3.11.53 \(\int \frac {A+B x}{(d+e x)^4 \sqrt {b x+c x^2}} \, dx\)
Optimal. Leaf size=331 \[ \frac {\sqrt {b x+c x^2} \left (B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )-A e \left (15 b^2 e^2-44 b c d e+44 c^2 d^2\right )\right )}{24 d^3 (d+e x) (c d-b e)^3}+\frac {\left (b^3 \left (-e^2\right ) (5 A e+B d)+2 b^2 c d e (9 A e+2 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{16 d^{7/2} (c d-b e)^{7/2}}-\frac {\sqrt {b x+c x^2} (5 A e (2 c d-b e)-B d (b e+4 c d))}{12 d^2 (d+e x)^2 (c d-b e)^2}+\frac {\sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)} \]
________________________________________________________________________________________
Rubi [A] time = 0.50, antiderivative size = 331, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used =
{834, 806, 724, 206} \begin {gather*} \frac {\sqrt {b x+c x^2} \left (B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )-A e \left (15 b^2 e^2-44 b c d e+44 c^2 d^2\right )\right )}{24 d^3 (d+e x) (c d-b e)^3}+\frac {\left (2 b^2 c d e (9 A e+2 B d)+b^3 \left (-e^2\right ) (5 A e+B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\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 )}{16 d^{7/2} (c d-b e)^{7/2}}-\frac {\sqrt {b x+c x^2} (5 A e (2 c d-b e)-B d (b e+4 c d))}{12 d^2 (d+e x)^2 (c d-b e)^2}+\frac {\sqrt {b x+c x^2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/((d + e*x)^4*Sqrt[b*x + c*x^2]),x]
[Out]
((B*d - A*e)*Sqrt[b*x + c*x^2])/(3*d*(c*d - b*e)*(d + e*x)^3) - ((5*A*e*(2*c*d - b*e) - B*d*(4*c*d + b*e))*Sqr
t[b*x + c*x^2])/(12*d^2*(c*d - b*e)^2*(d + e*x)^2) + ((B*d*(8*c^2*d^2 + 10*b*c*d*e - 3*b^2*e^2) - A*e*(44*c^2*
d^2 - 44*b*c*d*e + 15*b^2*e^2))*Sqrt[b*x + c*x^2])/(24*d^3*(c*d - b*e)^3*(d + e*x)) + ((16*A*c^3*d^3 - 8*b*c^2
*d^2*(B*d + 3*A*e) - b^3*e^2*(B*d + 5*A*e) + 2*b^2*c*d*e*(2*B*d + 9*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*S
qrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(16*d^(7/2)*(c*d - b*e)^(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 {b x+c x^2}} \, dx &=\frac {(B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^3}-\frac {\int \frac {\frac {1}{2} (b B d-6 A c d+5 A b e)-2 c (B d-A e) x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx}{3 d (c d-b e)}\\ &=\frac {(B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^3}-\frac {(5 A e (2 c d-b e)-B d (4 c d+b e)) \sqrt {b x+c x^2}}{12 d^2 (c d-b e)^2 (d+e x)^2}+\frac {\int \frac {\frac {1}{4} \left (24 A c^2 d^2+3 b^2 e (B d+5 A e)-2 b c d (4 B d+17 A e)\right )-\frac {1}{2} c (5 A e (2 c d-b e)-B d (4 c d+b e)) x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx}{6 d^2 (c d-b e)^2}\\ &=\frac {(B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^3}-\frac {(5 A e (2 c d-b e)-B d (4 c d+b e)) \sqrt {b x+c x^2}}{12 d^2 (c d-b e)^2 (d+e x)^2}+\frac {\left (B d \left (8 c^2 d^2+10 b c d e-3 b^2 e^2\right )-A e \left (44 c^2 d^2-44 b c d e+15 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{24 d^3 (c d-b e)^3 (d+e x)}+\frac {\left (16 A c^3 d^3-8 b c^2 d^2 (B d+3 A e)-b^3 e^2 (B d+5 A e)+2 b^2 c d e (2 B d+9 A e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{16 d^3 (c d-b e)^3}\\ &=\frac {(B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^3}-\frac {(5 A e (2 c d-b e)-B d (4 c d+b e)) \sqrt {b x+c x^2}}{12 d^2 (c d-b e)^2 (d+e x)^2}+\frac {\left (B d \left (8 c^2 d^2+10 b c d e-3 b^2 e^2\right )-A e \left (44 c^2 d^2-44 b c d e+15 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{24 d^3 (c d-b e)^3 (d+e x)}-\frac {\left (16 A c^3 d^3-8 b c^2 d^2 (B d+3 A e)-b^3 e^2 (B d+5 A e)+2 b^2 c d e (2 B d+9 A e)\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 )}{8 d^3 (c d-b e)^3}\\ &=\frac {(B d-A e) \sqrt {b x+c x^2}}{3 d (c d-b e) (d+e x)^3}-\frac {(5 A e (2 c d-b e)-B d (4 c d+b e)) \sqrt {b x+c x^2}}{12 d^2 (c d-b e)^2 (d+e x)^2}+\frac {\left (B d \left (8 c^2 d^2+10 b c d e-3 b^2 e^2\right )-A e \left (44 c^2 d^2-44 b c d e+15 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{24 d^3 (c d-b e)^3 (d+e x)}+\frac {\left (16 A c^3 d^3-8 b c^2 d^2 (B d+3 A e)-b^3 e^2 (B d+5 A e)+2 b^2 c d e (2 B d+9 A e)\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 )}{16 d^{7/2} (c d-b e)^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.72, size = 345, normalized size = 1.04 \begin {gather*} -\frac {\sqrt {x} \left (\frac {\sqrt {b+c x} (d+e x) \left (2 d^{3/2} \sqrt {x} \sqrt {b+c x} (c d-b e)^{3/2} (5 A e (b e-2 c d)+B d (b e+4 c d))-(d+e x) \left (-\sqrt {d} \sqrt {x} \sqrt {b+c x} \sqrt {c d-b e} \left (A e \left (-15 b^2 e^2+44 b c d e-44 c^2 d^2\right )+B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )\right )-3 (d+e x) \left (b^3 \left (-e^2\right ) (5 A e+B d)+2 b^2 c d e (9 A e+2 B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )}{d^{5/2} (c d-b e)^{5/2}}+8 \sqrt {x} (b+c x) (B d-A e)\right )}{24 d \sqrt {x (b+c x)} (d+e x)^3 (b e-c d)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/((d + e*x)^4*Sqrt[b*x + c*x^2]),x]
[Out]
-1/24*(Sqrt[x]*(8*(B*d - A*e)*Sqrt[x]*(b + c*x) + (Sqrt[b + c*x]*(d + e*x)*(2*d^(3/2)*(c*d - b*e)^(3/2)*(5*A*e
*(-2*c*d + b*e) + B*d*(4*c*d + b*e))*Sqrt[x]*Sqrt[b + c*x] - (d + e*x)*(-(Sqrt[d]*Sqrt[c*d - b*e]*(A*e*(-44*c^
2*d^2 + 44*b*c*d*e - 15*b^2*e^2) + B*d*(8*c^2*d^2 + 10*b*c*d*e - 3*b^2*e^2))*Sqrt[x]*Sqrt[b + c*x]) - 3*(16*A*
c^3*d^3 - 8*b*c^2*d^2*(B*d + 3*A*e) - b^3*e^2*(B*d + 5*A*e) + 2*b^2*c*d*e*(2*B*d + 9*A*e))*(d + e*x)*ArcTanh[(
Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])))/(d^(5/2)*(c*d - b*e)^(5/2))))/(d*(-(c*d) + b*e)*Sqrt[x*(b
+ c*x)]*(d + e*x)^3)
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 166.05, size = 12551, normalized size = 37.92 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x)/((d + e*x)^4*Sqrt[b*x + c*x^2]),x]
[Out]
Result too large to show
________________________________________________________________________________________
fricas [B] time = 0.49, size = 1781, normalized size = 5.38
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)^(1/2),x, algorithm="fricas")
[Out]
[-1/48*(3*(5*A*b^3*d^3*e^3 + 8*(B*b*c^2 - 2*A*c^3)*d^6 - 4*(B*b^2*c - 6*A*b*c^2)*d^5*e + (B*b^3 - 18*A*b^2*c)*
d^4*e^2 + (5*A*b^3*e^6 + 8*(B*b*c^2 - 2*A*c^3)*d^3*e^3 - 4*(B*b^2*c - 6*A*b*c^2)*d^2*e^4 + (B*b^3 - 18*A*b^2*c
)*d*e^5)*x^3 + 3*(5*A*b^3*d*e^5 + 8*(B*b*c^2 - 2*A*c^3)*d^4*e^2 - 4*(B*b^2*c - 6*A*b*c^2)*d^3*e^3 + (B*b^3 - 1
8*A*b^2*c)*d^2*e^4)*x^2 + 3*(5*A*b^3*d^2*e^4 + 8*(B*b*c^2 - 2*A*c^3)*d^5*e - 4*(B*b^2*c - 6*A*b*c^2)*d^4*e^2 +
(B*b^3 - 18*A*b^2*c)*d^3*e^3)*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*(24*B*c^3*d^7 + 33*A*b^3*d^3*e^4 - 36*(B*b*c^2 + 2*A*c^3)*d^6*e + 3*(5*B*b^2*c +
54*A*b*c^2)*d^5*e^2 - 3*(B*b^3 + 41*A*b^2*c)*d^4*e^3 + (8*B*c^3*d^5*e^2 + 15*A*b^3*d*e^6 + 2*(B*b*c^2 - 22*A*c
^3)*d^4*e^3 - (13*B*b^2*c - 88*A*b*c^2)*d^3*e^4 + (3*B*b^3 - 59*A*b^2*c)*d^2*e^5)*x^2 + 2*(12*B*c^3*d^6*e + 20
*A*b^3*d^2*e^5 - (5*B*b*c^2 + 54*A*c^3)*d^5*e^2 - (11*B*b^2*c - 113*A*b*c^2)*d^4*e^3 + (4*B*b^3 - 79*A*b^2*c)*
d^3*e^4)*x)*sqrt(c*x^2 + b*x))/(c^4*d^11 - 4*b*c^3*d^10*e + 6*b^2*c^2*d^9*e^2 - 4*b^3*c*d^8*e^3 + b^4*d^7*e^4
+ (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 + 6*b^2*c^2*d^6*e^5 - 4*b^3*c*d^5*e^6 + b^4*d^4*e^7)*x^3 + 3*(c^4*d^9*e^2 - 4
*b*c^3*d^8*e^3 + 6*b^2*c^2*d^7*e^4 - 4*b^3*c*d^6*e^5 + b^4*d^5*e^6)*x^2 + 3*(c^4*d^10*e - 4*b*c^3*d^9*e^2 + 6*
b^2*c^2*d^8*e^3 - 4*b^3*c*d^7*e^4 + b^4*d^6*e^5)*x), -1/24*(3*(5*A*b^3*d^3*e^3 + 8*(B*b*c^2 - 2*A*c^3)*d^6 - 4
*(B*b^2*c - 6*A*b*c^2)*d^5*e + (B*b^3 - 18*A*b^2*c)*d^4*e^2 + (5*A*b^3*e^6 + 8*(B*b*c^2 - 2*A*c^3)*d^3*e^3 - 4
*(B*b^2*c - 6*A*b*c^2)*d^2*e^4 + (B*b^3 - 18*A*b^2*c)*d*e^5)*x^3 + 3*(5*A*b^3*d*e^5 + 8*(B*b*c^2 - 2*A*c^3)*d^
4*e^2 - 4*(B*b^2*c - 6*A*b*c^2)*d^3*e^3 + (B*b^3 - 18*A*b^2*c)*d^2*e^4)*x^2 + 3*(5*A*b^3*d^2*e^4 + 8*(B*b*c^2
- 2*A*c^3)*d^5*e - 4*(B*b^2*c - 6*A*b*c^2)*d^4*e^2 + (B*b^3 - 18*A*b^2*c)*d^3*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arc
tan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (24*B*c^3*d^7 + 33*A*b^3*d^3*e^4 - 36*(B*b*c^2
+ 2*A*c^3)*d^6*e + 3*(5*B*b^2*c + 54*A*b*c^2)*d^5*e^2 - 3*(B*b^3 + 41*A*b^2*c)*d^4*e^3 + (8*B*c^3*d^5*e^2 + 15
*A*b^3*d*e^6 + 2*(B*b*c^2 - 22*A*c^3)*d^4*e^3 - (13*B*b^2*c - 88*A*b*c^2)*d^3*e^4 + (3*B*b^3 - 59*A*b^2*c)*d^2
*e^5)*x^2 + 2*(12*B*c^3*d^6*e + 20*A*b^3*d^2*e^5 - (5*B*b*c^2 + 54*A*c^3)*d^5*e^2 - (11*B*b^2*c - 113*A*b*c^2)
*d^4*e^3 + (4*B*b^3 - 79*A*b^2*c)*d^3*e^4)*x)*sqrt(c*x^2 + b*x))/(c^4*d^11 - 4*b*c^3*d^10*e + 6*b^2*c^2*d^9*e^
2 - 4*b^3*c*d^8*e^3 + b^4*d^7*e^4 + (c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 + 6*b^2*c^2*d^6*e^5 - 4*b^3*c*d^5*e^6 + b^4
*d^4*e^7)*x^3 + 3*(c^4*d^9*e^2 - 4*b*c^3*d^8*e^3 + 6*b^2*c^2*d^7*e^4 - 4*b^3*c*d^6*e^5 + b^4*d^5*e^6)*x^2 + 3*
(c^4*d^10*e - 4*b*c^3*d^9*e^2 + 6*b^2*c^2*d^8*e^3 - 4*b^3*c*d^7*e^4 + b^4*d^6*e^5)*x)]
________________________________________________________________________________________
giac [B] time = 0.39, size = 1667, normalized size = 5.04
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)^(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 + 24*A*b*c^2*d^2*e + B*b^3*d*e^2 - 18*A*b^2*c*d*e^2 + 5*A
*b^3*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^3*d^6 - 3*b*c^2*d^
5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*sqrt(-c*d^2 + b*d*e)) + 1/24*(64*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^4*
d^6 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c^3*d^5*e - 352*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^4*d^5*e +
96*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^(7/2)*d^6 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b*c^(5/2)*d^4*
e^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*c^(7/2)*d^4*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^2*c^(
5/2)*d^5*e - 528*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^(7/2)*d^5*e + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^
2*c^3*d^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^2*d^3*e^3 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^3*
d^3*e^3 + 168*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c^2*d^4*e^2 + 400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*
c^3*d^4*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*c^2*d^5*e - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x))*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))^4*B*b^2*c^(3/2)*d^3*e^3 + 360*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^4*A*b*c^(5/2)*d^3*e^3 + 54*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*c^(3/2)*d^4*e^2 + 756*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c^(5/2)*d^4*e^2 + 10*B*b^4*c^(3/2)*d^5*e - 44*A*b^3*c^(5/2)*d^5*e - 12*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c*d^2*e^4 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b*c^2*d^2*e^4 - 74*
(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*c*d^3*e^3 - 204*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^2*d^3*e^3 -
6*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*c*d^4*e^2 + 336*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*c^2*d^4*e^2 + 15
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*sqrt(c)*d^2*e^4 - 270*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2*c^(3/2)
*d^2*e^4 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4*sqrt(c)*d^3*e^3 - 498*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*
A*b^3*c^(3/2)*d^3*e^3 - 3*B*b^5*sqrt(c)*d^4*e^2 + 44*A*b^4*c^(3/2)*d^4*e^2 + 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^5*B*b^3*d*e^5 - 54*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c*d*e^5 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^
4*d^2*e^4 - 34*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c*d^2*e^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^5*d^3
*e^3 - 180*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*c*d^3*e^3 + 75*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*sqrt(c
)*d*e^5 + 120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^4*sqrt(c)*d^2*e^4 - 15*A*b^5*sqrt(c)*d^3*e^3 + 15*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^5*A*b^3*e^6 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*d*e^5 + 33*(sqrt(c)*x - sqrt(
c*x^2 + b*x))*A*b^5*d^2*e^4)/((c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3*d^3*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)^3)
________________________________________________________________________________________
maple [B] time = 0.13, size = 3242, normalized size = 9.79 \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)^(1/2),x)
[Out]
9/4*B/e/(b*e-c*d)^2/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^
2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*c+15/8*e/(b*e-c*d)^3/d^2/(-(b*e
-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2*c*A-5/2*e/(b*e-c*d)^3/d^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*
d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*c*A+5/2/(b*e-c*d)^3/d/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x
+d/e)/e)^(1/2)*c^2*A+1/3/e^2/(b*e-c*d)/d/(x+d/e)^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*A
+5/2/e/(b*e-c*d)^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^
(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*c^3*A+5/6/e^2/(b*e-c*d)^2/(x+d/e)^2*
((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c*B-5/2/e/(b*e-c*d)^3/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)
*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c^2*B+2/3*c/(b*e-c*d)^2/d^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*
d)*(x+d/e)/e)^(1/2)*A+5/12/(b*e-c*d)^2/d^2/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)
*b*A+1/2*B/e^2/(b*e-c*d)/d/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)+3/4*B/(b*e-c*d)
^2/d^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b-3/8*B/(b*e-c*d)^2/d^2/(-(b*e-c*d)*d
/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e
^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2-3*B/e^2/(b*e-c*d)^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/
e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)
)/(x+d/e))*c^2-1/3/e^3/(b*e-c*d)/(x+d/e)^3*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*B-5/6/e/(
b*e-c*d)^2/d/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c*A+5/8*e^2/(b*e-c*d)^3/d^3/(
x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*A+5/16*e/(b*e-c*d)^3/d^2/(-(b*e-c*d)*d/e^
2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+
(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^3*B-5/2/e^2/(b*e-c*d)^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e
^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))
/(x+d/e))*c^3*B*d-13/6*B/e/(b*e-c*d)^2/d/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c-1
/2*B/e^2*c/(b*e-c*d)/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e
^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))-15/4/(b*e-c*d)^3/d/(-(b*e-c*d)*d
/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e
^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*c^2*A-3/4/(b*e-c*d)^2/d^2*c/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)
^(1/2))/(x+d/e))*b*A-15/8/(b*e-c*d)^3/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+
2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2*c*B-5/16*e^
2/(b*e-c*d)^3/d^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(
1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^3*A+5/2/(b*e-c*d)^3/d/(x+d/e)*((x+d
/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*c*B-5/8*e/(b*e-c*d)^3/d^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d
)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*B+15/4/e/(b*e-c*d)^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+
(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x
+d/e))*b*c^2*B-5/12/e/(b*e-c*d)^2/d/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*B+3/
2/e/(b*e-c*d)^2/d*c^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^
2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*A
________________________________________________________________________________________
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)^(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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive, negative or zero?
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*x)/((b*x + c*x^2)^(1/2)*(d + e*x)^4),x)
[Out]
int((A + B*x)/((b*x + c*x^2)^(1/2)*(d + e*x)^4), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{4}}\, 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)**(1/2),x)
[Out]
Integral((A + B*x)/(sqrt(x*(b + c*x))*(d + e*x)**4), x)
________________________________________________________________________________________