3.23.31 \(\int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx\)
Optimal. Leaf size=444 \[ \frac {\sqrt {a+b x+c x^2} \left (B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )-A e \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {\left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )+b^3 e^2 (5 A e+B d)\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac {\sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (5 A e (2 c d-b e)-B \left (e (b d-6 a e)+4 c d^2\right )\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2} \]
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Rubi [A] time = 0.84, antiderivative size = 442, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used =
{834, 806, 724, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )-A e \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )\right )}{24 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {\left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )+b^3 e^2 (5 A e+B d)\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 )}{16 \left (a e^2-b d e+c d^2\right )^{7/2}}+\frac {\sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (B e (b d-6 a e)-5 A e (2 c d-b e)+4 B c d^2\right )}{12 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]
[Out]
((B*d - A*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) + ((4*B*c*d^2 + B*e*(b*d - 6*a*e)
- 5*A*e*(2*c*d - b*e))*Sqrt[a + b*x + c*x^2])/(12*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + ((B*(8*c^2*d^3 + 2*
c*d*e*(5*b*d - 26*a*e) - 3*b*e^2*(b*d - 6*a*e)) - A*e*(44*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(11*b*d + 4*a*e)))*Sqrt
[a + b*x + c*x^2])/(24*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - ((b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9
*A*c*d*e + 3*a*B*e^2) + 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^3) - 8*c*(A*c*d*(2*c*d^2 - 3*a*
e^2) + a*B*e*(4*c*d^2 - a*e^2)))*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])])/(16*(c*d^2 - b*d*e + a*e^2)^(7/2))
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 {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\int \frac {\frac {1}{2} (b B d-6 A c d+5 A b e-6 a B e)-2 c (B d-A e) x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx}{3 \left (c d^2-b d e+a e^2\right )}\\ &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\int \frac {\frac {1}{4} \left (3 b^2 e (B d+5 A e)+8 c \left (3 A c d^2+5 a B d e-2 a A e^2\right )-2 b \left (4 B c d^2+17 A c d e+9 a B e^2\right )\right )+\frac {1}{2} c \left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{6 \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{16 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\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 )}{8 \left (c d^2-b d e+a e^2\right )^3}\\ &=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\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 )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.93, size = 434, normalized size = 0.98 \begin {gather*} \frac {\frac {\sqrt {a+x (b+c x)} \left (A e \left (4 c e (4 a e+11 b d)-15 b^2 e^2-44 c^2 d^2\right )+B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )\right )}{4 (d+e x) \left (e (a e-b d)+c d^2\right )}+\frac {3 \left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (3 a e^2-2 c d^2\right )+a B e \left (a e^2-4 c d^2\right )\right )+b^3 e^2 (5 A e+B d)\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 )}{8 \left (e (a e-b d)+c d^2\right )^{3/2}}+\frac {2 \sqrt {a+x (b+c x)} (B d-A e) \left (e (a e-b d)+c d^2\right )}{(d+e x)^3}+\frac {\sqrt {a+x (b+c x)} \left (B e (b d-6 a e)+5 A e (b e-2 c d)+4 B c d^2\right )}{2 (d+e x)^2}}{6 \left (e (a e-b d)+c d^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]
[Out]
((2*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^3 + ((4*B*c*d^2 + B*e*(b*d - 6*a*e
) + 5*A*e*(-2*c*d + b*e))*Sqrt[a + x*(b + c*x)])/(2*(d + e*x)^2) + ((B*(8*c^2*d^3 + 2*c*d*e*(5*b*d - 26*a*e) -
3*b*e^2*(b*d - 6*a*e)) + A*e*(-44*c^2*d^2 - 15*b^2*e^2 + 4*c*e*(11*b*d + 4*a*e)))*Sqrt[a + x*(b + c*x)])/(4*(
c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (3*(b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9*A*c*d*e + 3*a*B*e^2)
+ 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^3) + 8*c*(a*B*e*(-4*c*d^2 + a*e^2) + A*c*d*(-2*c*d^2
+ 3*a*e^2)))*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))^(3/2)))/(6*(c*d^2 + e*(-(b*d) + a*e))^2)
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IntegrateAlgebraic [F] time = 180.30, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]
[Out]
$Aborted
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fricas [B] time = 85.51, size = 3684, normalized size = 8.30
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
[Out]
[-1/96*(3*(8*(B*b*c^2 - 2*A*c^3)*d^6 - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^5*e + (B*b^3 + 24*A*a*c^2 + 2*(10
*B*a*b - 9*A*b^2)*c)*d^4*e^2 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d^3*e^3 + (8*(B*b*c^2 - 2*A*c^3
)*d^3*e^3 - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^2*e^4 + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d*e^
5 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*e^6)*x^3 + 3*(8*(B*b*c^2 - 2*A*c^3)*d^4*e^2 - 4*(B*b^2*c +
2*(4*B*a - 3*A*b)*c^2)*d^3*e^3 + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^2*e^4 - (6*B*a*b^2 - 5*A*b
^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d*e^5)*x^2 + 3*(8*(B*b*c^2 - 2*A*c^3)*d^5*e - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2
)*d^4*e^2 + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^3*e^3 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*
a*b)*c)*d^2*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*(24*B*c^3*d^7 -
8*A*a^3*e^7 - 36*(B*b*c^2 + 2*A*c^3)*d^6*e + (15*B*b^2*c - 2*(8*B*a - 81*A*b)*c^2)*d^5*e^2 - (3*B*b^3 + 92*A*
a*c^2 - (44*B*a*b - 123*A*b^2)*c)*d^4*e^3 - (13*B*a*b^2 - 33*A*b^3 + 4*(11*B*a^2 - 34*A*a*b)*c)*d^3*e^4 + (20*
B*a^2*b - 59*A*a*b^2 - 28*A*a^2*c)*d^2*e^5 - 2*(2*B*a^3 - 17*A*a^2*b)*d*e^6 + (8*B*c^3*d^5*e^2 + 2*(B*b*c^2 -
22*A*c^3)*d^4*e^3 - (13*B*b^2*c + 44*(B*a - 2*A*b)*c^2)*d^3*e^4 + (3*B*b^3 - 28*A*a*c^2 + (80*B*a*b - 59*A*b^2
)*c)*d^2*e^5 - (21*B*a*b^2 - 15*A*b^3 + 4*(13*B*a^2 - 7*A*a*b)*c)*d*e^6 + (18*B*a^2*b - 15*A*a*b^2 + 16*A*a^2*
c)*e^7)*x^2 + 2*(12*B*c^3*d^6*e - (5*B*b*c^2 + 54*A*c^3)*d^5*e^2 - (11*B*b^2*c + (42*B*a - 113*A*b)*c^2)*d^4*e
^3 + (4*B*b^3 - 48*A*a*c^2 + (86*B*a*b - 79*A*b^2)*c)*d^3*e^4 - (29*B*a*b^2 - 20*A*b^3 + 2*(30*B*a^2 - 29*A*a*
b)*c)*d^2*e^5 + (31*B*a^2*b - 25*A*a*b^2 + 6*A*a^2*c)*d*e^6 - (6*B*a^3 - 5*A*a^2*b)*e^7)*x)*sqrt(c*x^2 + b*x +
a))/(c^4*d^11 - 4*b*c^3*d^10*e - 4*a^3*b*d^4*e^7 + a^4*d^3*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9*e^2 - 4*(b^3*c +
3*a*b*c^2)*d^8*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^5 + 2*(3*a^2*b^2 +
2*a^3*c)*d^5*e^6 + (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 - 4*a^3*b*d*e^10 + a^4*e^11 + 2*(3*b^2*c^2 + 2*a*c^3)*d^6*e^
5 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^7 - 4*(a*b^3 + 3*a^2*b*c)*d^3*e^8 + 2
*(3*a^2*b^2 + 2*a^3*c)*d^2*e^9)*x^3 + 3*(c^4*d^9*e^2 - 4*b*c^3*d^8*e^3 - 4*a^3*b*d^2*e^9 + a^4*d*e^10 + 2*(3*b
^2*c^2 + 2*a*c^3)*d^7*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^6 - 4*(a*b^3
+ 3*a^2*b*c)*d^4*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^8)*x^2 + 3*(c^4*d^10*e - 4*b*c^3*d^9*e^2 - 4*a^3*b*d^3*e^
8 + a^4*d^2*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*
c^2)*d^6*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^7)*x), -1/48*(3*(8*(B*b*c^2 - 2*A
*c^3)*d^6 - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^5*e + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^4*e^
2 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d^3*e^3 + (8*(B*b*c^2 - 2*A*c^3)*d^3*e^3 - 4*(B*b^2*c + 2*
(4*B*a - 3*A*b)*c^2)*d^2*e^4 + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d*e^5 - (6*B*a*b^2 - 5*A*b^3 -
4*(2*B*a^2 - 3*A*a*b)*c)*e^6)*x^3 + 3*(8*(B*b*c^2 - 2*A*c^3)*d^4*e^2 - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^3
*e^3 + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^2*e^4 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*
c)*d*e^5)*x^2 + 3*(8*(B*b*c^2 - 2*A*c^3)*d^5*e - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^4*e^2 + (B*b^3 + 24*A*a
*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^3*e^3 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d^2*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*(24*B*c^3*d^7 - 8*A*a^3*e^7 - 36*(B*b*c^2 + 2*A*c^3)*d^6*e + (15*B*b^2*c - 2*(8*B*a - 81*A*b)*c^2)*d^5*e^2 -
(3*B*b^3 + 92*A*a*c^2 - (44*B*a*b - 123*A*b^2)*c)*d^4*e^3 - (13*B*a*b^2 - 33*A*b^3 + 4*(11*B*a^2 - 34*A*a*b)*
c)*d^3*e^4 + (20*B*a^2*b - 59*A*a*b^2 - 28*A*a^2*c)*d^2*e^5 - 2*(2*B*a^3 - 17*A*a^2*b)*d*e^6 + (8*B*c^3*d^5*e^
2 + 2*(B*b*c^2 - 22*A*c^3)*d^4*e^3 - (13*B*b^2*c + 44*(B*a - 2*A*b)*c^2)*d^3*e^4 + (3*B*b^3 - 28*A*a*c^2 + (80
*B*a*b - 59*A*b^2)*c)*d^2*e^5 - (21*B*a*b^2 - 15*A*b^3 + 4*(13*B*a^2 - 7*A*a*b)*c)*d*e^6 + (18*B*a^2*b - 15*A*
a*b^2 + 16*A*a^2*c)*e^7)*x^2 + 2*(12*B*c^3*d^6*e - (5*B*b*c^2 + 54*A*c^3)*d^5*e^2 - (11*B*b^2*c + (42*B*a - 11
3*A*b)*c^2)*d^4*e^3 + (4*B*b^3 - 48*A*a*c^2 + (86*B*a*b - 79*A*b^2)*c)*d^3*e^4 - (29*B*a*b^2 - 20*A*b^3 + 2*(3
0*B*a^2 - 29*A*a*b)*c)*d^2*e^5 + (31*B*a^2*b - 25*A*a*b^2 + 6*A*a^2*c)*d*e^6 - (6*B*a^3 - 5*A*a^2*b)*e^7)*x)*s
qrt(c*x^2 + b*x + a))/(c^4*d^11 - 4*b*c^3*d^10*e - 4*a^3*b*d^4*e^7 + a^4*d^3*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^9
*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^8*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^7*e^4 - 4*(a*b^3 + 3*a^2*b*c)*d^6*e^5
+ 2*(3*a^2*b^2 + 2*a^3*c)*d^5*e^6 + (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 - 4*a^3*b*d*e^10 + a^4*e^11 + 2*(3*b^2*c^2
+ 2*a*c^3)*d^6*e^5 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^7 - 4*(a*b^3 + 3*a^2
*b*c)*d^3*e^8 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^9)*x^3 + 3*(c^4*d^9*e^2 - 4*b*c^3*d^8*e^3 - 4*a^3*b*d^2*e^9 + a^
4*d*e^10 + 2*(3*b^2*c^2 + 2*a*c^3)*d^7*e^4 - 4*(b^3*c + 3*a*b*c^2)*d^6*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^
5*e^6 - 4*(a*b^3 + 3*a^2*b*c)*d^4*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*d^3*e^8)*x^2 + 3*(c^4*d^10*e - 4*b*c^3*d^9*e^2
- 4*a^3*b*d^3*e^8 + a^4*d^2*e^9 + 2*(3*b^2*c^2 + 2*a*c^3)*d^8*e^3 - 4*(b^3*c + 3*a*b*c^2)*d^7*e^4 + (b^4 + 12
*a*b^2*c + 6*a^2*c^2)*d^6*e^5 - 4*(a*b^3 + 3*a^2*b*c)*d^5*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d^4*e^7)*x)]
________________________________________________________________________________________
giac [B] time = 0.74, size = 4249, normalized size = 9.57
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
[Out]
-1/8*(8*B*b*c^2*d^3 - 16*A*c^3*d^3 - 4*B*b^2*c*d^2*e - 32*B*a*c^2*d^2*e + 24*A*b*c^2*d^2*e + B*b^3*d*e^2 + 20*
B*a*b*c*d*e^2 - 18*A*b^2*c*d*e^2 + 24*A*a*c^2*d*e^2 - 6*B*a*b^2*e^3 + 5*A*b^3*e^3 + 8*B*a^2*c*e^3 - 12*A*a*b*c
*e^3)*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^3*d^6 - 3*
b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^
2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/24*(64*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*B*c^4*d^6 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b*c^3*d^5*e - 352*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^3*A*c^4*d^5*e + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b*c^(7/2)*d^6 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*B*b*c^(5/2)*d^4*e^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*c^(7/2)*d^4*e^2 + 24*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*B*b^2*c^(5/2)*d^5*e - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a*c^(7/2)*d^5*e - 528
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b*c^(7/2)*d^5*e + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^2*c^3*d^
6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b*c^2*d^3*e^3 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*c^3*
d^3*e^3 + 168*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^2*c^2*d^4*e^2 - 512*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^3*B*a*c^3*d^4*e^2 + 400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b*c^3*d^4*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*B*b^3*c^2*d^5*e - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*b*c^3*d^5*e - 264*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))*A*b^2*c^3*d^5*e + 8*B*b^3*c^(5/2)*d^6 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*b^2*c^(3/2
)*d^3*e^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*c^(5/2)*d^3*e^3 + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^4*A*b*c^(5/2)*d^3*e^3 + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b^3*c^(3/2)*d^4*e^2 - 696*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*B*a*b*c^(5/2)*d^4*e^2 + 756*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^2*c^(5/2)*d^4
*e^2 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*c^(7/2)*d^4*e^2 + 10*B*b^4*c^(3/2)*d^5*e - 48*B*a*b^2*c^(
5/2)*d^5*e - 44*A*b^3*c^(5/2)*d^5*e - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b^2*c*d^2*e^4 - 96*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*B*a*c^2*d^2*e^4 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b*c^2*d^2*e^4 - 74*(s
qrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^3*c*d^3*e^3 - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b*c^2*d^3*
e^3 - 204*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^2*c^2*d^3*e^3 + 656*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*
A*a*c^3*d^3*e^3 - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^4*c*d^4*e^2 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*B*a*b^2*c^2*d^4*e^2 + 336*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b^3*c^2*d^4*e^2 + 96*(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))*B*a^2*c^3*d^4*e^2 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b*c^3*d^4*e^2 + 15*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*B*b^3*sqrt(c)*d^2*e^4 + 300*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*b*c^(3/2)*d^2*
e^4 - 270*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*b^2*c^(3/2)*d^2*e^4 + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)^4*A*a*c^(5/2)*d^2*e^4 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b^4*sqrt(c)*d^3*e^3 + 252*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^2*B*a*b^2*c^(3/2)*d^3*e^3 - 498*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^3*c^(3/2)*d^3*e
^3 + 816*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*c^(5/2)*d^3*e^3 - 648*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^2*A*a*b*c^(5/2)*d^3*e^3 - 3*B*b^5*sqrt(c)*d^4*e^2 - 70*B*a*b^3*c^(3/2)*d^4*e^2 + 44*A*b^4*c^(3/2)*d^4*e^2 + 4
8*B*a^2*b*c^(5/2)*d^4*e^2 + 204*A*a*b^2*c^(5/2)*d^4*e^2 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b^3*d*e^5
+ 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b*c*d*e^5 - 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^2*c*d*
e^5 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*c^2*d*e^5 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b^4*d
^2*e^4 + 252*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b^2*c*d^2*e^4 - 34*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*A*b^3*c*d^2*e^4 + 624*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*c^2*d^2*e^4 - 264*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*A*a*b*c^2*d^2*e^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b^5*d^3*e^3 + 186*(sqrt(c)*x - sqrt(
c*x^2 + b*x + a))*B*a*b^3*c*d^3*e^3 - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*b^4*c*d^3*e^3 + 864*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*B*a^2*b*c^2*d^3*e^3 - 900*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^2*c^2*d^3*e^3 -
480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*c^3*d^3*e^3 - 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a*b^2*s
qrt(c)*d*e^5 + 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*b^3*sqrt(c)*d*e^5 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^4*B*a^2*c^(3/2)*d*e^5 - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*A*a*b*c^(3/2)*d*e^5 - 24*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*B*a*b^3*sqrt(c)*d^2*e^4 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^4*sqrt(c)*d
^2*e^4 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^2*b*c^(3/2)*d^2*e^4 + 432*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^2*A*a*b^2*c^(3/2)*d^2*e^4 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a^2*c^(5/2)*d^2*e^4 + 24*B*a*b^
4*sqrt(c)*d^3*e^3 - 15*A*b^5*sqrt(c)*d^3*e^3 + 240*B*a^2*b^2*c^(3/2)*d^3*e^3 - 206*A*a*b^3*c^(3/2)*d^3*e^3 - 1
6*B*a^3*c^(5/2)*d^3*e^3 - 240*A*a^2*b*c^(5/2)*d^3*e^3 - 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b^2*e^6 +
15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^3*e^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2*c*e^6 - 3
6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b*c*e^6 - 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b^3*d*e^5 +
40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b^4*d*e^5 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*b*c*d*
e^5 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b^2*c*d*e^5 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a
^2*c^2*d*e^5 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*b^4*d^2*e^4 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*A*b^5*d^2*e^4 - 432*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b^2*c*d^2*e^4 + 450*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))*A*a*b^3*c*d^2*e^4 - 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3*c^2*d^2*e^4 + 432*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*A*a^2*b*c^2*d^2*e^4 - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*a*b^3*sqrt(c)*d*e^5 - 192
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^3*c^(3/2)*d*e^5 - 87*B*a^2*b^3*sqrt(c)*d^2*e^4 + 78*A*a*b^4*sqrt(c)
*d^2*e^4 - 284*B*a^3*b*c^(3/2)*d^2*e^4 + 222*A*a^2*b^2*c^(3/2)*d^2*e^4 + 88*A*a^3*c^(5/2)*d^2*e^4 + 48*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^3*B*a^2*b^2*e^6 - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a*b^3*e^6 + 96*(sqrt
(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b*c*e^6 + 57*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^2*b^3*d*e^5 - 66*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a*b^4*d*e^5 + 276*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3*b*c*d*e^5 - 3
06*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b^2*c*d*e^5 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*c^2*d
*e^5 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a^3*b*sqrt(c)*e^6 + 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2
*A*a^3*c^(3/2)*e^6 + 114*B*a^3*b^2*sqrt(c)*d*e^5 - 111*A*a^2*b^3*sqrt(c)*d*e^5 + 104*B*a^4*c^(3/2)*d*e^5 - 28*
A*a^3*b*c^(3/2)*d*e^5 - 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3*b^2*e^6 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))*A*a^2*b^3*e^6 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^4*c*e^6 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))*A*a^3*b*c*e^6 - 48*B*a^4*b*sqrt(c)*e^6 + 48*A*a^3*b^2*sqrt(c)*e^6 - 32*A*a^4*c^(3/2)*e^6)/((c^3*d^6*e -
3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*
c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7)*((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)^3)
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maple [B] time = 0.07, size = 3898, normalized size = 8.78 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x)
[Out]
-1/3/e^2/(a*e^2-b*d*e+c*d^2)/(x+d/e)^3*((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)*A-1/2
*B/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)-3/4/(a*
e^2-b*d*e+c*d^2)^2*c/((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*A-5
/8*e^2/(a*e^2-b*d*e+c*d^2)^3/(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*A+5
/16*e^2/(a*e^2-b*d*e+c*d^2)^3/((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*A+1/2*B/e^2*c/(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))+1/3/e^3/(a*e^2-b*d*e+c*d^2)/(x+d/e)^3*((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*d-5/2/(a*e^2-b*d*e+c*d^2)^3/(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*A+9/4*B/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*d-15/4/e/(a*e^2-b*d*e+c*d^2)^3/((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^3*B+5/2*e/(a*e^2-b*d*e+c*d^2)^3/(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*A-15/8*e/(a*e^2-b*d*e+c*d^2)^3/((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*A-5/6/e/(a*e^2-b*d*e+c*d^2)^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*A-3*B/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^2*d^2+3/4*B/(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+5/12/(a*e^2-b*d*e+c*d^2)^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*A-13/6*B/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*d+5/6/e^2/(a*e^2-b*d*e+c*d^2)^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^2*B+5/8*e/(a*e^2-b*d*e+c*d^2)^3/(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*B*d+5/2/e/(a*e^2-b*d*e+c*d^2)^3/(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^3*B-5/16*e/(a*e^2-b*d*e+c*d^2)^3/((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*B*d+2/3/(a*e^2-b*d*e+c*d^2)^2*c/(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)*A-3/8*B/(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+5/2/e^2/(a*e^2-b*d*e+c*d^2)^3/((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^4*B+3/2/e/(a*e^2-b*d*e+c*d^2)
^2*c^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*A-5/12/e/(a*e^2-b
*d*e+c*d^2)^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*B*d+15/4/(a*e^2-b*
d*e+c*d^2)^3/((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*A-5
/2/(a*e^2-b*d*e+c*d^2)^3/(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^2*B+1
5/8/(a*e^2-b*d*e+c*d^2)^3/((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^2*B-5/2/e/(a*e^2-b*d*e+c*d^2)^3/((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*A
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)^4/(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?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (d+e\,x\right )}^4\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((d + e*x)^4*(a + b*x + c*x^2)^(1/2)),x)
[Out]
int((A + B*x)/((d + e*x)^4*(a + b*x + c*x^2)^(1/2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (d + e x\right )^{4} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)
[Out]
Integral((A + B*x)/((d + e*x)**4*sqrt(a + b*x + c*x**2)), x)
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