3.24.99 \(\int \frac {1}{(d+e x)^3 (a+b x+c x^2)^{5/2}} \, dx\) [2399]

Optimal. Leaf size=621 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{9/2}} \]

[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)^2/(c*x^2+b*x+a)^(3/2)+5/8
*e^4*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*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)^(9/2)-2/3*(8*a*c*e*(-b*e+2*c*d)^2-(2*a*c*e-b^2*e+b*c*d)*(20*a*c*e^2-
7*b^2*e^2+8*c^2*d^2)-c*(-b*e+2*c*d)*(8*c^2*d^2-7*b^2*e^2-4*c*e*(-9*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)^2/(c*x^2+b*x+a)^(1/2)+1/6*e*(64*c^4*d^4-35*b^4*e^4-128*c^3*d^2*e*(-3*a*e+b*d)-48*a*c^2*e^3*(5
*a*e+8*b*d)+8*b^2*c*e^3*(27*a*e+8*b*d))*(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)^2+1/1
2*e*(-b*e+2*c*d)*(64*c^4*d^4-105*b^4*e^4-64*c^3*d^2*e*(-7*a*e+2*b*d)+40*b^2*c*e^3*(19*a*e+2*b*d)-16*c^2*e^2*(8
1*a^2*e^2+28*a*b*d*e+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)^4/(e*x+d)

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 621, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {754, 836, 848, 820, 738, 212} \begin {gather*} \frac {e \sqrt {a+b x+c x^2} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{12 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^4}-\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (20 a c e^2-7 b^2 e^2+8 c^2 d^2\right )+8 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac {5 e^4 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\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 )}{8 \left (a e^2-b d e+c d^2\right )^{9/2}}-\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)^2 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {e \sqrt {a+b x+c x^2} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{6 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(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)^2*(a + b
*x + c*x^2)^(3/2)) - (2*(8*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 7*b^2*e^2 + 20*a*c*e
^2) - c*(2*c*d - b*e)*(8*c^2*d^2 - 7*b^2*e^2 - 4*c*e*(2*b*d - 9*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e +
a*e^2)^2*(d + e*x)^2*Sqrt[a + b*x + c*x^2]) + (e*(64*c^4*d^4 - 35*b^4*e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a
*c^2*e^3*(8*b*d + 5*a*e) + 8*b^2*c*e^3*(8*b*d + 27*a*e))*Sqrt[a + b*x + c*x^2])/(6*(b^2 - 4*a*c)^2*(c*d^2 - b*
d*e + a*e^2)^3*(d + e*x)^2) + (e*(2*c*d - b*e)*(64*c^4*d^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2*b*d - 7*a*e) + 40*b
^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b*d*e + 81*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(12*(b^2 -
4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)) + (5*e^4*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*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])])/(8*(c*d^2 - b*d*e + a*
e^2)^(9/2))

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])

Rule 848

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 {1}{(d+e x)^3 \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)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )+4 c e (2 c d-b e) x}{(d+e x)^3 \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)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 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^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 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)^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} e \left (8 c e (2 c d-b e) \left (b^2 d-12 a c d+4 a b e\right )+\left (4 b c d-5 b^2 e+12 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )\right )+c e (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x}{(d+e x)^3 \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)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 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^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 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)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {2 \int \frac {\frac {1}{8} \left (-220 b^4 c d e^4+105 b^5 e^5+8 b^3 c e^3 \left (6 c d^2-95 a e^2\right )+64 a c^3 d e^2 \left (2 c d^2-33 a e^2\right )+96 b^2 c^2 d e^2 \left (c d^2+16 a e^2\right )-16 b c^2 e \left (4 c^2 d^4+36 a c d^2 e^2-81 a^2 e^4\right )\right )-\frac {1}{4} c e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 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 )^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)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 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^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 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)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {\left (5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\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)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 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^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 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)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}-\frac {\left (5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\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 )}{4 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\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)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 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^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 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)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\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 )}{8 \left (c d^2-b d e+a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 11.75, size = 626, normalized size = 1.01 \begin {gather*} \frac {2 \left (\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x)^2 (a+x (b+c x))^{3/2}}+\frac {-7 b^4 e^3+7 b^3 c e^2 (d-e x)-4 b c^2 \left (a e^2 (13 d-9 e x)+2 c d^2 (d-3 e x)\right )+2 b^2 c e \left (21 a e^2+c d (4 d+3 e x)\right )-8 c^2 \left (5 a^2 e^3+2 c^2 d^3 x+a c d e (-2 d+9 e x)\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (d+e x)^2 \sqrt {a+x (b+c x)}}+\frac {e \left (\frac {4 \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+x (b+c x)}}{(d+e x)^2}+\frac {2 (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+x (b+c x)}}{\left (c d^2+e (-b d+a e)\right ) (d+e x)}-\frac {15 \left (b^2-4 a c\right )^2 e^3 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \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 )}{\left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{16 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^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)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*((b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^2*(a + x*(b + c*x))^(3/2)) + (-7*b^4*e^3 + 7*b^3*c
*e^2*(d - e*x) - 4*b*c^2*(a*e^2*(13*d - 9*e*x) + 2*c*d^2*(d - 3*e*x)) + 2*b^2*c*e*(21*a*e^2 + c*d*(4*d + 3*e*x
)) - 8*c^2*(5*a^2*e^3 + 2*c^2*d^3*x + a*c*d*e*(-2*d + 9*e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d +
e*x)^2*Sqrt[a + x*(b + c*x)]) + (e*((4*(64*c^4*d^4 - 35*b^4*e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a*c^2*e^3*(
8*b*d + 5*a*e) + 8*b^2*c*e^3*(8*b*d + 27*a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^2 + (2*(2*c*d - b*e)*(64*c^4*d
^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2*b*d - 7*a*e) + 40*b^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b
*d*e + 81*a^2*e^2))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) - (15*(b^2 - 4*a*c)^2*e^3*(2
4*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*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)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(16*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) +
a*e))^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. \(2042\) vs. \(2(595)=1190\).
time = 0.82, size = 2043, normalized size = 3.29

method	result	size
default	\(\text {Expression too large to display}\)	\(2043\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e^3*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/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)-7/4*e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^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)))-5/2*c/(a*e^2-b*d*e+c*d^2)*e^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)))))

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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)^3/(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. 7138 vs. \(2 (612) = 1224\).
time = 87.87, size = 14319, normalized size = 23.06 \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)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(15*sqrt(c*d^2 - b*d*e + a*e^2)*(((7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*x^6 + 2*(7*
b^7*c - 60*a*b^5*c^2 + 144*a^2*b^3*c^3 - 64*a^3*b*c^4)*x^5 + (7*b^8 - 46*a*b^6*c + 24*a^2*b^4*c^2 + 224*a^3*b^
2*c^3 - 128*a^4*c^4)*x^4 + 2*(7*a*b^7 - 60*a^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4*b*c^3)*x^3 + (7*a^2*b^6 - 60*a
^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*x^2)*e^8 - 2*(12*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d*x^6 + (17*b
^6*c^2 - 132*a*b^4*c^3 + 240*a^2*b^2*c^4 + 64*a^3*c^5)*d*x^5 - 2*(b^7*c - 24*a*b^5*c^2 + 144*a^2*b^3*c^3 - 256
*a^3*b*c^4)*d*x^4 - (7*b^8 - 70*a*b^6*c + 216*a^2*b^4*c^2 - 160*a^3*b^2*c^3 - 128*a^4*c^4)*d*x^3 - 2*(7*a*b^7
- 66*a^2*b^5*c + 192*a^3*b^3*c^2 - 160*a^4*b*c^3)*d*x^2 - (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5
*c^3)*d*x)*e^7 + (24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^2*x^6 - (65*b^6*c^2 - 564*a*b^4*c^3 + 1392*a^2*b^2
*c^4 - 704*a^3*c^5)*d^2*x^4 - 2*(17*b^7*c - 108*a*b^5*c^2 + 48*a^2*b^3*c^3 + 448*a^3*b*c^4)*d^2*x^3 + (7*b^8 -
142*a*b^6*c + 816*a^2*b^4*c^2 - 1504*a^3*b^2*c^3 + 256*a^4*c^4)*d^2*x^2 + 14*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b
^3*c^2 - 64*a^4*b*c^3)*d^2*x + (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*d^2)*e^6 + 24*(2*(b^4
*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^3*x^5 + 3*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^3*x^4 + 4*(a*b^4*c^3 - 8
*a^2*b^2*c^4 + 16*a^3*c^5)*d^3*x^3 - (b^7*c - 10*a*b^5*c^2 + 32*a^2*b^3*c^3 - 32*a^3*b*c^4)*d^3*x^2 - 2*(a*b^6
*c - 9*a^2*b^4*c^2 + 24*a^3*b^2*c^3 - 16*a^4*c^4)*d^3*x - (a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^3)*e^5
+ 24*((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^4*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^4*x^3 + (b^6*c
^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^4*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^4*x + (a^2*b^4*c^2 - 8
*a^3*b^2*c^3 + 16*a^4*c^4)*d^4)*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*(128*c^8*d^9*x^3 + 192*b*
c^7*d^9*x^2 + 48*(b^2*c^6 + 4*a*c^7)*d^9*x - 8*(b^3*c^5 - 12*a*b*c^6)*d^9 - (6*a^4*b^4 - 48*a^5*b^2*c + 96*a^6
*c^2 - (105*a*b^5*c^2 - 760*a^2*b^3*c^3 + 1296*a^3*b*c^4)*x^5 - 6*(35*a*b^6*c - 265*a^2*b^4*c^2 + 504*a^3*b^2*
c^3 - 80*a^4*c^4)*x^4 - 3*(35*a*b^7 - 230*a^2*b^5*c + 232*a^3*b^3*c^2 + 448*a^4*b*c^3)*x^3 - 4*(35*a^2*b^6 - 2
79*a^3*b^4*c + 588*a^4*b^2*c^2 - 160*a^5*c^3)*x^2 - 21*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x)*e^9 - ((105*b
^6*c^2 - 470*a*b^4*c^3 - 672*a^2*b^2*c^4 + 2592*a^3*c^5)*d*x^5 + (210*b^7*c - 1185*a*b^5*c^2 + 152*a^2*b^3*c^3
+ 3696*a^3*b*c^4)*d*x^4 + 3*(35*b^8 - 250*a*b^6*c + 516*a^2*b^4*c^2 - 848*a^3*b^2*c^3 + 1536*a^4*c^4)*d*x^3 -
(35*a*b^7 - 402*a^2*b^5*c + 1608*a^3*b^3*c^2 - 2176*a^4*b*c^3)*d*x^2 - (217*a^2*b^6 - 1770*a^3*b^4*c + 4032*a
^4*b^2*c^2 - 1952*a^5*c^3)*d*x - 45*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d)*e^8 + ((395*b^5*c^3 - 2552*a*b^3
*c^4 + 2544*a^2*b*c^5)*d^2*x^5 + (615*b^6*c^2 - 4598*a*b^4*c^3 + 7680*a^2*b^2*c^4 - 2784*a^3*c^5)*d^2*x^4 + (4
5*b^7*c - 582*a*b^5*c^2 + 2072*a^2*b^3*c^3 - 3264*a^3*b*c^4)*d^2*x^3 - (175*b^8 - 1282*a*b^6*c + 2724*a^2*b^4*
c^2 - 3344*a^3*b^2*c^3 + 4736*a^4*c^4)*d^2*x^2 - (182*a*b^7 - 951*a^2*b^5*c - 624*a^3*b^3*c^2 + 5360*a^4*b*c^3
)*d^2*x + (41*a^2*b^6 - 390*a^3*b^4*c + 1296*a^4*b^2*c^2 - 1696*a^5*c^3)*d^2)*e^7 - (2*(233*b^4*c^4 - 1704*a*b
^2*c^5 + 848*a^2*c^6)*d^3*x^5 + (269*b^5*c^3 - 2888*a*b^3*c^4 + 3216*a^2*b*c^5)*d^3*x^4 - 4*(201*b^6*c^2 - 139
3*a*b^4*c^3 + 2340*a^2*b^2*c^4 - 384*a^3*c^5)*d^3*x^3 - (551*b^7*c - 3894*a*b^5*c^2 + 5992*a^2*b^3*c^3 - 1408*
a^3*b*c^4)*d^3*x^2 + 2*(28*b^8 - 535*a*b^6*c + 2805*a^2*b^4*c^2 - 4064*a^3*b^2*c^3 - 208*a^4*c^4)*d^3*x + (88*
a*b^7 - 543*a^2*b^5*c + 480*a^3*b^3*c^2 + 688*a^4*b*c^3)*d^3)*e^6 + 8*(10*(b^3*c^5 - 32*a*b*c^6)*d^4*x^5 - (79
*b^4*c^4 - 192*a*b^2*c^5 + 64*a^2*c^6)*d^4*x^4 - 7*(23*b^5*c^3 - 167*a*b^3*c^4 + 204*a^2*b*c^5)*d^4*x^3 - 2*(2
2*b^6*c^2 - 160*a*b^4*c^3 + 57*a^2*b^2*c^4 + 28*a^3*c^5)*d^4*x^2 + (29*b^7*c - 294*a*b^5*c^2 + 1015*a^2*b^3*c^
3 - 1156*a^3*b*c^4)*d^4*x + (b^8 + 38*a*b^6*c - 300*a^2*b^4*c^2 + 430*a^3*b^2*c^3 + 56*a^4*c^4)*d^4)*e^5 + 8*(
4*(13*b^2*c^6 + 32*a*c^7)*d^5*x^5 + 14*(7*b^3*c^5 - 32*a*b*c^6)*d^5*x^4 + (25*b^4*c^4 - 678*a*b^2*c^5 + 520*a^
2*c^6)*d^5*x^3 - (55*b^5*c^3 - 235*a*b^3*c^4 + 732*a^2*b*c^5)*d^5*x^2 - 10*(4*b^6*c^2 - 31*a*b^4*c^3 + 57*a^2*
b^2*c^4 - 44*a^3*c^5)*d^5*x - 5*(b^7*c + 6*a*b^5*c^2 - 85*a^2*b^3*c^3 + 124*a^3*b*c^4)*d^5)*e^4 - 8*(56*b*c^7*
d^6*x^5 - 4*(5*b^2*c^6 + 64*a*c^7)*d^6*x^4 - 5*(29*b^3*c^5 - 4*a*b*c^6)*d^6*x^3 - (65*b^4*c^4 - 330*a*b^2*c^5
+ 504*a^2*c^6)*d^6*x^2 - (10*b^5*c^3 - 145*a*b^3*c^4 - 84*a^2*b*c^5)*d^6*x - 10*(b^6*c^2 - 4*a*b^4*c^3 - 21*a^
2*b^2*c^4 + 28*a^3*c^5)*d^6)*e^3 + 8*(16*c^8*d^7*x^5 - 88*b*c^7*d^7*x^4 - 2*(55*b^2*c^6 - 76*a*c^7)*d^7*x^3 +
(35*b^3*c^5 + 36*a*b*c^6)*d^7*x^2 + (25*b^4*c^4 + 54*a*b^2*c^5 + 168*a^2*c^6)*d^7*x - (10*b^5*c^3 - 85*a*b^3*c
^4 + 12*a^2*b*c^5)*d^7)*e^2 + 8*(32*c^8*d^8*x^4...

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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 )^{3} \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)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d + e*x)**3*(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. 24663 vs. \(2 (612) = 1224\).
time = 3.75, size = 24663, normalized size = 39.71 \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)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

5/4*(24*c^2*d^2*e^4 - 24*b*c*d*e^5 + 7*b^2*e^6 - 4*a*c*e^6)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + s
qrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^
3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12
*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 + a^4*e^8)*sqrt(-c*d^2 + b*d*e - a*e^2)
) + 2/3*((((16*c^19*d^29 - 232*b*c^18*d^28*e + 1548*b^2*c^17*d^27*e^2 + 304*a*c^18*d^27*e^2 - 6282*b^3*c^16*d^
26*e^3 - 4104*a*b*c^17*d^26*e^3 + 17220*b^4*c^15*d^25*e^4 + 25572*a*b^2*c^16*d^25*e^4 + 2208*a^2*c^17*d^25*e^4
- 33315*b^5*c^14*d^24*e^5 - 97350*a*b^3*c^15*d^24*e^5 - 27600*a^2*b*c^16*d^24*e^5 + 45608*b^6*c^13*d^23*e^6 +
252264*a*b^4*c^14*d^23*e^6 + 159144*a^2*b^2*c^15*d^23*e^6 + 8608*a^3*c^16*d^23*e^6 - 41492*b^7*c^12*d^22*e^7
- 468096*a*b^5*c^13*d^22*e^7 - 560556*a^2*b^3*c^14*d^22*e^7 - 98992*a^3*b*c^15*d^22*e^7 + 17424*b^8*c^11*d^21*
e^8 + 634040*a*b^6*c^12*d^21*e^8 + 1344816*a^2*b^4*c^13*d^21*e^8 + 524568*a^3*b^2*c^14*d^21*e^8 + 19888*a^4*c^
15*d^21*e^8 + 14135*b^9*c^10*d^20*e^9 - 620334*a*b^7*c^11*d^20*e^9 - 2315082*a^2*b^5*c^12*d^20*e^9 - 1696772*a
^3*b^3*c^13*d^20*e^9 - 208824*a^4*b*c^14*d^20*e^9 - 34628*b^10*c^9*d^19*e^10 + 409860*a*b^8*c^10*d^19*e^10 + 2
924460*a^2*b^6*c^11*d^19*e^10 + 3736040*a^3*b^4*c^12*d^19*e^10 + 1011780*a^4*b^2*c^13*d^19*e^10 + 25872*a^5*c^
14*d^19*e^10 + 35910*b^11*c^8*d^18*e^11 - 132088*a*b^9*c^9*d^18*e^11 - 2704878*a^2*b^7*c^10*d^18*e^11 - 589881
6*a^3*b^5*c^11*d^18*e^11 - 2999150*a^4*b^3*c^12*d^18*e^11 - 245784*a^5*b*c^13*d^18*e^11 - 24540*b^12*c^7*d^17*
e^12 - 57420*a*b^10*c^8*d^17*e^12 + 1762992*a^2*b^8*c^9*d^17*e^12 + 6826644*a^3*b^6*c^10*d^17*e^12 + 6064740*a
^4*b^4*c^11*d^17*e^12 + 1093356*a^5*b^2*c^12*d^17*e^12 + 8448*a^6*c^13*d^17*e^12 + 11883*b^13*c^6*d^16*e^13 +
108222*a*b^11*c^7*d^16*e^13 - 702372*a^2*b^9*c^8*d^16*e^13 - 5776056*a^3*b^7*c^9*d^16*e^13 - 8797041*a^4*b^5*c
^10*d^16*e^13 - 3026034*a^5*b^3*c^11*d^16*e^13 - 71808*a^6*b*c^12*d^16*e^13 - 4080*b^14*c^5*d^15*e^14 - 75888*
a*b^12*c^6*d^15*e^14 + 44880*a^2*b^10*c^7*d^15*e^14 + 3446784*a^3*b^8*c^8*d^15*e^14 + 9317088*a^4*b^6*c^9*d^15
*e^14 + 5789520*a^5*b^4*c^10*d^15*e^14 + 350064*a^6*b^2*c^11*d^15*e^14 - 35904*a^7*c^12*d^15*e^14 + 952*b^15*c
^4*d^14*e^15 + 32640*a*b^13*c^5*d^14*e^15 + 144840*a^2*b^11*c^6*d^14*e^15 - 1286560*a^3*b^9*c^7*d^14*e^15 - 71
35920*a^4*b^7*c^8*d^14*e^15 - 7970688*a^5*b^5*c^9*d^14*e^15 - 1189320*a^6*b^3*c^10*d^14*e^15 + 269280*a^7*b*c^
11*d^14*e^15 - 136*b^16*c^3*d^13*e^16 - 8976*a*b^14*c^4*d^13*e^16 - 102816*a^2*b^12*c^5*d^13*e^16 + 146608*a^3
*b^10*c^6*d^13*e^16 + 3769920*a^4*b^8*c^7*d^13*e^16 + 7916832*a^5*b^6*c^8*d^13*e^16 + 2764608*a^6*b^4*c^9*d^13
*e^16 - 780912*a^7*b^2*c^10*d^13*e^16 - 80784*a^8*c^11*d^13*e^16 + 9*b^17*c^2*d^12*e^17 + 1462*a*b^15*c^3*d^12
*e^17 + 36414*a^2*b^13*c^4*d^12*e^17 + 129948*a^3*b^11*c^5*d^12*e^17 - 1191190*a^4*b^9*c^6*d^12*e^17 - 5513508
*a^5*b^7*c^7*d^12*e^17 - 4288284*a^6*b^5*c^8*d^12*e^17 + 991848*a^7*b^3*c^9*d^12*e^17 + 525096*a^8*b*c^10*d^12
*e^17 - 108*a*b^16*c^2*d^11*e^18 - 7044*a^2*b^14*c^3*d^11*e^18 - 79912*a^3*b^12*c^4*d^11*e^18 + 89628*a^4*b^10
*c^5*d^11*e^18 + 2500344*a^5*b^8*c^6*d^11*e^18 + 4359432*a^6*b^6*c^7*d^11*e^18 - 121968*a^7*b^4*c^8*d^11*e^18
- 1365804*a^8*b^2*c^9*d^11*e^18 - 93104*a^9*c^10*d^11*e^18 + 594*a^2*b^15*c^2*d^10*e^19 + 19888*a^3*b^13*c^3*d
^10*e^19 + 90486*a^4*b^11*c^4*d^10*e^19 - 595320*a^5*b^9*c^5*d^10*e^19 - 2798004*a^6*b^7*c^6*d^10*e^19 - 12545
28*a^7*b^5*c^7*d^10*e^19 + 1735866*a^8*b^3*c^8*d^10*e^19 + 512072*a^9*b*c^9*d^10*e^19 - 1980*a^3*b^14*c^2*d^9*
e^20 - 35860*a^4*b^12*c^3*d^9*e^20 - 8844*a^5*b^10*c^4*d^9*e^20 + 1021680*a^6*b^8*c^5*d^9*e^20 + 1661880*a^7*b
^6*c^6*d^9*e^20 - 924660*a^8*b^4*c^7*d^9*e^20 - 1106820*a^9*b^2*c^8*d^9*e^20 - 69344*a^10*c^9*d^9*e^20 + 4455*
a^4*b^13*c^2*d^8*e^21 + 41382*a^5*b^11*c^3*d^8*e^21 - 138468*a^6*b^9*c^4*d^8*e^21 - 957528*a^7*b^7*c^5*d^8*e^2
1 - 193941*a^8*b^5*c^6*d^8*e^21 + 1140150*a^9*b^3*c^7*d^8*e^21 + 312048*a^10*b*c^8*d^8*e^21 - 7128*a^5*b^12*c^
2*d^7*e^22 - 26664*a^6*b^10*c^3*d^7*e^22 + 234432*a^7*b^8*c^4*d^7*e^22 + 488664*a^8*b^6*c^5*d^7*e^22 - 479160*
a^9*b^4*c^6*d^7*e^22 - 528792*a^10*b^2*c^7*d^7*e^22 - 34656*a^11*c^8*d^7*e^22 + 8316*a^6*b^11*c^2*d^6*e^23 + 5
28*a^7*b^9*c^3*d^6*e^23 - 206316*a^8*b^7*c^4*d^6*e^23 - 59136*a^9*b^5*c^5*d^6*e^23 + 394548*a^10*b^3*c^6*d^6*e
^23 + 121296*a^11*b*c^7*d^6*e^23 - 7128*a^7*b^10*c^2*d^5*e^24 + 17424*a^8*b^8*c^3*d^5*e^24 + 106568*a^9*b^6*c^
4*d^5*e^24 - 92400*a^10*b^4*c^5*d^5*e^24 - 148008*a^11*b^2*c^6*d^5*e^24 - 11312*a^12*c^7*d^5*e^24 + 4455*a^8*b
^9*c^2*d^4*e^25 - 18590*a^9*b^7*c^3*d^4*e^25 - 27258*a^10*b^5*c^4*d^4*e^25 + 66780*a^11*b^3*c^5*d^4*e^25 + 282
80*a^12*b*c^6*d^4*e^25 - 1980*a^9*b^8*c^2*d^3*e^26 + 10604*a^10*b^6*c^3*d^3*e^26 - 1656*a^11*b^4*c^4*d^3*e^26
- 21156*a^12*b^2*c^5*d^3*e^26 - 2192*a^13*c^6*d^3*e^26 + 594*a^10*b^7*c^2*d^2*e^27 - 3648*a^11*b^5*c^3*d^2*e^2
7 + 3454*a^12*b^3*c^4*d^2*e^27 + 3288*a^13*b*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 )}^3\,{\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)^3*(a + b*x + c*x^2)^(5/2)),x)

[Out]

int(1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)), x)

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