{\frac {3}{2}} d^{2} e^{2} - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {5}{2}} d^{2} e^{2} - 4 \, b^{3} c^{\frac {3}{2}} d^{3} e - 16 \, a b c^{\frac {5}{2}} d^{3} e + 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} c d e^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c^{2} d e^{3} + 22 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} c d^{2} e^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c^{2} d^{2} e^{2} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{3} \sqrt {c} d e^{3} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b c^{\frac {3}{2}} d e^{3} + 3 \, b^{4} \sqrt {c} d^{2} e^{2} + 10 \, a b^{2} c^{\frac {3}{2}} d^{2} e^{2} + 8 \, a^{2} c^{\frac {5}{2}} d^{2} e^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{3} e^{4} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b c e^{4} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{4} d e^{3} - 22 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} c d e^{3} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c^{2} d e^{3} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} c^{\frac {3}{2}} e^{4} - 11 \, a b^{3} \sqrt {c} d e^{3} + 12 \, a^{2} b c^{\frac {3}{2}} d e^{3} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{3} e^{4} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b c e^{4} + 8 \, a^{2} b^{2} \sqrt {c} e^{4} - 16 \, a^{3} c^{\frac {3}{2}} e^{4}}{4 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} d + b d - a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(2*b^2*c*d*e - 8*a*c^2*d*e - b^3*e^2 + 4*a*b*c*e^2)*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^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2
*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/4*(16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(7/2)*d^4 - 32*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))^2*b*c^(5/2)*d^3*e + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^3*d^4 - 24*(sqrt(c)*
x - sqrt(c*x^2 + b*x + a))*b^2*c^2*d^3*e - 32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^3*d^3*e + 4*b^2*c^(5/2)*
d^4 + 34*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(3/2)*d^2*e^2 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*
a*c^(5/2)*d^2*e^2 - 4*b^3*c^(3/2)*d^3*e - 16*a*b*c^(5/2)*d^3*e + 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c
*d*e^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^2*d*e^3 + 22*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c*d
^2*e^2 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^2*d^2*e^2 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*s
qrt(c)*d*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(3/2)*d*e^3 + 3*b^4*sqrt(c)*d^2*e^2 + 10*a*b^2*c^
(3/2)*d^2*e^2 + 8*a^2*c^(5/2)*d^2*e^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*e^4 + 12*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^3*a*b*c*e^4 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4*d*e^3 - 22*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*a*b^2*c*d*e^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^2*d*e^3 + 16*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^2*a^2*c^(3/2)*e^4 - 11*a*b^3*sqrt(c)*d*e^3 + 12*a^2*b*c^(3/2)*d*e^3 + 5*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*a*b^3*e^4 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*c*e^4 + 8*a^2*b^2*sqrt(c)*e^4 - 16*a^3*c^(3/
2)*e^4)/((c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*((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)^2)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 1588, normalized size = 7.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?`
for more details)Is a*e^2-b*d*e                            +c*d^2    positive, negative or zero?

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {b+2\,c\,x}{{\left (d+e\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b + 2 c x}{\left (d + e x\right )^{3} \sqrt {a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________