3.2478 \(\int \frac{A+B x}{(d+e x)^2 (a+b x+c x^2)^{3/2}} \, dx\)
Optimal. Leaf size=334 \[ \frac{e \sqrt{a+b x+c x^2} \left (2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-2 a A e^2+3 a B d e+A c d^2\right )+b^2 e (B d-3 A e)\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{e \left (3 A e (2 c d-b e)-B \left (4 c d^2-e (2 a e+b d)\right )\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}} \]
[Out]
(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*x))/((b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(b^2*e*(B*d - 3*A*e) - 4*c*(A*c*d^2 + 3*a*B*d*e
- 2*a*A*e^2) + 2*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*
e^2)^2*(d + e*x)) + (e*(3*A*e*(2*c*d - b*e) - B*(4*c*d^2 - e*(b*d + 2*a*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])])/(2*(c*d^2 - b*d*e + a*e^2)^(5/2))
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Rubi [A] time = 0.424941, antiderivative size = 332, normalized size of antiderivative =
0.99, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules
used = {822, 806, 724, 206} \[ \frac{e \sqrt{a+b x+c x^2} \left (2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-2 a A e^2+3 a B d e+A c d^2\right )+b^2 e (B d-3 A e)\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{e \left (-B e (2 a e+b d)-3 A e (2 c d-b e)+4 B c d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)),x]
[Out]
(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*x))/((b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(b^2*e*(B*d - 3*A*e) - 4*c*(A*c*d^2 + 3*a*B*d*e
- 2*a*A*e^2) + 2*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*
e^2)^2*(d + e*x)) - (e*(4*B*c*d^2 - B*e*(b*d + 2*a*e) - 3*A*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])])/(2*(c*d^2 - b*d*e + a*e^2)^(5/2))
Rule 822
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])
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 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 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])
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}}-\frac{2 \int \frac{\frac{1}{2} e \left (b^2 (B d-3 A e)-8 a c (B d-A e)+2 b (A c d+a B e)\right )-c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}}+\frac{e \left (b^2 e (B d-3 A e)-4 c \left (A c d^2+3 a B d e-2 a A e^2\right )+2 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{\left (e \left (4 B c d^2-B e (b d+2 a e)-3 A e (2 c d-b e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx}{2 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}}+\frac{e \left (b^2 e (B d-3 A e)-4 c \left (A c d^2+3 a B d e-2 a A e^2\right )+2 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{\left (e \left (4 B c d^2-B e (b d+2 a e)-3 A e (2 c d-b e)\right )\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 )}{\left (c d^2-b d e+a e^2\right )^2}\\ &=\frac{2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt{a+b x+c x^2}}+\frac{e \left (b^2 e (B d-3 A e)-4 c \left (A c d^2+3 a B d e-2 a A e^2\right )+2 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{e \left (4 B c d^2-B e (b d+2 a e)-3 A e (2 c d-b e)\right ) \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 )}{2 \left (c d^2-b d e+a e^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 1.10722, size = 326, normalized size = 0.98 \[ \frac{e \sqrt{a+x (b+c x)} \left (2 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-2 a A e^2+3 a B d e+A c d^2\right )+b^2 e (B d-3 A e)\right )}{\left (b^2-4 a c\right ) (d+e x) \left (e (a e-b d)+c d^2\right )^2}+\frac{2 \left (-2 A c (a e+c d x)+a B (2 c (d-e x)-b e)+A b^2 e+A b c (e x-d)+b B c d x\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )}+\frac{e \left (-B e (2 a e+b d)+3 A e (b e-2 c d)+4 B c d^2\right ) \tanh ^{-1}\left (\frac{2 a e-b d+b e x-2 c d x}{2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}}\right )}{2 \left (e (a e-b d)+c d^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)),x]
[Out]
(e*(b^2*e*(B*d - 3*A*e) - 4*c*(A*c*d^2 + 3*a*B*d*e - 2*a*A*e^2) + 2*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a
+ x*(b + c*x)])/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + (2*(A*b^2*e + b*B*c*d*x - 2*A*c*(a*e
+ c*d*x) + A*b*c*(-d + e*x) + a*B*(-(b*e) + 2*c*(d - e*x))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))*(d + e*
x)*Sqrt[a + x*(b + c*x)]) + (e*(4*B*c*d^2 - B*e*(b*d + 2*a*e) + 3*A*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)])])/(2*(c*d^2 + e*(-(b*d) + a*e))^(5
/2))
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Maple [B] time = 0.013, size = 3090, normalized size = 9.3 \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)/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x)
[Out]
3*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^
2*c*A-12/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
*x*c^3*d^3*B+12*B/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2
)^(1/2)*x*c^2*d-6*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e
^2)^(1/2)*b^2*c*d*A-6/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*b*c^2*d^3*B+12/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*x*b*c^2*d^2*B-B/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/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)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*c*d^2*B-3/2*e^2/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
*b*A+6*B/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b
*c*d-8*c^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x
*A+3/(a*e^2-b*d*e+c*d^2)^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)
+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))
*c*d^2*B+3*e/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d*A-4*c
/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*A+1/e/(a*
e^2-b*d*e+c*d^2)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*B*d+3/2*e/(a*e^2-b*
d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*B*d+3/2*e^2/(a*e^2-b*d*e+c*d^
2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*A+3/2*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((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b*A-3*e/(a*e^2-b*
d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*c*B*d-12*e/(a
*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c^2*d*A-
1/(a*e^2-b*d*e+c*d^2)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*A-B/(a*e^2-b*d
*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+
c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))+B/(a*e^2-b*d*e+c
*d^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+6/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/((
x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*c*d^2*B+6/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2
)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c^2*d^2*A-2*B/(a*e^2-b*d*e+c*d^2)/(4*a*c
-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c-3/2*e/(a*e^2-b*d*e+c*d^2)^2/(4*a
*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*B*d-3/2*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((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2
)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b*B*d-3*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((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*c*d*A+12/(a*e^2-b*d*
e+c*d^2)^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^3*d^2*A
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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)/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 123.349, size = 8431, normalized size = 25.24 \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)/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
[Out]
[1/4*((4*(B*a*b^2*c - 4*B*a^2*c^2)*d^3*e - (B*a*b^3 - 24*A*a^2*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^2 - (2
*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*d*e^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d^2*e^2 - (B*b^3*c - 24
*A*a*c^3 - 2*(2*B*a*b - 3*A*b^2)*c^2)*d*e^3 + (4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*e^4)*x^3 +
(4*(B*b^2*c^2 - 4*B*a*c^3)*d^3*e + 3*(B*b^3*c + 8*A*a*c^3 - 2*(2*B*a*b + A*b^2)*c^2)*d^2*e^2 - (B*b^4 - 4*(2*
B*a^2 + 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*d*e^3 - (2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*e^
4)*x^2 + (4*(B*b^3*c - 4*B*a*b*c^2)*d^3*e - (B*b^4 + 8*(2*B*a^2 - 3*A*a*b)*c^2 - 2*(4*B*a*b^2 - 3*A*b^3)*c)*d^
2*e^2 - 3*(B*a*b^3 - A*b^4 - 8*A*a^2*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d*e^3 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(
2*B*a^3 - 3*A*a^2*b)*c)*e^4)*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*(2*
(2*B*a - A*b)*c^3*d^5 - 2*(4*A*a*c^3 + (4*B*a*b - 3*A*b^2)*c^2)*d^4*e - (4*(B*a^2 - 3*A*a*b)*c^2 - (7*B*a*b^2
- 6*A*b^3)*c)*d^3*e^2 - (3*B*a*b^3 - 2*A*b^4 + 4*A*a^2*c^2 - (4*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^3 + (3*B*a^2*b^2
- A*a*b^3 - 2*(4*B*a^3 - A*a^2*b)*c)*d*e^4 - (A*a^2*b^2 - 4*A*a^3*c)*e^5 + (2*(B*b*c^3 - 2*A*c^4)*d^4*e - (B*
b^2*c^2 + 4*(3*B*a - 2*A*b)*c^3)*d^3*e^2 - (B*b^3*c - 4*A*a*c^3 - (16*B*a*b - 7*A*b^2)*c^2)*d^2*e^3 - (4*(3*B*
a^2 + A*a*b)*c^2 + (B*a*b^2 - 3*A*b^3)*c)*d*e^4 + (8*A*a^2*c^2 + (2*B*a^2*b - 3*A*a*b^2)*c)*e^5)*x^2 + (2*(B*b
*c^3 - 2*A*c^4)*d^5 - 2*(B*b^2*c^2 + (2*B*a - 3*A*b)*c^3)*d^4*e + (B*b^3*c - 8*A*a*c^3)*d^3*e^2 - (B*b^4 + 8*(
B*a^2 - 2*A*a*b)*c^2 - (8*B*a*b^2 - 5*A*b^3)*c)*d^2*e^3 - (B*a*b^3 - 3*A*b^4 + 4*A*a^2*c^2 + 2*(B*a^2*b + 4*A*
a*b^2)*c)*d*e^4 + (2*B*a^2*b^2 - 3*A*a*b^3 - 2*(2*B*a^3 - 5*A*a^2*b)*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))/((a*b^2
*c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5*e^2 -
(a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3*(a^3*b^3 - 4*
a^4*b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^2 +
3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^4*c - 3*
a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e^7)*x^3 + ((b^
2*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^5*c - 11*a*b^3
*c^2 + 12*a^2*b*c^3)*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 - 4*a^2*b^3*c)
*d^2*e^5 - (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b^3*c^3 - 4*a*b*c
^4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^4*c - 15
*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^2*c - 4*a^4*c^
2)*d^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x), -1/2*((4*(B*a*b^2*c - 4*B*a^2*c^2)*d
^3*e - (B*a*b^3 - 24*A*a^2*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^2 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3
- 3*A*a^2*b)*c)*d*e^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d^2*e^2 - (B*b^3*c - 24*A*a*c^3 - 2*(2*B*a*b - 3*A*b^2)*c^2
)*d*e^3 + (4*(2*B*a^2 - 3*A*a*b)*c^2 - (2*B*a*b^2 - 3*A*b^3)*c)*e^4)*x^3 + (4*(B*b^2*c^2 - 4*B*a*c^3)*d^3*e +
3*(B*b^3*c + 8*A*a*c^3 - 2*(2*B*a*b + A*b^2)*c^2)*d^2*e^2 - (B*b^4 - 4*(2*B*a^2 + 3*A*a*b)*c^2 - (2*B*a*b^2 -
3*A*b^3)*c)*d*e^3 - (2*B*a*b^3 - 3*A*b^4 - 4*(2*B*a^2*b - 3*A*a*b^2)*c)*e^4)*x^2 + (4*(B*b^3*c - 4*B*a*b*c^2)*
d^3*e - (B*b^4 + 8*(2*B*a^2 - 3*A*a*b)*c^2 - 2*(4*B*a*b^2 - 3*A*b^3)*c)*d^2*e^2 - 3*(B*a*b^3 - A*b^4 - 8*A*a^2
*c^2 - 2*(2*B*a^2*b - 3*A*a*b^2)*c)*d*e^3 - (2*B*a^2*b^2 - 3*A*a*b^3 - 4*(2*B*a^3 - 3*A*a^2*b)*c)*e^4)*x)*sqrt
(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*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)/(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*(2*(2*B*a - A*b)*c^3*d^5 - 2*(4*A*a*c^3 + (4*B*a*b - 3*A*b^2)*c^2)*d^4*e - (4*(B*a^2 - 3*A*a*b)*c^2 - (7*
B*a*b^2 - 6*A*b^3)*c)*d^3*e^2 - (3*B*a*b^3 - 2*A*b^4 + 4*A*a^2*c^2 - (4*B*a^2*b - 3*A*a*b^2)*c)*d^2*e^3 + (3*B
*a^2*b^2 - A*a*b^3 - 2*(4*B*a^3 - A*a^2*b)*c)*d*e^4 - (A*a^2*b^2 - 4*A*a^3*c)*e^5 + (2*(B*b*c^3 - 2*A*c^4)*d^4
*e - (B*b^2*c^2 + 4*(3*B*a - 2*A*b)*c^3)*d^3*e^2 - (B*b^3*c - 4*A*a*c^3 - (16*B*a*b - 7*A*b^2)*c^2)*d^2*e^3 -
(4*(3*B*a^2 + A*a*b)*c^2 + (B*a*b^2 - 3*A*b^3)*c)*d*e^4 + (8*A*a^2*c^2 + (2*B*a^2*b - 3*A*a*b^2)*c)*e^5)*x^2 +
(2*(B*b*c^3 - 2*A*c^4)*d^5 - 2*(B*b^2*c^2 + (2*B*a - 3*A*b)*c^3)*d^4*e + (B*b^3*c - 8*A*a*c^3)*d^3*e^2 - (B*b
^4 + 8*(B*a^2 - 2*A*a*b)*c^2 - (8*B*a*b^2 - 5*A*b^3)*c)*d^2*e^3 - (B*a*b^3 - 3*A*b^4 + 4*A*a^2*c^2 + 2*(B*a^2*
b + 4*A*a*b^2)*c)*d*e^4 + (2*B*a^2*b^2 - 3*A*a*b^3 - 2*(2*B*a^3 - 5*A*a^2*b)*c)*e^5)*x)*sqrt(c*x^2 + b*x + a))
/((a*b^2*c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^
5*e^2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3*(a^3*
b^3 - 4*a^4*b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c^4)*d^
5*e^2 + 3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^
4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e^7)*x^
3 + ((b^2*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^5*c -
11*a*b^3*c^2 + 12*a^2*b*c^3)*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 - 4*a^
2*b^3*c)*d^2*e^5 - (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b^3*c^3 -
4*a*b*c^4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^
4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^2*c -
4*a^4*c^2)*d^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x)]
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Sympy [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)/(e*x+d)**2/(c*x**2+b*x+a)**(3/2),x)
[Out]
Timed out
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
[Out]
Exception raised: RuntimeError