3.24.98 \(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^{5/2}} \, dx\) [2398]

Optimal. Leaf size=473 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}} \]

[Out]

-2/3*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)/(c*x^2+b*x+a)^(3/2)+5/2*e
^4*(-b*e+2*c*d)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b
*d*e+c*d^2)^(7/2)-2/3*(6*a*c*e*(-b*e+2*c*d)^2-(2*a*c*e-b^2*e+b*c*d)*(8*c^2*d^2-5*b^2*e^2-2*c*e*(-8*a*e+b*d))-c
*(-b*e+2*c*d)*(8*c^2*d^2-5*b^2*e^2-4*c*e*(-7*a*e+2*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)/(c*x^
2+b*x+a)^(1/2)+1/3*e*(32*c^4*d^4-15*b^4*e^4-16*c^3*d^2*e*(-9*a*e+4*b*d)+20*b^2*c*e^3*(5*a*e+b*d)+4*c^2*e^2*(-3
2*a^2*e^2-36*a*b*d*e+3*b^2*d^2))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)

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Rubi [A]
time = 0.39, antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {754, 836, 820, 738, 212} \begin {gather*} \frac {e \sqrt {a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {5 e^4 (2 c d-b e) \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 )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x
+ c*x^2)^(3/2)) - (2*(6*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(b*d
- 8*a*e)) - c*(2*c*d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 - b
*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e)
+ 20*b^2*c*e^3*(b*d + 5*a*e) + 4*c^2*e^2*(3*b^2*d^2 - 36*a*b*d*e - 32*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(3*(b^2
- 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (5*e^4*(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)^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

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 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 820

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)/(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[S
implify[m + 2*p + 3], 0]

Rule 836

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])

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )+3 c e (2 c d-b e) x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} e \left (10 b^3 c d e^2-15 b^4 e^3+32 a c^2 e \left (c d^2-4 a e^2\right )-8 b c^2 d \left (2 c d^2+11 a e^2\right )+4 b^2 c e \left (4 c d^2+25 a e^2\right )\right )+\frac {1}{2} c e (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\left (5 e^4 (2 c d-b e)\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 )^3}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (5 e^4 (2 c d-b e)\right ) \text {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 )^3}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (6 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-5 b^2 e^2-2 c e (b d-8 a e)\right )-c (2 c d-b e) \left (8 c^2 d^2-5 b^2 e^2-4 c e (2 b d-7 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)+4 c^2 e^2 \left (3 b^2 d^2-36 a b d e-32 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {5 e^4 (2 c d-b e) \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 )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 11.14, size = 482, normalized size = 1.02 \begin {gather*} \frac {2 \left (\frac {e \left (32 c^4 d^4-15 b^4 e^4-16 c^3 d^2 e (4 b d-9 a e)+20 b^2 c e^3 (b d+5 a e)-4 c^2 e^2 \left (-3 b^2 d^2+36 a b d e+32 a^2 e^2\right )\right ) \sqrt {a+x (b+c x)}}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x) (a+x (b+c x))^{3/2}}+\frac {-5 b^4 e^3+b^3 c e^2 (3 d-5 e x)-8 c^2 \left (4 a^2 e^3+2 c^2 d^3 x-a c d e (d-7 e x)\right )-4 b c^2 \left (a e^2 (9 d-7 e x)+2 c d^2 (d-3 e x)\right )+2 b^2 c e \left (16 a e^2+c d (5 d+e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (d+e x) \sqrt {a+x (b+c x)}}+\frac {15 \left (b^2-4 a c\right ) e^4 (-2 c d+b e) \tanh ^{-1}\left (\frac {-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)}}\right )}{4 \left (c d^2+e (-b d+a e)\right )^{5/2}}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*((e*(32*c^4*d^4 - 15*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^2*c*e^3*(b*d + 5*a*e) - 4*c^2*e^2*(-3*b^
2*d^2 + 36*a*b*d*e + 32*a^2*e^2))*Sqrt[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*
x)) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)*(a + x*(b + c*x))^(3/2)) + (-5*b^4*e^3 + b^3*c*e
^2*(3*d - 5*e*x) - 8*c^2*(4*a^2*e^3 + 2*c^2*d^3*x - a*c*d*e*(d - 7*e*x)) - 4*b*c^2*(a*e^2*(9*d - 7*e*x) + 2*c*
d^2*(d - 3*e*x)) + 2*b^2*c*e*(16*a*e^2 + c*d*(5*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)]) + (15*(b^2 - 4*a*c)*e^4*(-2*c*d + b*e)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*S
qrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*(c*d^2 + e*(-(b*d) + a*e))^(5/2))))/(3*(b^2 - 4*a*c)
*(c*d^2 + e*(-(b*d) + a*e)))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1142\) vs. \(2(451)=902\).
time = 1.04, size = 1143, normalized size = 2.42

method	result	size
default	\(\frac {-\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}-\frac {5 e \left (b e -2 c d \right ) \left (\frac {e^{2}}{3 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}-\frac {e \left (b e -2 c d \right ) \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}+\frac {2 \left (b e -2 c d \right )}{3 e}}{\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{3 \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right )}+\frac {e^{2} \left (\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a -b d e +c \,d^{2}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right )}-\frac {4 c \,e^{2} \left (\frac {\frac {4 c \left (x +\frac {d}{e}\right )}{3}+\frac {2 \left (b e -2 c d \right )}{3 e}}{\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}+\frac {16 c \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{3 \left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right )^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a -b d e +c \,d^{2}}}{e^{2}}\)	\(1143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e^2*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-
5/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(1/3/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2
-b*d*e+c*d^2)/e^2)^(3/2)-1/2*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(2/3*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(4*c*(a*e^2-
b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+16/3
*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)^2*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(c*(x+d/e)^2+1/e*(b*e-2*c
*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))+1/(a*e^2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b*d*e+c*d^2)*e^2/(c*(x+d/e)^2+
1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+1/e*(b*e
-2*c*d))/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^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+1/e*(
b*e-2*c*d)*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/
e^2)^(1/2))/(x+d/e))))-4*c/(a*e^2-b*d*e+c*d^2)*e^2*(2/3*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(4*c*(a*e^2-b*d*e+c*d^2)
/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+16/3*c/(4*c*(a*e
^2-b*d*e+c*d^2)/e^2-1/e^2*(b*e-2*c*d)^2)^2*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)+
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)))

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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(1/(e*x+d)^2/(c*x^2+b*x+a)^(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(c*d^2-%e*b*d+%e^2*a>0)', see `
assume?` for

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4375 vs. \(2 (467) = 934\).
time = 26.90, size = 8793, normalized size = 18.59 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(15*sqrt(c*d^2 - b*d*e + a*e^2)*(((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^5 + 2*(b^6*c - 8*a*b^4*c^2 +
16*a^2*b^2*c^3)*x^4 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^3 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*x^2 + (a
^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x)*e^6 - (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*x^5 + 3*(b^5*c^2 - 8*a
*b^3*c^3 + 16*a^2*b*c^4)*d*x^4 + 4*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d*x^3 - (b^7 - 10*a*b^5*c + 32*a^2
*b^3*c^2 - 32*a^3*b*c^3)*d*x^2 - 2*(a*b^6 - 9*a^2*b^4*c + 24*a^3*b^2*c^2 - 16*a^4*c^3)*d*x - (a^2*b^5 - 8*a^3*
b^3*c + 16*a^4*b*c^2)*d)*e^5 - 2*((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16
*a^2*b*c^4)*d^2*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d^2*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*
d^2*x + (a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^2)*e^4)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2 + 4*a*c)*d
^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (b^2 + 4
*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)) + 4*(32
*c^7*d^7*x^3 + 48*b*c^6*d^7*x^2 + 12*(b^2*c^5 + 4*a*c^6)*d^7*x - 2*(b^3*c^4 - 12*a*b*c^5)*d^7 - (3*a^3*b^4 - 2
4*a^4*b^2*c + 48*a^5*c^2 + (15*a*b^4*c^2 - 100*a^2*b^2*c^3 + 128*a^3*c^4)*x^4 + 6*(5*a*b^5*c - 35*a^2*b^3*c^2
+ 52*a^3*b*c^3)*x^3 + 3*(5*a*b^6 - 30*a^2*b^4*c + 16*a^3*b^2*c^2 + 64*a^4*c^3)*x^2 + 4*(5*a^2*b^5 - 37*a^3*b^3
*c + 64*a^4*b*c^2)*x)*e^7 + ((15*b^5*c^2 - 80*a*b^3*c^3 - 16*a^2*b*c^4)*d*x^4 + 2*(15*b^6*c - 90*a*b^4*c^2 + 3
8*a^2*b^2*c^3 + 56*a^3*c^4)*d*x^3 + 3*(5*b^7 - 30*a*b^5*c + 18*a^2*b^3*c^2 + 8*a^3*b*c^3)*d*x^2 + 2*(5*a*b^6 -
32*a^2*b^4*c + 20*a^3*b^2*c^2 + 64*a^4*c^3)*d*x - (11*a^2*b^5 - 76*a^3*b^3*c + 112*a^4*b*c^2)*d)*e^6 - ((35*b
^4*c^3 - 256*a*b^2*c^4 - 16*a^2*c^5)*d^2*x^4 + 4*(15*b^5*c^2 - 121*a*b^3*c^3 + 88*a^2*b*c^4)*d^2*x^3 + 3*(5*b^
6*c - 42*a*b^4*c^2 + 28*a^2*b^2*c^3 - 48*a^3*c^4)*d^2*x^2 - 2*(5*b^7 - 43*a*b^5*c + 125*a^2*b^3*c^2 - 156*a^3*
b*c^3)*d^2*x - (16*a*b^6 - 91*a^2*b^4*c + 16*a^3*b^2*c^2 + 176*a^4*c^3)*d^2)*e^5 + 2*(4*(b^3*c^4 - 44*a*b*c^5)
*d^3*x^4 - 4*(b^4*c^3 + 49*a*b^2*c^4 - 32*a^2*c^5)*d^3*x^3 - 3*(7*b^5*c^2 - 34*a*b^3*c^3 + 64*a^2*b*c^4)*d^3*x
^2 - 2*(7*b^6*c - 51*a*b^4*c^2 + 61*a^2*b^2*c^3 - 76*a^3*c^4)*d^3*x - (b^7 + 16*a*b^5*c - 155*a^2*b^3*c^2 + 18
0*a^3*b*c^3)*d^3)*e^4 + 2*(2*(19*b^2*c^5 + 44*a*c^6)*d^4*x^4 + (61*b^3*c^4 - 44*a*b*c^5)*d^4*x^3 + 12*(2*b^4*c
^3 - 17*a*b^2*c^4 + 16*a^2*c^5)*d^4*x^2 + 2*(3*b^5*c^2 - 46*a*b^3*c^3 - 44*a^2*b*c^4)*d^4*x + 4*(b^6*c - 3*a*b
^4*c^2 - 32*a^2*b^2*c^3 + 32*a^3*c^4)*d^4)*e^3 - 2*(48*b*c^6*d^5*x^4 + 2*(17*b^2*c^5 - 44*a*c^6)*d^5*x^3 - 3*(
13*b^3*c^4 + 20*a*b*c^5)*d^5*x^2 - 16*(b^4*c^3 + 4*a*b^2*c^4 + 7*a^2*c^5)*d^5*x + 2*(3*b^5*c^2 - 26*a*b^3*c^3
- 4*a^2*b*c^4)*d^5)*e^2 + 2*(16*c^7*d^6*x^4 - 24*b*c^6*d^6*x^3 - 6*(11*b^2*c^5 - 4*a*c^6)*d^6*x^2 - (19*b^3*c^
4 + 60*a*b*c^5)*d^6*x + 4*(b^4*c^3 - 11*a*b^2*c^4 + 4*a^2*c^5)*d^6)*e)*sqrt(c*x^2 + b*x + a))/((b^4*c^6 - 8*a*
b^2*c^7 + 16*a^2*c^8)*d^9*x^4 + 2*(b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*d^9*x^3 + (b^6*c^4 - 6*a*b^4*c^5 + 32
*a^3*c^7)*d^9*x^2 + 2*(a*b^5*c^4 - 8*a^2*b^3*c^5 + 16*a^3*b*c^6)*d^9*x + (a^2*b^4*c^4 - 8*a^3*b^2*c^5 + 16*a^4
*c^6)*d^9 + ((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*x^5 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^4
+ (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*x^3 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^2 + (a^6*b^4 - 8*a^7*
b^2*c + 16*a^8*c^2)*x)*e^9 - (4*(a^3*b^5*c^2 - 8*a^4*b^3*c^3 + 16*a^5*b*c^4)*d*x^5 + (8*a^3*b^6*c - 65*a^4*b^4
*c^2 + 136*a^5*b^2*c^3 - 16*a^6*c^4)*d*x^4 + 2*(2*a^3*b^7 - 13*a^4*b^5*c + 8*a^5*b^3*c^2 + 48*a^6*b*c^3)*d*x^3
+ (7*a^4*b^6 - 58*a^5*b^4*c + 128*a^6*b^2*c^2 - 32*a^7*c^3)*d*x^2 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*
d*x - (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*d)*e^8 + 2*((3*a^2*b^6*c^2 - 22*a^3*b^4*c^3 + 32*a^4*b^2*c^4 + 32*a
^5*c^5)*d^2*x^5 + 2*(3*a^2*b^7*c - 23*a^3*b^5*c^2 + 40*a^4*b^3*c^3 + 16*a^5*b*c^4)*d^2*x^4 + (3*a^2*b^8 - 20*a
^3*b^6*c + 20*a^4*b^4*c^2 + 32*a^5*b^2*c^3 + 64*a^6*c^4)*d^2*x^3 + 4*(a^3*b^7 - 8*a^4*b^5*c + 16*a^5*b^3*c^2)*
d^2*x^2 - (a^4*b^6 - 10*a^5*b^4*c + 32*a^6*b^2*c^2 - 32*a^7*c^3)*d^2*x - 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c
^2)*d^2)*e^7 - 2*(2*(a*b^7*c^2 - 5*a^2*b^5*c^3 - 8*a^3*b^3*c^4 + 48*a^4*b*c^5)*d^3*x^5 + (4*a*b^8*c - 23*a^2*b
^6*c^2 - 10*a^3*b^4*c^3 + 160*a^4*b^2*c^4 - 32*a^5*c^5)*d^3*x^4 + 2*(a*b^9 - 6*a^2*b^7*c + 4*a^3*b^5*c^2 + 64*
a^5*b*c^4)*d^3*x^3 + (a^2*b^8 - 4*a^3*b^6*c - 20*a^4*b^4*c^2 + 96*a^5*b^2*c^3 - 64*a^6*c^4)*d^3*x^2 - 2*(2*a^3
*b^7 - 17*a^4*b^5*c + 40*a^5*b^3*c^2 - 16*a^6*b*c^3)*d^3*x - (3*a^4*b^6 - 22*a^5*b^4*c + 32*a^6*b^2*c^2 + 32*a
^7*c^3)*d^3)*e^6 + ((b^8*c^2 + 4*a*b^6*c^3 - 74*a^2*b^4*c^4 + 144*a^3*b^2*c^5 + 96*a^4*c^6)*d^4*x^5 + 2*(b^9*c
+ 2*a*b^7*c^2 - 64*a^2*b^5*c^3 + 160*a^3*b^3*c^4)*d^4*x^4 + (b^10 - 2*a*b^8*c - 26*a^2*b^6*c^2 + 60*a^3*b^4*c
^3 + 192*a^5*c^5)*d^4*x^3 - 2*(a*b^9 - 10*a^2*b^7*c + 38*a^3*b^5*c^2 - 80*a^4*b^3*c^3 + 96*a^5*b*c^4)*d^4*x^2
- (7*a^2*b^8 - 44*a^3*b^6*c + 10*a^4*b^4*c^2 + ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d + e*x)**2*(a + b*x + c*x**2)**(5/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 5122 vs. \(2 (467) = 934\).
time = 2.29, size = 5122, normalized size = 10.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

-1/6*((64*sqrt(c*d^2 - b*d*e + a*e^2)*c^5*d^4*e^2 - 128*sqrt(c*d^2 - b*d*e + a*e^2)*b*c^4*d^3*e^3 + 24*sqrt(c*
d^2 - b*d*e + a*e^2)*b^2*c^3*d^2*e^4 + 288*sqrt(c*d^2 - b*d*e + a*e^2)*a*c^4*d^2*e^4 - 30*b^4*c^(3/2)*d*e^6*lo
g(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) + 240*a*b^2*c^(5/2)*d*e^6*log(abs(-2*c*d + b*e +
2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 480*a^2*c^(7/2)*d*e^6*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e +
a*e^2)*sqrt(c))) + 40*sqrt(c*d^2 - b*d*e + a*e^2)*b^3*c^2*d*e^5 - 288*sqrt(c*d^2 - b*d*e + a*e^2)*a*b*c^3*d*e^
5 + 15*b^5*sqrt(c)*e^7*log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 120*a*b^3*c^(3/2)*e^7*
log(abs(-2*c*d + b*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) + 240*a^2*b*c^(5/2)*e^7*log(abs(-2*c*d + b*e +
2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c))) - 30*sqrt(c*d^2 - b*d*e + a*e^2)*b^4*c*e^6 + 200*sqrt(c*d^2 - b*d*e +
a*e^2)*a*b^2*c^2*e^6 - 256*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*c^3*e^6)*sgn(1/(x*e + d))/(sqrt(c*d^2 - b*d*e + a*e
^2)*b^4*c^(7/2)*d^6 - 8*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^2*c^(9/2)*d^6 + 16*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*c^(
11/2)*d^6 - 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^5*c^(5/2)*d^5*e + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^3*c^(7/2)*d^5
*e - 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b*c^(9/2)*d^5*e + 3*sqrt(c*d^2 - b*d*e + a*e^2)*b^6*c^(3/2)*d^4*e^2 -
21*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^4*c^(5/2)*d^4*e^2 + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^2*c^(7/2)*d^4*e^2
+ 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*c^(9/2)*d^4*e^2 - sqrt(c*d^2 - b*d*e + a*e^2)*b^7*sqrt(c)*d^3*e^3 + 2*sqr
t(c*d^2 - b*d*e + a*e^2)*a*b^5*c^(3/2)*d^3*e^3 + 32*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^3*c^(5/2)*d^3*e^3 - 96*s
qrt(c*d^2 - b*d*e + a*e^2)*a^3*b*c^(7/2)*d^3*e^3 + 3*sqrt(c*d^2 - b*d*e + a*e^2)*a*b^6*sqrt(c)*d^2*e^4 - 21*sq
rt(c*d^2 - b*d*e + a*e^2)*a^2*b^4*c^(3/2)*d^2*e^4 + 24*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*b^2*c^(5/2)*d^2*e^4 + 4
8*sqrt(c*d^2 - b*d*e + a*e^2)*a^4*c^(7/2)*d^2*e^4 - 3*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*b^5*sqrt(c)*d*e^5 + 24*s
qrt(c*d^2 - b*d*e + a*e^2)*a^3*b^3*c^(3/2)*d*e^5 - 48*sqrt(c*d^2 - b*d*e + a*e^2)*a^4*b*c^(5/2)*d*e^5 + sqrt(c
*d^2 - b*d*e + a*e^2)*a^3*b^4*sqrt(c)*e^6 - 8*sqrt(c*d^2 - b*d*e + a*e^2)*a^4*b^2*c^(3/2)*e^6 + 16*sqrt(c*d^2
- b*d*e + a*e^2)*a^5*c^(5/2)*e^6) + 2*((((4*(8*c^6*d^7*e^16 - 28*b*c^5*d^6*e^17 + 30*b^2*c^4*d^5*e^18 + 48*a*c
^5*d^5*e^18 - 5*b^3*c^3*d^4*e^19 - 120*a*b*c^4*d^4*e^19 - 18*b^4*c^2*d^3*e^20 + 164*a*b^2*c^3*d^3*e^20 - 88*a^
2*c^4*d^3*e^20 + 18*b^5*c*d^2*e^21 - 126*a*b^3*c^2*d^2*e^21 + 132*a^2*b*c^3*d^2*e^21 - 5*b^6*d*e^22 + 24*a*b^4
*c*d*e^22 + 30*a^2*b^2*c^2*d*e^22 - 128*a^3*c^3*d*e^22 + 5*a*b^5*e^23 - 37*a^2*b^3*c*e^23 + 64*a^3*b*c^2*e^23)
/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d)) - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d)) + 16*a^2*c^5*d^6*e^11*sgn(1/(x*e +
d)) - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d)) + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d)) - 48*a^2*b*c^4*d^5*e^12*sg
n(1/(x*e + d)) + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d)) - 21*a*b^4*c^2*d^4*e^13*sgn(1/(x*e + d)) + 24*a^2*b^2*c^3*d
^4*e^13*sgn(1/(x*e + d)) + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d)) - b^7*d^3*e^14*sgn(1/(x*e + d)) + 2*a*b^5*c*d^
3*e^14*sgn(1/(x*e + d)) + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d)) - 96*a^3*b*c^3*d^3*e^14*sgn(1/(x*e + d)) +
3*a*b^6*d^2*e^15*sgn(1/(x*e + d)) - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d)) + 24*a^3*b^2*c^2*d^2*e^15*sgn(1/(x*
e + d)) + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d)) - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d)) + 24*a^3*b^3*c*d*e^16*sgn(1
/(x*e + d)) - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d)) + a^3*b^4*e^17*sgn(1/(x*e + d)) - 8*a^4*b^2*c*e^17*sgn(1/(x
*e + d)) + 16*a^5*c^2*e^17*sgn(1/(x*e + d))) + 3*(b^4*c^2*d^4*e^21 - 8*a*b^2*c^3*d^4*e^21 + 16*a^2*c^4*d^4*e^2
1 - 2*b^5*c*d^3*e^22 + 16*a*b^3*c^2*d^3*e^22 - 32*a^2*b*c^3*d^3*e^22 + b^6*d^2*e^23 - 6*a*b^4*c*d^2*e^23 + 32*
a^3*c^3*d^2*e^23 - 2*a*b^5*d*e^24 + 16*a^2*b^3*c*d*e^24 - 32*a^3*b*c^2*d*e^24 + a^2*b^4*e^25 - 8*a^3*b^2*c*e^2
5 + 16*a^4*c^2*e^25)*e^(-1)/((b^4*c^3*d^6*e^11*sgn(1/(x*e + d)) - 8*a*b^2*c^4*d^6*e^11*sgn(1/(x*e + d)) + 16*a
^2*c^5*d^6*e^11*sgn(1/(x*e + d)) - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d)) + 24*a*b^3*c^3*d^5*e^12*sgn(1/(x*e + d)
) - 48*a^2*b*c^4*d^5*e^12*sgn(1/(x*e + d)) + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d)) - 21*a*b^4*c^2*d^4*e^13*sgn(1/(
x*e + d)) + 24*a^2*b^2*c^3*d^4*e^13*sgn(1/(x*e + d)) + 48*a^3*c^4*d^4*e^13*sgn(1/(x*e + d)) - b^7*d^3*e^14*sgn
(1/(x*e + d)) + 2*a*b^5*c*d^3*e^14*sgn(1/(x*e + d)) + 32*a^2*b^3*c^2*d^3*e^14*sgn(1/(x*e + d)) - 96*a^3*b*c^3*
d^3*e^14*sgn(1/(x*e + d)) + 3*a*b^6*d^2*e^15*sgn(1/(x*e + d)) - 21*a^2*b^4*c*d^2*e^15*sgn(1/(x*e + d)) + 24*a^
3*b^2*c^2*d^2*e^15*sgn(1/(x*e + d)) + 48*a^4*c^3*d^2*e^15*sgn(1/(x*e + d)) - 3*a^2*b^5*d*e^16*sgn(1/(x*e + d))
+ 24*a^3*b^3*c*d*e^16*sgn(1/(x*e + d)) - 48*a^4*b*c^2*d*e^16*sgn(1/(x*e + d)) + a^3*b^4*e^17*sgn(1/(x*e + d))
- 8*a^4*b^2*c*e^17*sgn(1/(x*e + d)) + 16*a^5*c^2*e^17*sgn(1/(x*e + d)))*(x*e + d)))*e^(-1)/(x*e + d) - 3*(32*
c^6*d^6*e^15 - 96*b*c^5*d^5*e^16 + 80*b^2*c^4*d^4*e^17 + 160*a*c^5*d^4*e^17 - 320*a*b*c^4*d^3*e^18 - 46*b^4*c^
2*d^2*e^19 + 368*a*b^2*c^3*d^2*e^19 - 256*a^2*c...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)),x)

[Out]

int(1/((d + e*x)^2*(a + b*x + c*x^2)^(5/2)), x)

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