3.58 \(\int \frac{A+B x+C x^2}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x}} \, dx\)
Optimal. Leaf size=254 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (-3 a^2 b C (c f+d e)+2 a^3 C d f+a b^2 (-2 A d f+B c f+B d e+4 c C e)-b^3 (-A c f-A d e+2 B c e)\right )}{b^2 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}+\frac{2 C \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{b^2 \sqrt{d} \sqrt{f}} \]
[Out]
-(((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x))) + (2*C*ArcTanh[
(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b^2*Sqrt[d]*Sqrt[f]) + ((2*a^3*C*d*f - 3*a^2*b*C*(d*e + c*f
) - b^3*(2*B*c*e - A*d*e - A*c*f) + a*b^2*(4*c*C*e + B*d*e + B*c*f - 2*A*d*f))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c
+ d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(b^2*(b*c - a*d)^(3/2)*(b*e - a*f)^(3/2))
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Rubi [A] time = 0.638489, antiderivative size = 254, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used =
{1613, 157, 63, 217, 206, 93, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (-3 a^2 b C (c f+d e)+2 a^3 C d f+a b^2 (-2 A d f+B c f+B d e+4 c C e)-b^3 (-A c f-A d e+2 B c e)\right )}{b^2 (b c-a d)^{3/2} (b e-a f)^{3/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}+\frac{2 C \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{b^2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x + C*x^2)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
[Out]
-(((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)*(b*e - a*f)*(a + b*x))) + (2*C*ArcTanh[
(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b^2*Sqrt[d]*Sqrt[f]) + ((2*a^3*C*d*f - 3*a^2*b*C*(d*e + c*f
) - b^3*(2*B*c*e - A*d*e - A*c*f) + a*b^2*(4*c*C*e + B*d*e + B*c*f - 2*A*d*f))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c
+ d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(b^2*(b*c - a*d)^(3/2)*(b*e - a*f)^(3/2))
Rule 1613
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)
*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && ILtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]
Rule 157
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
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 93
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
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{A+B x+C x^2}{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x}} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{b (b c-a d) (b e-a f) (a+b x)}-\frac{\int \frac{-\frac{a^2 C (d e+c f)+b^2 (2 B c e-A d e-A c f)-a b (2 c C e+B d e+B c f-2 A d f)}{2 b}-\frac{C (b c-a d) (b e-a f) x}{b}}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{(b c-a d) (b e-a f)}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{b (b c-a d) (b e-a f) (a+b x)}+\frac{C \int \frac{1}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{b^2}-\frac{\left (2 a^3 C d f-3 a^2 b C (d e+c f)-b^3 (2 B c e-A d e-A c f)+a b^2 (4 c C e+B d e+B c f-2 A d f)\right ) \int \frac{1}{(a+b x) \sqrt{c+d x} \sqrt{e+f x}} \, dx}{2 b^2 (b c-a d) (b e-a f)}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{b (b c-a d) (b e-a f) (a+b x)}+\frac{(2 C) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e-\frac{c f}{d}+\frac{f x^2}{d}}} \, dx,x,\sqrt{c+d x}\right )}{b^2 d}-\frac{\left (2 a^3 C d f-3 a^2 b C (d e+c f)-b^3 (2 B c e-A d e-A c f)+a b^2 (4 c C e+B d e+B c f-2 A d f)\right ) \operatorname{Subst}\left (\int \frac{1}{-b c+a d-(-b e+a f) x^2} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{b^2 (b c-a d) (b e-a f)}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{b (b c-a d) (b e-a f) (a+b x)}+\frac{\left (2 a^3 C d f-3 a^2 b C (d e+c f)-b^3 (2 B c e-A d e-A c f)+a b^2 (4 c C e+B d e+B c f-2 A d f)\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{b^2 (b c-a d)^{3/2} (b e-a f)^{3/2}}+\frac{(2 C) \operatorname{Subst}\left (\int \frac{1}{1-\frac{f x^2}{d}} \, dx,x,\frac{\sqrt{c+d x}}{\sqrt{e+f x}}\right )}{b^2 d}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{b (b c-a d) (b e-a f) (a+b x)}+\frac{2 C \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right )}{b^2 \sqrt{d} \sqrt{f}}+\frac{\left (2 a^3 C d f-3 a^2 b C (d e+c f)-b^3 (2 B c e-A d e-A c f)+a b^2 (4 c C e+B d e+B c f-2 A d f)\right ) \tanh ^{-1}\left (\frac{\sqrt{b e-a f} \sqrt{c+d x}}{\sqrt{b c-a d} \sqrt{e+f x}}\right )}{b^2 (b c-a d)^{3/2} (b e-a f)^{3/2}}\\ \end{align*}
Mathematica [A] time = 2.01084, size = 324, normalized size = 1.28 \[ \frac{-\frac{b \sqrt{c+d x} \sqrt{e+f x} \left (a (a C-b B)+A b^2\right )}{(a+b x) (b c-a d) (b e-a f)}+\frac{\left (a (a C-b B)+A b^2\right ) (-2 a d f+b c f+b d e) \tan ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{a d-b c}}\right )}{(a d-b c)^{3/2} (b e-a f)^{3/2}}+\frac{2 (b B-2 a C) \tan ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{a d-b c}}\right )}{\sqrt{a d-b c} \sqrt{b e-a f}}+\frac{2 C \sqrt{e+f x} \sinh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d e-c f}}\right )}{\sqrt{f} \sqrt{d e-c f} \sqrt{\frac{d (e+f x)}{d e-c f}}}}{b^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x + C*x^2)/((a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
[Out]
(-((b*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x))) + (2*C*Sqrt
[e + f*x]*ArcSinh[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[d*e - c*f]])/(Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[(d*(e + f*x))/(d*e -
c*f)]) + (2*(b*B - 2*a*C)*ArcTan[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/(Sqrt[-
(b*c) + a*d]*Sqrt[b*e - a*f]) + ((A*b^2 + a*(-(b*B) + a*C))*(b*d*e + b*c*f - 2*a*d*f)*ArcTan[(Sqrt[b*e - a*f]*
Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(b*e - a*f)^(3/2)))/b^2
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Maple [B] time = 0.052, size = 2973, normalized size = 11.7 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((C*x^2+B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)
[Out]
-1/2*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-2*B*a*b^3*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c
)*(f*x+e))^(1/2)-B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*
x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*x*a*b^3*c*f*(d*f)^(1/2)-B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*x*a*b^3*d*e*(d*
f)^(1/2)-2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*x*a^2*b^2*d*f*((a^2*d
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f
)^(1/2))*x*a*b^3*c*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/
2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*x*a*b^3*d*e*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-2*C*ln((-2*a*d*
f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*
c*e)/(b*x+a))*x*a^3*b*d*f*(d*f)^(1/2)+3*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/
b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*x*a^2*b^2*c*f*(d*f)^(1/2)+3*C*ln((-2*a*d*f*
x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*
e)/(b*x+a))*x*a^2*b^2*d*e*(d*f)^(1/2)-4*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/
b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*x*a*b^3*c*e*(d*f)^(1/2)+2*A*ln((-2*a*d*f*x+
b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)
/(b*x+a))*x*a*b^3*d*f*(d*f)^(1/2)+2*A*b^4*(d*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(
f*x+e))^(1/2)-A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e
))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*x*b^4*c*f*(d*f)^(1/2)-A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*
b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*x*b^4*d*e*(d*f)^(1/2
)+2*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b
-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*x*b^4*c*e*(d*f)^(1/2)-2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)
+c*f+d*e)/(d*f)^(1/2))*x*b^4*c*e*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*A*ln((-2*a*d*f*x+b*c*f*x+b*d*
e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^
2*b^2*d*f*(d*f)^(1/2)-A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c
)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b^3*c*f*(d*f)^(1/2)-A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^
2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b^3*d*e*(d
*f)^(1/2)-B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(
1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b^2*c*f*(d*f)^(1/2)-B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*
c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b^2*d*e*(d*f)^(1/2
)+2*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b
-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b^3*c*e*(d*f)^(1/2)-2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)
+c*f+d*e)/(d*f)^(1/2))*a^3*b*d*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*
(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*b^2*c*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*C
*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*b^2*d*e*((a^2*d*f-a*b*c*f-a*b
*d*e+b^2*c*e)/b^2)^(1/2)-2*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a*b^3
*c*e*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+3*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*
d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^3*b*c*f*(d*f)^(1/2)+3*C*ln((
-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d
*e+2*b*c*e)/(b*x+a))*a^3*b*d*e*(d*f)^(1/2)-4*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*
c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a^2*b^2*c*e*(d*f)^(1/2)+2*C*a^2*b^2*(d
*f)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)-2*C*ln((-2*a*d*f*x+b*c*f*x+b*d
*e*x+2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a
^4*d*f*(d*f)^(1/2))/((d*x+c)*(f*x+e))^(1/2)/(a*d-b*c)/(d*f)^(1/2)/(a*f-b*e)/((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)
/b^2)^(1/2)/(b*x+a)/b^3
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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((C*x^2+B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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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((C*x^2+B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")
[Out]
Timed out
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Sympy [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((C*x**2+B*x+A)/(b*x+a)**2/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
[Out]
Exception raised: ValueError
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Giac [B] time = 11.1543, size = 1831, normalized size = 7.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((C*x^2+B*x+A)/(b*x+a)^2/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")
[Out]
(3*sqrt(d*f)*C*a^2*b*c*d^2*f - sqrt(d*f)*B*a*b^2*c*d^2*f - sqrt(d*f)*A*b^3*c*d^2*f - 2*sqrt(d*f)*C*a^3*d^3*f +
2*sqrt(d*f)*A*a*b^2*d^3*f - 4*sqrt(d*f)*C*a*b^2*c*d^2*e + 2*sqrt(d*f)*B*b^3*c*d^2*e + 3*sqrt(d*f)*C*a^2*b*d^3
*e - sqrt(d*f)*B*a*b^2*d^3*e - sqrt(d*f)*A*b^3*d^3*e)*arctan(-1/2*(b*c*d*f - 2*a*d^2*f + b*d^2*e - (sqrt(d*f)*
sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*b)/(sqrt(a*b*c*d*f^2 - a^2*d^2*f^2 - b^2*c*d*f*e + a*b*
d^2*f*e)*d))/((a*b^3*c*f*abs(d) - a^2*b^2*d*f*abs(d) - b^4*c*abs(d)*e + a*b^3*d*abs(d)*e)*sqrt(a*b*c*d*f^2 - a
^2*d^2*f^2 - b^2*c*d*f*e + a*b*d^2*f*e)*d) + 2*(sqrt(d*f)*C*a^2*b*c^2*d^3*f^2 - sqrt(d*f)*B*a*b^2*c^2*d^3*f^2
+ sqrt(d*f)*A*b^3*c^2*d^3*f^2 - 2*sqrt(d*f)*C*a^2*b*c*d^4*f*e + 2*sqrt(d*f)*B*a*b^2*c*d^4*f*e - 2*sqrt(d*f)*A*
b^3*c*d^4*f*e - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^2*b*c*d^2*f +
sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a*b^2*c*d^2*f - sqrt(d*f)*(sqrt(
d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*b^3*c*d^2*f + 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x +
c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^3*d^3*f - 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c
)*d*f - c*d*f + d^2*e))^2*B*a^2*b*d^3*f + 2*sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f +
d^2*e))^2*A*a*b^2*d^3*f + sqrt(d*f)*C*a^2*b*d^5*e^2 - sqrt(d*f)*B*a*b^2*d^5*e^2 + sqrt(d*f)*A*b^3*d^5*e^2 - sq
rt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*C*a^2*b*d^3*e + sqrt(d*f)*(sqrt(d*f)
*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*B*a*b^2*d^3*e - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - s
qrt((d*x + c)*d*f - c*d*f + d^2*e))^2*A*b^3*d^3*e)/((b*c^2*d^2*f^2 - 2*b*c*d^3*f*e - 2*(sqrt(d*f)*sqrt(d*x + c
) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*b*c*d*f + 4*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f +
d^2*e))^2*a*d^2*f + b*d^4*e^2 - 2*(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^2*b*d^2*e +
(sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f + d^2*e))^4*b)*(a*b^3*c*f*abs(d) - a^2*b^2*d*f*abs(d) -
b^4*c*abs(d)*e + a*b^3*d*abs(d)*e)) - sqrt(d*f)*C*log((sqrt(d*f)*sqrt(d*x + c) - sqrt((d*x + c)*d*f - c*d*f +
d^2*e))^2)/(b^2*f*abs(d))