3.1168 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=235 \[ -\frac{\left (A b^2 e^3+B d \left (3 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{3/2} e^3 (c d-b e)^{3/2}}+\frac{\sqrt{b x+c x^2} \left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e x (B d (6 c d-5 b e)-A e (2 c d-b e))\right )}{4 d e^2 (d+e x)^2 (c d-b e)}+\frac{2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^3} \]

[Out]

((d*(A*b*e^2 - B*d*(4*c*d - 3*b*e)) - e*(B*d*(6*c*d - 5*b*e) - A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(4*d*e
^2*(c*d - b*e)*(d + e*x)^2) + (2*B*Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e^3 - ((A*b^2*e^3 + B*d*(8*
c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2
])])/(8*d^(3/2)*e^3*(c*d - b*e)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.322618, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {810, 843, 620, 206, 724} \[ -\frac{\left (A b^2 e^3+B d \left (3 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{3/2} e^3 (c d-b e)^{3/2}}+\frac{\sqrt{b x+c x^2} \left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e x (B d (6 c d-5 b e)-A e (2 c d-b e))\right )}{4 d e^2 (d+e x)^2 (c d-b e)}+\frac{2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d*(A*b*e^2 - B*d*(4*c*d - 3*b*e)) - e*(B*d*(6*c*d - 5*b*e) - A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(4*d*e
^2*(c*d - b*e)*(d + e*x)^2) + (2*B*Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/e^3 - ((A*b^2*e^3 + B*d*(8*
c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2
])])/(8*d^(3/2)*e^3*(c*d - b*e)^(3/2))

Rule 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], 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] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rule 620

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

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]

Rubi steps

\begin{align*} \int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^3} \, dx &=\frac{\left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e (B d (6 c d-5 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{4 d e^2 (c d-b e) (d+e x)^2}-\frac{\int \frac{\frac{1}{2} b \left (A b e^2-B d (4 c d-3 b e)\right )-4 B c d (c d-b e) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{4 d e^2 (c d-b e)}\\ &=\frac{\left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e (B d (6 c d-5 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{4 d e^2 (c d-b e) (d+e x)^2}+\frac{(B c) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{e^3}-\frac{\left (A b^2 e^3+B d \left (8 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 d e^3 (c d-b e)}\\ &=\frac{\left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e (B d (6 c d-5 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{4 d e^2 (c d-b e) (d+e x)^2}+\frac{(2 B c) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{e^3}+\frac{\left (A b^2 e^3+B d \left (8 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{4 d e^3 (c d-b e)}\\ &=\frac{\left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e (B d (6 c d-5 b e)-A e (2 c d-b e)) x\right ) \sqrt{b x+c x^2}}{4 d e^2 (c d-b e) (d+e x)^2}+\frac{2 B \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^3}-\frac{\left (A b^2 e^3+B d \left (8 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{8 d^{3/2} e^3 (c d-b e)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 2.96439, size = 363, normalized size = 1.54 \[ \frac{\sqrt{x (b+c x)} \left (\frac{\frac{\sqrt{d} \left (A b^2 e^3+B d \left (3 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x} \sqrt{b e-c d}}-\frac{e \sqrt{x} \left (A e^2 \left (b^2 e-b c d+b c e x-2 c^2 d x\right )+B d \left (3 b^2 e^2+b c e (3 e x-7 d)+2 c^2 d (2 d-e x)\right )\right )}{b e-c d}+\frac{8 B \sqrt{c} d^2 (b e-c d) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}}{2 d e^3}+\frac{x^{3/2} (b+c x) (A e (2 c d-b e)+B d (2 c d-3 b e))}{2 d (d+e x) (c d-b e)}+\frac{x^{3/2} (b+c x) (A e-B d)}{(d+e x)^2}\right )}{2 d \sqrt{x} (b e-c d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(((-(B*d) + A*e)*x^(3/2)*(b + c*x))/(d + e*x)^2 + ((B*d*(2*c*d - 3*b*e) + A*e*(2*c*d - b*e)
)*x^(3/2)*(b + c*x))/(2*d*(c*d - b*e)*(d + e*x)) + (-((e*Sqrt[x]*(A*e^2*(-(b*c*d) + b^2*e - 2*c^2*d*x + b*c*e*
x) + B*d*(3*b^2*e^2 + 2*c^2*d*(2*d - e*x) + b*c*e*(-7*d + 3*e*x))))/(-(c*d) + b*e)) + (8*B*Sqrt[c]*d^2*(-(c*d)
+ b*e)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b]) + (Sqrt[d]*(A*b^2*e^3 + B*d*(8*c^2*d^2
- 12*b*c*d*e + 3*b^2*e^2))*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[-(c*d) + b*e]*
Sqrt[b + c*x]))/(2*d*e^3)))/(2*d*(-(c*d) + b*e)*Sqrt[x])

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 4316, normalized size = 18.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/d/(b*e-c*d)^2*c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*b*B+1/2/e^3*d^2/(b*e-c*d)^2/(-
d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b
*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*c^3*A+1/4*e/d^2/(b*e-c*d)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c
*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*b*A-1/2/e^4*d^3/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e
^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))
/(x+d/e))*c^3*B-3/2/e^4*c^2*d^2/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e
)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*B-1/4/e*c^(1/
2)/d/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1
/2))*b*A-1/2/e^2*c/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-
c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b*A+1/2/e^3*c^2*d/(b*e-c*d
)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*
c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*A+3/4/e^2/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c
)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b*B*d+5/8/e/(b*e-c*d)^2/(-d*(b*e-
c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*
d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2*c*A-B/e*c/d/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(
b*e-c*d)/e^2)^(1/2)*x+1/e^3*d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d
/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b*c^2*B+1/2
/e/d/(b*e-c*d)/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*A-1/4*e/d^2/(b*e-c*d)^2*((x
+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*A+1/2/e^2/(b*e-c*d)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x
+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*B*d-3/4/e/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(
b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b*A+1/2/e^2*d/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c
)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(5/2)*A-1/2/e^3*d^2/(b*e-c*d)^2*ln((1/2
*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(5/2)*B-1/2/e*c
/d/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A-3/2/e^3*c^(3/2)/(b*e-c*d)*ln((1/2*(b*
e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*B*d-1/2/e/(b*e-c*d)^2
*c^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*B+5/4/e^2*c^(1/2)/(b*e-c*d)*ln((1/2*(b*e-2*c*
d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*b*B+B/e/d/(b*e-c*d)/(x+d/e)
*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)-B/e/d/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-
d*(b*e-c*d)/e^2)^(1/2)*b-1/2*B/e^2/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+
d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2+1/2/d/
(b*e-c*d)^2*c^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*A-1/4/d/(b*e-c*d)^2/(x+d/e)*((x+d/
e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*b*B-1/2/d/(b*e-c*d)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(
x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c*A+3/4/d/(b*e-c*d)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*
b*c*A+1/4/d/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d
)/e^2)^(1/2))*c^(1/2)*b^2*A-1/8/d/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x
+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^3*A+1/2
/e/(b*e-c*d)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c*B-3/4/e/(b*e-c*d)^2*((x+d/e
)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*c*B-1/4/e/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^
(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^2*B+1/8/e/(b*e-c*d)^2/(-d*(b*e-c*d)
/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^3*B+2/e^3*c/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/
e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)
)/(x+d/e))*b*B*d-1/4*e/d^2/(b*e-c*d)^2*c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*b*A-5/8/e
^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2
)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2*c*B*d-1/e^2*d/(b*e-c*d)^2/(-d*(b*e-c
*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d
)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b*c^2*A+3/2/e^2*c/(b*e-c*d)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*
(b*e-c*d)/e^2)^(1/2)*B-1/2/e/(b*e-c*d)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A+1/2/e
^2*c^(3/2)/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e
^2)^(1/2))*A+1/4/d/(b*e-c*d)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*B-1/2/e^2/(b*e-c*
d)/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*B

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 6.83854, size = 4652, normalized size = 19.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(8*(B*c^2*d^6 - 2*B*b*c*d^5*e + B*b^2*d^4*e^2 + (B*c^2*d^4*e^2 - 2*B*b*c*d^3*e^3 + B*b^2*d^2*e^4)*x^2 + 2
*(B*c^2*d^5*e - 2*B*b*c*d^4*e^2 + B*b^2*d^3*e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (8*
B*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3*e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d
*e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e - 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*sqrt(c*d^2 - b
*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(4*B*c^2*d^5*e - 7*
B*b*c*d^4*e^2 + A*b^2*d^2*e^4 + (3*B*b^2 - A*b*c)*d^3*e^3 + (6*B*c^2*d^4*e^2 - A*b^2*d*e^5 - (11*B*b*c + 2*A*c
^2)*d^3*e^3 + (5*B*b^2 + 3*A*b*c)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^6*e^3 - 2*b*c*d^5*e^4 + b^2*d^4*e^5 +
(c^2*d^4*e^5 - 2*b*c*d^3*e^6 + b^2*d^2*e^7)*x^2 + 2*(c^2*d^5*e^4 - 2*b*c*d^4*e^5 + b^2*d^3*e^6)*x), -1/4*((8*B
*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3*e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d*
e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e - 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*sqrt(-c*d^2 + b
*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - 4*(B*c^2*d^6 - 2*B*b*c*d^5*e + B*b^2*d
^4*e^2 + (B*c^2*d^4*e^2 - 2*B*b*c*d^3*e^3 + B*b^2*d^2*e^4)*x^2 + 2*(B*c^2*d^5*e - 2*B*b*c*d^4*e^2 + B*b^2*d^3*
e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + (4*B*c^2*d^5*e - 7*B*b*c*d^4*e^2 + A*b^2*d^2*e^
4 + (3*B*b^2 - A*b*c)*d^3*e^3 + (6*B*c^2*d^4*e^2 - A*b^2*d*e^5 - (11*B*b*c + 2*A*c^2)*d^3*e^3 + (5*B*b^2 + 3*A
*b*c)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^6*e^3 - 2*b*c*d^5*e^4 + b^2*d^4*e^5 + (c^2*d^4*e^5 - 2*b*c*d^3*e^6
+ b^2*d^2*e^7)*x^2 + 2*(c^2*d^5*e^4 - 2*b*c*d^4*e^5 + b^2*d^3*e^6)*x), -1/8*(16*(B*c^2*d^6 - 2*B*b*c*d^5*e +
B*b^2*d^4*e^2 + (B*c^2*d^4*e^2 - 2*B*b*c*d^3*e^3 + B*b^2*d^2*e^4)*x^2 + 2*(B*c^2*d^5*e - 2*B*b*c*d^4*e^2 + B*b
^2*d^3*e^3)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (8*B*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3
*e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d*e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e
- 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqr
t(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(4*B*c^2*d^5*e - 7*B*b*c*d^4*e^2 + A*b^2*d^2*e^4 + (3*B*b^2
- A*b*c)*d^3*e^3 + (6*B*c^2*d^4*e^2 - A*b^2*d*e^5 - (11*B*b*c + 2*A*c^2)*d^3*e^3 + (5*B*b^2 + 3*A*b*c)*d^2*e^
4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^6*e^3 - 2*b*c*d^5*e^4 + b^2*d^4*e^5 + (c^2*d^4*e^5 - 2*b*c*d^3*e^6 + b^2*d^2*e
^7)*x^2 + 2*(c^2*d^5*e^4 - 2*b*c*d^4*e^5 + b^2*d^3*e^6)*x), -1/4*((8*B*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3*
e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d*e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e
- 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(
c*x^2 + b*x)/((c*d - b*e)*x)) + 8*(B*c^2*d^6 - 2*B*b*c*d^5*e + B*b^2*d^4*e^2 + (B*c^2*d^4*e^2 - 2*B*b*c*d^3*e^
3 + B*b^2*d^2*e^4)*x^2 + 2*(B*c^2*d^5*e - 2*B*b*c*d^4*e^2 + B*b^2*d^3*e^3)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x
)*sqrt(-c)/(c*x)) + (4*B*c^2*d^5*e - 7*B*b*c*d^4*e^2 + A*b^2*d^2*e^4 + (3*B*b^2 - A*b*c)*d^3*e^3 + (6*B*c^2*d^
4*e^2 - A*b^2*d*e^5 - (11*B*b*c + 2*A*c^2)*d^3*e^3 + (5*B*b^2 + 3*A*b*c)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d
^6*e^3 - 2*b*c*d^5*e^4 + b^2*d^4*e^5 + (c^2*d^4*e^5 - 2*b*c*d^3*e^6 + b^2*d^2*e^7)*x^2 + 2*(c^2*d^5*e^4 - 2*b*
c*d^4*e^5 + b^2*d^3*e^6)*x)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**3, x)

________________________________________________________________________________________

Giac [B]  time = 1.41309, size = 1106, normalized size = 4.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*sqrt(c)*e^(-3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b)) - 1/4*(8*B*c^2*d^3 - 12*B*b*c*d^2*e
+ 3*B*b^2*d*e^2 + A*b^2*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c
*d^2*e^3 - b*d*e^4)*sqrt(-c*d^2 + b*d*e)) - 1/4*(16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^(5/2)*d^3*e + 24*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^2*B*c^3*d^4 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^2*d^3*e - 8*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^2*A*c^3*d^3*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^(5/2)*d^4 - 20*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^3*B*b*c^(3/2)*d^2*e^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(5/2)*d^2*e^2 - 24*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*B*b^2*c^(3/2)*d^3*e - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*c^(5/2)*d^3*e + 6*B*b^2*c^2*d^4
- (sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^2*c*d^2*e^2 - 5*B*b^3*c*d^3*e - 2*A*b^2*c^2*d^3*e + 5*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^3*B*b^2*sqrt(c)*d*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^(3/2)*d*e^3 + 3*(sqrt(c)*x
- sqrt(c*x^2 + b*x))*B*b^3*sqrt(c)*d^2*e^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*c^(3/2)*d^2*e^2 + 5*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c*d*e^3 + A*b^3*c*d^2*e^2 - (sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*sqrt(c)
*e^4 + (sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*sqrt(c)*d*e^3)/((c^(3/2)*d^2*e^3 - b*sqrt(c)*d*e^4)*((sqrt(c)*x -
sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2)