3.1167 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^2} \, dx\)
Optimal. Leaf size=185 \[ \frac{\sqrt{b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (-2 A c e-b B e+4 B c d)}{\sqrt{c} e^3}+\frac{(B d (4 c d-3 b e)-A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 \sqrt{d} e^3 \sqrt{c d-b e}} \]
[Out]
((2*B*d - A*e + B*e*x)*Sqrt[b*x + c*x^2])/(e^2*(d + e*x)) - ((4*B*c*d - b*B*e - 2*A*c*e)*ArcTanh[(Sqrt[c]*x)/S
qrt[b*x + c*x^2]])/(Sqrt[c]*e^3) + ((B*d*(4*c*d - 3*b*e) - A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/
(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*Sqrt[d]*e^3*Sqrt[c*d - b*e])
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Rubi [A] time = 0.185601, antiderivative size = 185, 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 =
{812, 843, 620, 206, 724} \[ \frac{\sqrt{b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (-2 A c e-b B e+4 B c d)}{\sqrt{c} e^3}+\frac{(B d (4 c d-3 b e)-A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 \sqrt{d} e^3 \sqrt{c d-b e}} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^2,x]
[Out]
((2*B*d - A*e + B*e*x)*Sqrt[b*x + c*x^2])/(e^2*(d + e*x)) - ((4*B*c*d - b*B*e - 2*A*c*e)*ArcTanh[(Sqrt[c]*x)/S
qrt[b*x + c*x^2]])/(Sqrt[c]*e^3) + ((B*d*(4*c*d - 3*b*e) - A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/
(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*Sqrt[d]*e^3*Sqrt[c*d - b*e])
Rule 812
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + b*x + c*x^2)^p)/(e^2*(m + 1)*(m
+ 2*p + 2)), x] + Dist[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(
b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p
+ 2))*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] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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)^2} \, dx &=\frac{(2 B d-A e+B e x) \sqrt{b x+c x^2}}{e^2 (d+e x)}-\frac{\int \frac{b (2 B d-A e)+(4 B c d-b B e-2 A c e) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 e^2}\\ &=\frac{(2 B d-A e+B e x) \sqrt{b x+c x^2}}{e^2 (d+e x)}-\frac{(4 B c d-b B e-2 A c e) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{2 e^3}+\frac{(B d (4 c d-3 b e)-A e (2 c d-b e)) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 e^3}\\ &=\frac{(2 B d-A e+B e x) \sqrt{b x+c x^2}}{e^2 (d+e x)}-\frac{(4 B c d-b B e-2 A c e) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{e^3}-\frac{(B d (4 c d-3 b e)-A e (2 c d-b e)) \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 )}{e^3}\\ &=\frac{(2 B d-A e+B e x) \sqrt{b x+c x^2}}{e^2 (d+e x)}-\frac{(4 B c d-b B e-2 A c e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c} e^3}+\frac{(B d (4 c d-3 b e)-A e (2 c d-b e)) \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 )}{2 \sqrt{d} e^3 \sqrt{c d-b e}}\\ \end{align*}
Mathematica [A] time = 1.13256, size = 187, normalized size = 1.01 \[ \frac{\sqrt{x (b+c x)} \left (\frac{(A e (b e-2 c d)+B d (4 c d-3 b e)) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{d} \sqrt{b+c x} \sqrt{b e-c d}}+\frac{\sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right ) (2 A c e+b B e-4 B c d)}{\sqrt{b} \sqrt{c} \sqrt{\frac{c x}{b}+1}}+\frac{e \sqrt{x} (-A e+2 B d+B e x)}{d+e x}\right )}{e^3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^2,x]
[Out]
(Sqrt[x*(b + c*x)]*((e*Sqrt[x]*(2*B*d - A*e + B*e*x))/(d + e*x) + ((-4*B*c*d + b*B*e + 2*A*c*e)*ArcSinh[(Sqrt[
c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]) + ((B*d*(4*c*d - 3*b*e) + A*e*(-2*c*d + b*e))*ArcTan
[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[b + c*x])))/(e^3*Sqrt
[x])
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Maple [B] time = 0.013, size = 2285, normalized size = 12.4 \begin{align*} \text{result 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)^2,x)
[Out]
B/e^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/2*B/e^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-B/e^3*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)*d+B/e^3*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/e^4*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+1/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)*A-1/e/(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-1/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*A+1/e/(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*B+1/e/(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)*c*A-1/e^2/(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)*c*B*d+1/e/(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))*c^(1/2)*b*A-1/e^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))*c^(1/2)*b*B*d-1/e^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))*c^(3/2)*A+1/e^3*d^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))*c^(3/2)*B-1/2/e/(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*A+1/2/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*B*d+3/2/e^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))
*b*c*A-3/2/e^3*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*c*B-1/e^3*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))*c^2*A+1/e^4*d^3/(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))*c^2*B-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*A+1/
e*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)*x*B
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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)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 3.58049, size = 3182, normalized size = 17.2 \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)^2,x, algorithm="fricas")
[Out]
[-1/2*((4*B*c^2*d^4 - (5*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 2*A*b*c)*d^2*e^2 + (4*B*c^2*d^3*e - (5*B*b*c + 2*A*
c^2)*d^2*e^2 + (B*b^2 + 2*A*b*c)*d*e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (4*B*c^2*d^3
+ A*b*c*d*e^2 - (3*B*b*c + 2*A*c^2)*d^2*e + (4*B*c^2*d^2*e + A*b*c*e^3 - (3*B*b*c + 2*A*c^2)*d*e^2)*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*(2*B*c^2*d^
3*e + A*b*c*d*e^3 - (2*B*b*c + A*c^2)*d^2*e^2 + (B*c^2*d^2*e^2 - B*b*c*d*e^3)*x)*sqrt(c*x^2 + b*x))/(c^2*d^3*e
^3 - b*c*d^2*e^4 + (c^2*d^2*e^4 - b*c*d*e^5)*x), 1/2*(2*(4*B*c^2*d^3 + A*b*c*d*e^2 - (3*B*b*c + 2*A*c^2)*d^2*e
+ (4*B*c^2*d^2*e + A*b*c*e^3 - (3*B*b*c + 2*A*c^2)*d*e^2)*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^4 - (5*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 2*A*b*c)*d^2*e^2 +
(4*B*c^2*d^3*e - (5*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 2*A*b*c)*d*e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^
2 + b*x)*sqrt(c)) + 2*(2*B*c^2*d^3*e + A*b*c*d*e^3 - (2*B*b*c + A*c^2)*d^2*e^2 + (B*c^2*d^2*e^2 - B*b*c*d*e^3)
*x)*sqrt(c*x^2 + b*x))/(c^2*d^3*e^3 - b*c*d^2*e^4 + (c^2*d^2*e^4 - b*c*d*e^5)*x), 1/2*(2*(4*B*c^2*d^4 - (5*B*b
*c + 2*A*c^2)*d^3*e + (B*b^2 + 2*A*b*c)*d^2*e^2 + (4*B*c^2*d^3*e - (5*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 2*A*
b*c)*d*e^3)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (4*B*c^2*d^3 + A*b*c*d*e^2 - (3*B*b*c + 2*A
*c^2)*d^2*e + (4*B*c^2*d^2*e + A*b*c*e^3 - (3*B*b*c + 2*A*c^2)*d*e^2)*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*(2*B*c^2*d^3*e + A*b*c*d*e^3 - (2*B*b*c +
A*c^2)*d^2*e^2 + (B*c^2*d^2*e^2 - B*b*c*d*e^3)*x)*sqrt(c*x^2 + b*x))/(c^2*d^3*e^3 - b*c*d^2*e^4 + (c^2*d^2*e^4
- b*c*d*e^5)*x), ((4*B*c^2*d^3 + A*b*c*d*e^2 - (3*B*b*c + 2*A*c^2)*d^2*e + (4*B*c^2*d^2*e + A*b*c*e^3 - (3*B*
b*c + 2*A*c^2)*d*e^2)*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^4 - (5*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 2*A*b*c)*d^2*e^2 + (4*B*c^2*d^3*e - (5*B*b*c + 2*A*c^2)*
d^2*e^2 + (B*b^2 + 2*A*b*c)*d*e^3)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (2*B*c^2*d^3*e + A*b
*c*d*e^3 - (2*B*b*c + A*c^2)*d^2*e^2 + (B*c^2*d^2*e^2 - B*b*c*d*e^3)*x)*sqrt(c*x^2 + b*x))/(c^2*d^3*e^3 - b*c*
d^2*e^4 + (c^2*d^2*e^4 - b*c*d*e^5)*x)]
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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 )^{2}}\, 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)**2,x)
[Out]
Integral(sqrt(x*(b + c*x))*(A + B*x)/(d + e*x)**2, x)
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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((B*x+A)*(c*x^2+b*x)^(1/2)/(e*x+d)^2,x, algorithm="giac")
[Out]
Timed out