3.1783 \(\int \frac {A+B x}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=460 \[ -\frac {e^3 (a+b x) (a B e-5 A b e+4 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}-\frac {e^3 (a+b x) (B d-A e)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}-\frac {5 b e^3 (a+b x) \log (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac {5 b e^3 (a+b x) \log (d+e x) (a B e-3 A b e+2 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}-\frac {2 b e^2 (2 a B e-5 A b e+3 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}+\frac {3 b e (a B e-2 A b e+b B d)}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {b (2 a B e-3 A b e+b B d)}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {b (A b-a B)}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

[Out]

-2*b*e^2*(-5*A*b*e+2*B*a*e+3*B*b*d)/(-a*e+b*d)^6/((b*x+a)^2)^(1/2)-1/4*b*(A*b-B*a)/(-a*e+b*d)^3/(b*x+a)^3/((b*
x+a)^2)^(1/2)-1/3*b*(-3*A*b*e+2*B*a*e+B*b*d)/(-a*e+b*d)^4/(b*x+a)^2/((b*x+a)^2)^(1/2)+3/2*b*e*(-2*A*b*e+B*a*e+
B*b*d)/(-a*e+b*d)^5/(b*x+a)/((b*x+a)^2)^(1/2)-1/2*e^3*(-A*e+B*d)*(b*x+a)/(-a*e+b*d)^5/(e*x+d)^2/((b*x+a)^2)^(1
/2)-e^3*(-5*A*b*e+B*a*e+4*B*b*d)*(b*x+a)/(-a*e+b*d)^6/(e*x+d)/((b*x+a)^2)^(1/2)-5*b*e^3*(-3*A*b*e+B*a*e+2*B*b*
d)*(b*x+a)*ln(b*x+a)/(-a*e+b*d)^7/((b*x+a)^2)^(1/2)+5*b*e^3*(-3*A*b*e+B*a*e+2*B*b*d)*(b*x+a)*ln(e*x+d)/(-a*e+b
*d)^7/((b*x+a)^2)^(1/2)

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Rubi [A]  time = 0.56, antiderivative size = 460, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac {e^3 (a+b x) (a B e-5 A b e+4 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^6}-\frac {e^3 (a+b x) (B d-A e)}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^5}-\frac {2 b e^2 (2 a B e-5 A b e+3 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}-\frac {5 b e^3 (a+b x) \log (a+b x) (a B e-3 A b e+2 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac {5 b e^3 (a+b x) \log (d+e x) (a B e-3 A b e+2 b B d)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^7}+\frac {3 b e (a B e-2 A b e+b B d)}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac {b (2 a B e-3 A b e+b B d)}{3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac {b (A b-a B)}{4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-2*b*e^2*(3*b*B*d - 5*A*b*e + 2*a*B*e))/((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(A*b - a*B))/(4*(b
*d - a*e)^3*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(b*B*d - 3*A*b*e + 2*a*B*e))/(3*(b*d - a*e)^4*(a +
b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b*e*(b*B*d - 2*A*b*e + a*B*e))/(2*(b*d - a*e)^5*(a + b*x)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) - (e^3*(B*d - A*e)*(a + b*x))/(2*(b*d - a*e)^5*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (e^3*(4*b*B*d - 5*A*b*e + a*B*e)*(a + b*x))/((b*d - a*e)^6*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*
b*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x)*Log[a + b*x])/((b*d - a*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*b
*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x)*Log[d + e*x])/((b*d - a*e)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {A+B x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {A+B x}{\left (a b+b^2 x\right )^5 (d+e x)^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac {A b-a B}{b^3 (b d-a e)^3 (a+b x)^5}+\frac {b B d-3 A b e+2 a B e}{b^3 (b d-a e)^4 (a+b x)^4}+\frac {3 e (-b B d+2 A b e-a B e)}{b^3 (b d-a e)^5 (a+b x)^3}-\frac {2 e^2 (-3 b B d+5 A b e-2 a B e)}{b^3 (b d-a e)^6 (a+b x)^2}+\frac {5 e^3 (-2 b B d+3 A b e-a B e)}{b^3 (b d-a e)^7 (a+b x)}-\frac {e^4 (-B d+A e)}{b^5 (b d-a e)^5 (d+e x)^3}-\frac {e^4 (-4 b B d+5 A b e-a B e)}{b^5 (b d-a e)^6 (d+e x)^2}-\frac {5 e^4 (-2 b B d+3 A b e-a B e)}{b^4 (b d-a e)^7 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 b e^2 (3 b B d-5 A b e+2 a B e)}{(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (A b-a B)}{4 (b d-a e)^3 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (b B d-3 A b e+2 a B e)}{3 (b d-a e)^4 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 b e (b B d-2 A b e+a B e)}{2 (b d-a e)^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (B d-A e) (a+b x)}{2 (b d-a e)^5 (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (4 b B d-5 A b e+a B e) (a+b x)}{(b d-a e)^6 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 b e^3 (2 b B d-3 A b e+a B e) (a+b x) \log (a+b x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 b e^3 (2 b B d-3 A b e+a B e) (a+b x) \log (d+e x)}{(b d-a e)^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 302, normalized size = 0.66 \[ \frac {\frac {6 e^3 (a+b x)^3 (b d-a e)^2 (A e-B d)}{(d+e x)^2}+\frac {12 e^3 (a+b x)^3 (b d-a e) (-a B e+5 A b e-4 b B d)}{d+e x}-60 b e^3 (a+b x)^3 \log (a+b x) (a B e-3 A b e+2 b B d)+60 b e^3 (a+b x)^3 \log (d+e x) (a B e-3 A b e+2 b B d)+24 b e^2 (a+b x)^2 (b d-a e) (-2 a B e+5 A b e-3 b B d)-\frac {3 b (A b-a B) (b d-a e)^4}{a+b x}-18 b e (a+b x) (b d-a e)^2 (-a B e+2 A b e-b B d)-4 b (b d-a e)^3 (2 a B e-3 A b e+b B d)}{12 \left ((a+b x)^2\right )^{3/2} (b d-a e)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-4*b*(b*d - a*e)^3*(b*B*d - 3*A*b*e + 2*a*B*e) - (3*b*(A*b - a*B)*(b*d - a*e)^4)/(a + b*x) - 18*b*e*(b*d - a*
e)^2*(-(b*B*d) + 2*A*b*e - a*B*e)*(a + b*x) + 24*b*e^2*(b*d - a*e)*(-3*b*B*d + 5*A*b*e - 2*a*B*e)*(a + b*x)^2
+ (6*e^3*(b*d - a*e)^2*(-(B*d) + A*e)*(a + b*x)^3)/(d + e*x)^2 + (12*e^3*(b*d - a*e)*(-4*b*B*d + 5*A*b*e - a*B
*e)*(a + b*x)^3)/(d + e*x) - 60*b*e^3*(2*b*B*d - 3*A*b*e + a*B*e)*(a + b*x)^3*Log[a + b*x] + 60*b*e^3*(2*b*B*d
- 3*A*b*e + a*B*e)*(a + b*x)^3*Log[d + e*x])/(12*(b*d - a*e)^7*((a + b*x)^2)^(3/2))

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fricas [B]  time = 1.50, size = 2466, normalized size = 5.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(6*A*a^6*e^6 - (B*a*b^5 + 3*A*b^6)*d^6 + 2*(5*B*a^2*b^4 + 12*A*a*b^5)*d^5*e - 30*(2*B*a^3*b^3 + 3*A*a^2*b
^4)*d^4*e^2 - 80*(B*a^4*b^2 - 3*A*a^3*b^3)*d^3*e^3 + 5*(25*B*a^5*b - 21*A*a^4*b^2)*d^2*e^4 + 6*(B*a^6 - 12*A*a
^5*b)*d*e^5 - 60*(2*B*b^6*d^2*e^4 - (B*a*b^5 + 3*A*b^6)*d*e^5 - (B*a^2*b^4 - 3*A*a*b^5)*e^6)*x^5 - 30*(6*B*b^6
*d^3*e^3 + (11*B*a*b^5 - 9*A*b^6)*d^2*e^4 - 2*(5*B*a^2*b^4 + 6*A*a*b^5)*d*e^5 - 7*(B*a^3*b^3 - 3*A*a^2*b^4)*e^
6)*x^4 - 20*(2*B*b^6*d^4*e^2 + (31*B*a*b^5 - 3*A*b^6)*d^3*e^3 + 9*(B*a^2*b^4 - 5*A*a*b^5)*d^2*e^4 - (29*B*a^3*
b^3 - 9*A*a^2*b^4)*d*e^5 - 13*(B*a^4*b^2 - 3*A*a^3*b^3)*e^6)*x^3 + 5*(2*B*b^6*d^5*e - (31*B*a*b^5 + 3*A*b^6)*d
^4*e^2 - 4*(37*B*a^2*b^4 - 12*A*a*b^5)*d^3*e^3 + 2*(23*B*a^3*b^3 + 99*A*a^2*b^4)*d^2*e^4 + 2*(53*B*a^4*b^2 - 8
4*A*a^3*b^3)*d*e^5 + 25*(B*a^5*b - 3*A*a^4*b^2)*e^6)*x^2 - 2*(2*B*b^6*d^6 - (19*B*a*b^5 + 3*A*b^6)*d^5*e + 10*
(11*B*a^2*b^4 + 3*A*a*b^5)*d^4*e^2 + 20*(8*B*a^3*b^3 - 9*A*a^2*b^4)*d^3*e^3 - 10*(14*B*a^4*b^2 + 15*A*a^3*b^3)
*d^2*e^4 - (107*B*a^5*b - 285*A*a^4*b^2)*d*e^5 - 6*(B*a^6 - 3*A*a^5*b)*e^6)*x - 60*(2*B*a^4*b^2*d^3*e^3 + (B*a
^5*b - 3*A*a^4*b^2)*d^2*e^4 + (2*B*b^6*d*e^5 + (B*a*b^5 - 3*A*b^6)*e^6)*x^6 + 2*(2*B*b^6*d^2*e^4 + (5*B*a*b^5
- 3*A*b^6)*d*e^5 + 2*(B*a^2*b^4 - 3*A*a*b^5)*e^6)*x^5 + (2*B*b^6*d^3*e^3 + (17*B*a*b^5 - 3*A*b^6)*d^2*e^4 + 4*
(5*B*a^2*b^4 - 6*A*a*b^5)*d*e^5 + 6*(B*a^3*b^3 - 3*A*a^2*b^4)*e^6)*x^4 + 4*(2*B*a*b^5*d^3*e^3 + (7*B*a^2*b^4 -
3*A*a*b^5)*d^2*e^4 + (5*B*a^3*b^3 - 9*A*a^2*b^4)*d*e^5 + (B*a^4*b^2 - 3*A*a^3*b^3)*e^6)*x^3 + (12*B*a^2*b^4*d
^3*e^3 + 2*(11*B*a^3*b^3 - 9*A*a^2*b^4)*d^2*e^4 + 2*(5*B*a^4*b^2 - 12*A*a^3*b^3)*d*e^5 + (B*a^5*b - 3*A*a^4*b^
2)*e^6)*x^2 + 2*(4*B*a^3*b^3*d^3*e^3 + 2*(2*B*a^4*b^2 - 3*A*a^3*b^3)*d^2*e^4 + (B*a^5*b - 3*A*a^4*b^2)*d*e^5)*
x)*log(b*x + a) + 60*(2*B*a^4*b^2*d^3*e^3 + (B*a^5*b - 3*A*a^4*b^2)*d^2*e^4 + (2*B*b^6*d*e^5 + (B*a*b^5 - 3*A*
b^6)*e^6)*x^6 + 2*(2*B*b^6*d^2*e^4 + (5*B*a*b^5 - 3*A*b^6)*d*e^5 + 2*(B*a^2*b^4 - 3*A*a*b^5)*e^6)*x^5 + (2*B*b
^6*d^3*e^3 + (17*B*a*b^5 - 3*A*b^6)*d^2*e^4 + 4*(5*B*a^2*b^4 - 6*A*a*b^5)*d*e^5 + 6*(B*a^3*b^3 - 3*A*a^2*b^4)*
e^6)*x^4 + 4*(2*B*a*b^5*d^3*e^3 + (7*B*a^2*b^4 - 3*A*a*b^5)*d^2*e^4 + (5*B*a^3*b^3 - 9*A*a^2*b^4)*d*e^5 + (B*a
^4*b^2 - 3*A*a^3*b^3)*e^6)*x^3 + (12*B*a^2*b^4*d^3*e^3 + 2*(11*B*a^3*b^3 - 9*A*a^2*b^4)*d^2*e^4 + 2*(5*B*a^4*b
^2 - 12*A*a^3*b^3)*d*e^5 + (B*a^5*b - 3*A*a^4*b^2)*e^6)*x^2 + 2*(4*B*a^3*b^3*d^3*e^3 + 2*(2*B*a^4*b^2 - 3*A*a^
3*b^3)*d^2*e^4 + (B*a^5*b - 3*A*a^4*b^2)*d*e^5)*x)*log(e*x + d))/(a^4*b^7*d^9 - 7*a^5*b^6*d^8*e + 21*a^6*b^5*d
^7*e^2 - 35*a^7*b^4*d^6*e^3 + 35*a^8*b^3*d^5*e^4 - 21*a^9*b^2*d^4*e^5 + 7*a^10*b*d^3*e^6 - a^11*d^2*e^7 + (b^1
1*d^7*e^2 - 7*a*b^10*d^6*e^3 + 21*a^2*b^9*d^5*e^4 - 35*a^3*b^8*d^4*e^5 + 35*a^4*b^7*d^3*e^6 - 21*a^5*b^6*d^2*e
^7 + 7*a^6*b^5*d*e^8 - a^7*b^4*e^9)*x^6 + 2*(b^11*d^8*e - 5*a*b^10*d^7*e^2 + 7*a^2*b^9*d^6*e^3 + 7*a^3*b^8*d^5
*e^4 - 35*a^4*b^7*d^4*e^5 + 49*a^5*b^6*d^3*e^6 - 35*a^6*b^5*d^2*e^7 + 13*a^7*b^4*d*e^8 - 2*a^8*b^3*e^9)*x^5 +
(b^11*d^9 + a*b^10*d^8*e - 29*a^2*b^9*d^7*e^2 + 91*a^3*b^8*d^6*e^3 - 119*a^4*b^7*d^5*e^4 + 49*a^5*b^6*d^4*e^5
+ 49*a^6*b^5*d^3*e^6 - 71*a^7*b^4*d^2*e^7 + 34*a^8*b^3*d*e^8 - 6*a^9*b^2*e^9)*x^4 + 4*(a*b^10*d^9 - 4*a^2*b^9*
d^8*e + a^3*b^8*d^7*e^2 + 21*a^4*b^7*d^6*e^3 - 49*a^5*b^6*d^5*e^4 + 49*a^6*b^5*d^4*e^5 - 21*a^7*b^4*d^3*e^6 -
a^8*b^3*d^2*e^7 + 4*a^9*b^2*d*e^8 - a^10*b*e^9)*x^3 + (6*a^2*b^9*d^9 - 34*a^3*b^8*d^8*e + 71*a^4*b^7*d^7*e^2 -
49*a^5*b^6*d^6*e^3 - 49*a^6*b^5*d^5*e^4 + 119*a^7*b^4*d^4*e^5 - 91*a^8*b^3*d^3*e^6 + 29*a^9*b^2*d^2*e^7 - a^1
0*b*d*e^8 - a^11*e^9)*x^2 + 2*(2*a^3*b^8*d^9 - 13*a^4*b^7*d^8*e + 35*a^5*b^6*d^7*e^2 - 49*a^6*b^5*d^6*e^3 + 35
*a^7*b^4*d^5*e^4 - 7*a^8*b^3*d^4*e^5 - 7*a^9*b^2*d^3*e^6 + 5*a^10*b*d^2*e^7 - a^11*d*e^8)*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x

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maple [B]  time = 0.08, size = 2420, normalized size = 5.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/12*(-3*A*b^6*d^6+6*A*a^6*e^6-B*a*b^5*d^6+6*B*a^6*d*e^5-4*B*x*b^6*d^6+12*B*x*a^6*e^6+180*A*ln(b*x+a)*x^6*b^6
*e^6-180*A*ln(e*x+d)*x^6*b^6*e^6-780*A*x^3*a^3*b^3*e^6+60*A*x^3*b^6*d^3*e^3-105*A*a^4*b^2*d^2*e^4+240*A*a^3*b^
3*d^3*e^3-90*A*a^2*b^4*d^4*e^2+24*A*a*b^5*d^5*e+125*B*a^5*b*d^2*e^4-80*B*a^4*b^2*d^3*e^3-60*B*a^3*b^3*d^4*e^2+
10*B*a^2*b^4*d^5*e-72*A*a^5*b*d*e^5+260*B*x^3*a^4*b^2*e^6-40*B*x^3*b^6*d^4*e^2-375*A*x^2*a^4*b^2*e^6-15*A*x^2*
b^6*d^4*e^2+125*B*x^2*a^5*b*e^6+10*B*x^2*b^6*d^5*e+6*A*x*b^6*d^5*e-630*A*x^4*a^2*b^4*e^6+270*A*x^4*b^6*d^2*e^4
+210*B*x^4*a^3*b^3*e^6-180*B*x^4*b^6*d^3*e^3-180*A*x^5*a*b^5*e^6+180*A*x^5*b^6*d*e^5+60*B*x^5*a^2*b^4*e^6-120*
B*x^5*b^6*d^2*e^4-36*A*x*a^5*b*e^6-600*B*ln(b*x+a)*x^5*a*b^5*d*e^5+600*B*ln(e*x+d)*x^5*a*b^5*d*e^5+1200*B*ln(e
*x+d)*x^3*a^3*b^3*d*e^5+1680*B*ln(e*x+d)*x^3*a^2*b^4*d^2*e^4+480*B*ln(e*x+d)*x^3*a*b^5*d^3*e^3+1440*A*ln(b*x+a
)*x^2*a^3*b^3*d*e^5+1080*A*ln(b*x+a)*x^2*a^2*b^4*d^2*e^4-1440*A*ln(e*x+d)*x^2*a^3*b^3*d*e^5-1080*A*ln(e*x+d)*x
^2*a^2*b^4*d^2*e^4-600*B*ln(b*x+a)*x^2*a^4*b^2*d*e^5-1320*B*ln(b*x+a)*x^2*a^3*b^3*d^2*e^4-720*B*ln(b*x+a)*x^2*
a^2*b^4*d^3*e^3+600*B*ln(e*x+d)*x^2*a^4*b^2*d*e^5+1440*A*ln(b*x+a)*x^4*a*b^5*d*e^5+480*B*ln(e*x+d)*x*a^4*b^2*d
^2*e^4+480*B*ln(e*x+d)*x*a^3*b^3*d^3*e^3-1440*A*ln(e*x+d)*x^4*a*b^5*d*e^5-1200*B*ln(b*x+a)*x^4*a^2*b^4*d*e^5-1
020*B*ln(b*x+a)*x^4*a*b^5*d^2*e^4+1200*B*ln(e*x+d)*x^4*a^2*b^4*d*e^5+1020*B*ln(e*x+d)*x^4*a*b^5*d^2*e^4+2160*A
*ln(b*x+a)*x^3*a^2*b^4*d*e^5+720*A*ln(b*x+a)*x^3*a*b^5*d^2*e^4-2160*A*ln(e*x+d)*x^3*a^2*b^4*d*e^5-720*A*ln(e*x
+d)*x^3*a*b^5*d^2*e^4-1200*B*ln(b*x+a)*x^3*a^3*b^3*d*e^5-1680*B*ln(b*x+a)*x^3*a^2*b^4*d^2*e^4-480*B*ln(b*x+a)*
x^3*a*b^5*d^3*e^3+1320*B*ln(e*x+d)*x^2*a^3*b^3*d^2*e^4+720*B*ln(e*x+d)*x^2*a^2*b^4*d^3*e^3+360*A*ln(b*x+a)*x*a
^4*b^2*d*e^5+720*A*ln(b*x+a)*x*a^3*b^3*d^2*e^4-360*A*ln(e*x+d)*x*a^4*b^2*d*e^5-720*A*ln(e*x+d)*x*a^3*b^3*d^2*e
^4-120*B*ln(b*x+a)*x*a^5*b*d*e^5-480*B*ln(b*x+a)*x*a^4*b^2*d^2*e^4-480*B*ln(b*x+a)*x*a^3*b^3*d^3*e^3+120*B*ln(
e*x+d)*x*a^5*b*d*e^5-180*A*x^3*a^2*b^4*d*e^5+900*A*x^3*a*b^5*d^2*e^4+580*B*x^3*a^3*b^3*d*e^5-180*B*x^3*a^2*b^4
*d^2*e^4-620*B*x^3*a*b^5*d^3*e^3-840*A*x^2*a^3*b^3*d*e^5+990*A*x^2*a^2*b^4*d^2*e^4+240*A*x^2*a*b^5*d^3*e^3+530
*B*x^2*a^4*b^2*d*e^5+230*B*x^2*a^3*b^3*d^2*e^4-740*B*x^2*a^2*b^4*d^3*e^3-155*B*x^2*a*b^5*d^4*e^2-570*A*x*a^4*b
^2*d*e^5+300*A*x*a^3*b^3*d^2*e^4+360*A*x*a^2*b^4*d^3*e^3-60*A*x*a*b^5*d^4*e^2+214*B*x*a^5*b*d*e^5+280*B*x*a^4*
b^2*d^2*e^4-320*B*x*a^3*b^3*d^3*e^3-220*B*x*a^2*b^4*d^4*e^2+38*B*x*a*b^5*d^5*e+360*A*x^4*a*b^5*d*e^5+300*B*x^4
*a^2*b^4*d*e^5-330*B*x^4*a*b^5*d^2*e^4+60*B*x^5*a*b^5*d*e^5+180*A*ln(b*x+a)*a^4*b^2*d^2*e^4-180*A*ln(e*x+d)*a^
4*b^2*d^2*e^4-60*B*ln(b*x+a)*a^5*b*d^2*e^4-120*B*ln(b*x+a)*a^4*b^2*d^3*e^3+60*B*ln(e*x+d)*a^5*b*d^2*e^4+120*B*
ln(e*x+d)*a^4*b^2*d^3*e^3-720*A*ln(e*x+d)*x^5*a*b^5*e^6-360*A*ln(e*x+d)*x^5*b^6*d*e^5-240*B*ln(b*x+a)*x^5*a^2*
b^4*e^6-240*B*ln(b*x+a)*x^5*b^6*d^2*e^4+240*B*ln(e*x+d)*x^5*a^2*b^4*e^6+240*B*ln(e*x+d)*x^5*b^6*d^2*e^4+1080*A
*ln(b*x+a)*x^4*a^2*b^4*e^6+180*A*ln(b*x+a)*x^4*b^6*d^2*e^4-1080*A*ln(e*x+d)*x^4*a^2*b^4*e^6-180*A*ln(e*x+d)*x^
4*b^6*d^2*e^4-360*B*ln(b*x+a)*x^4*a^3*b^3*e^6-120*B*ln(b*x+a)*x^4*b^6*d^3*e^3+360*B*ln(e*x+d)*x^4*a^3*b^3*e^6+
120*B*ln(e*x+d)*x^4*b^6*d^3*e^3+720*A*ln(b*x+a)*x^3*a^3*b^3*e^6-720*A*ln(e*x+d)*x^3*a^3*b^3*e^6-240*B*ln(b*x+a
)*x^3*a^4*b^2*e^6+240*B*ln(e*x+d)*x^3*a^4*b^2*e^6-60*B*ln(b*x+a)*x^6*a*b^5*e^6-120*B*ln(b*x+a)*x^6*b^6*d*e^5+6
0*B*ln(e*x+d)*x^6*a*b^5*e^6+120*B*ln(e*x+d)*x^6*b^6*d*e^5+720*A*ln(b*x+a)*x^5*a*b^5*e^6+360*A*ln(b*x+a)*x^5*b^
6*d*e^5+180*A*ln(b*x+a)*x^2*a^4*b^2*e^6-180*A*ln(e*x+d)*x^2*a^4*b^2*e^6-60*B*ln(b*x+a)*x^2*a^5*b*e^6+60*B*ln(e
*x+d)*x^2*a^5*b*e^6)*(b*x+a)/(e*x+d)^2/(a*e-b*d)^7/((b*x+a)^2)^(5/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(5/2),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-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x}{{\left (d+e\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)

[Out]

int((A + B*x)/((d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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