3.1198 \(\int \frac{A+B x}{(d+e x)^4 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=331 \[ \frac{\sqrt{b x+c x^2} \left (B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )-A e \left (15 b^2 e^2-44 b c d e+44 c^2 d^2\right )\right )}{24 d^3 (d+e x) (c d-b e)^3}+\frac{\left (2 b^2 c d e (9 A e+2 B d)+b^3 \left (-e^2\right ) (5 A e+B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{7/2} (c d-b e)^{7/2}}-\frac{\sqrt{b x+c x^2} (5 A e (2 c d-b e)-B d (b e+4 c d))}{12 d^2 (d+e x)^2 (c d-b e)^2}+\frac{\sqrt{b x+c x^2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)} \]

[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))

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Rubi [A]  time = 0.497411, antiderivative size = 331, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{b x+c x^2} \left (B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )-A e \left (15 b^2 e^2-44 b c d e+44 c^2 d^2\right )\right )}{24 d^3 (d+e x) (c d-b e)^3}+\frac{\left (2 b^2 c d e (9 A e+2 B d)+b^3 \left (-e^2\right ) (5 A e+B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{7/2} (c d-b e)^{7/2}}-\frac{\sqrt{b x+c x^2} (5 A e (2 c d-b e)-B d (b e+4 c d))}{12 d^2 (d+e x)^2 (c d-b e)^2}+\frac{\sqrt{b x+c x^2} (B d-A e)}{3 d (d+e x)^3 (c d-b e)} \]

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 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])

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)^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 = 1.29643, size = 322, normalized size = 0.97 \[ \frac{\sqrt{x} \left (-\frac{\sqrt{x} (b+c x) \left (A e \left (-15 b^2 e^2+44 b c d e-44 c^2 d^2\right )+B d \left (-3 b^2 e^2+10 b c d e+8 c^2 d^2\right )\right )}{d^2 (d+e x) (c d-b e)^2}+\frac{3 \sqrt{b+c x} \left (-2 b^2 c d e (9 A e+2 B d)+b^3 e^2 (5 A e+B d)+8 b c^2 d^2 (3 A e+B d)-16 A c^3 d^3\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{5/2} (b e-c d)^{5/2}}-\frac{2 \sqrt{x} (b+c x) (5 A e (b e-2 c d)+B d (b e+4 c d))}{d (d+e x)^2 (c d-b e)}+\frac{8 \sqrt{x} (b+c x) (A e-B d)}{(d+e x)^3}\right )}{24 d \sqrt{x (b+c x)} (b e-c d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^4*Sqrt[b*x + c*x^2]),x]

[Out]

(Sqrt[x]*((8*(-(B*d) + A*e)*Sqrt[x]*(b + c*x))/(d + e*x)^3 - (2*(5*A*e*(-2*c*d + b*e) + B*d*(4*c*d + b*e))*Sqr
t[x]*(b + c*x))/(d*(c*d - b*e)*(d + e*x)^2) - ((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]*(b + c*x))/(d^2*(c*d - b*e)^2*(d + e*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))*Sqrt[b + c*x]*ArcTan[(Sqrt[-(c*d) + b*e]*S
qrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(d^(5/2)*(-(c*d) + b*e)^(5/2))))/(24*d*(-(c*d) + b*e)*Sqrt[x*(b + c*x)])

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 3242, normalized size = 9.8 \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)^4/(c*x^2+b*x)^(1/2),x)

[Out]

-15/8/d/(b*e-c*d)^3/(-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*c*B-15/4/d/(b*e-c*d)^3/(-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^2*A-5/2/e^2/(b*e-c*d)^3/(-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^3*B*d+3/2/e/d/(b*e-c*d)^2*c^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))*A-5
/12/e/d/(b*e-c*d)^2/(x+d/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)*b*B-5/6/e/d/(b*e-c*d)^
2/(x+d/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)*c*A+5/8*e^2/d^3/(b*e-c*d)^3/(x+d/e)*((x+
d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*A-5/8*e/d^2/(b*e-c*d)^3/(x+d/e)*((x+d/e)^2*c+(b*e-2*
c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*B-5/16*e^2/d^3/(b*e-c*d)^3/(-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^3*A+5/16*e/d^2/(b*e-c*d)^3/(-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^3*B+1/2*B/
e^2/d/(b*e-c*d)/(x+d/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)+3/4*B/d^2/(b*e-c*d)^2/(x+d
/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b-3/8*B/d^2/(b*e-c*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))*b^2-3/e^2/(b*e-c*d)^2*c^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)
)*B+5/6/e^2/(b*e-c*d)^2/(x+d/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)*c*B-5/2/e/(b*e-c*d
)^3/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*B+1/3/e^2/d/(b*e-c*d)/(x+d/e)^3*((x+
d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A+5/2/e/(b*e-c*d)^3/(-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^3*A+2/3*c/d^2/(b*e-c*d)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/
2)*A+5/12/d^2/(b*e-c*d)^2/(x+d/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)*b*A+5/2/d/(b*e-c
*d)^3/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A-1/2*B/e^2*c/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))-3/4/d^2/(b*e-c*d)^2*c/(-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*A+15/4/e/(b*e-c*d)^3/(-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^2*B-13/6/e*c/d
/(b*e-c*d)^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*B+5/2/d/(b*e-c*d)^3/(x+d/e)*((x
+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*c*B-1/3/e^3/(b*e-c*d)/(x+d/e)^3*((x+d/e)^2*c+(b*e-2*c
*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*B+9/4/e/d/(b*e-c*d)^2*c/(-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+15/8*e/d^2/(b*e-c*d)^3/(-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*c*A-5/2*e/d^2/(b*e
-c*d)^3/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*c*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)^4/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.82264, size = 3686, normalized size = 11.14 \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)^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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{4}}\, dx \end{align*}

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)

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Giac [B]  time = 1.57949, size = 2250, normalized size = 6.8 \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)^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)