3.24.89 \(\int \frac {1}{(d+e x)^3 (a+b x+c x^2)^{3/2}} \, dx\) [2389]

Optimal. Leaf size=371 \[ -\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}-\frac {e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{7/2}} \]

[Out]

3/8*e^2*(16*c^2*d^2+5*b^2*e^2-4*c*e*(a*e+4*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)^(7/2)-2*(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)^(1/2)-1/2*e*(8*c^2*d^2+5*b^2*e^2-4*c*e*(3*a*e+2*b*d))*(c*x^2+b*x+a)^(1/
2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^2-1/4*e*(-b*e+2*c*d)*(8*c^2*d^2+15*b^2*e^2-4*c*e*(13*a*e+2*b*d))
*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)

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Rubi [A]
time = 0.30, antiderivative size = 371, 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, 848, 820, 738, 212} \begin {gather*} -\frac {e \sqrt {a+b x+c x^2} (2 c d-b e) \left (-4 c e (13 a e+2 b d)+15 b^2 e^2+8 c^2 d^2\right )}{4 \left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac {e \sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}+\frac {3 e^2 \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 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 )^{7/2}}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*Sqrt[a +
b*x + c*x^2]) - (e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/(2*(b^2 - 4*a*c)*(c
*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt
[a + b*x + c*x^2])/(4*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (3*e^2*(16*c^2*d^2 + 5*b^2*e^2 - 4*
c*e*(4*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)^(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 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 )^{3/2}} \, dx &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} e \left (4 b c d-5 b^2 e+12 a c e\right )+2 c e (2 c d-b e) x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx}{\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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}-\frac {e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\int \frac {\frac {1}{4} e \left (28 b^2 c d e-80 a c^2 d e-15 b^3 e^2-4 b c \left (2 c d^2-13 a e^2\right )\right )-\frac {1}{2} c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right ) \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 )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}-\frac {e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {\left (3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 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 )^3}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}-\frac {e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 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 )^3}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}-\frac {e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 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 )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 10.89, size = 353, normalized size = 0.95 \begin {gather*} \frac {2 \left (-\frac {e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x)^2 \sqrt {a+x (b+c x)}}+\frac {1}{16} e \left (-\frac {2 (2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+x (b+c x)}}{\left (c d^2+e (-b d+a e)\right )^2 (d+e x)}-\frac {3 \left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 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 )^{5/2}}\right )\right )}{\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)^(3/2)),x]

[Out]

(2*(-1/4*(e*(8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))
*(d + e*x)^2) + (b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^2*Sqrt[a + x*(b + c*x)]) + (e*((-2*(2*
c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2*b*d + 13*a*e))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e)
)^2*(d + e*x)) - (3*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*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))^(5/2)))/
16))/((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. \(1188\) vs. \(2(351)=702\).
time = 0.81, size = 1189, normalized size = 3.20

method	result	size
default	\(\frac {-\frac {e^{2}}{2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\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}}}}-\frac {5 e \left (b e -2 c d \right ) \left (-\frac {e^{2}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\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 {3 e \left (b e -2 c d \right ) \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 )}{2 \left (e^{2} a -b d e +c \,d^{2}\right )}-\frac {4 c \,e^{2} \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}}}}\right )}{4 \left (e^{2} a -b d e +c \,d^{2}\right )}-\frac {3 c \,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 )}{2 \left (e^{2} a -b d e +c \,d^{2}\right )}}{e^{3}}\)	\(1189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+b*x+a)^(3/2),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)^(1
/2)-5/4*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)^(1/2)-3/2*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/(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*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))-3/2*c/(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)^(3/2),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. 2802 vs. \(2 (364) = 728\).
time = 19.10, size = 5647, normalized size = 15.22 \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)^(3/2),x, algorithm="fricas")

[Out]

[-1/16*(3*sqrt(c*d^2 - b*d*e + a*e^2)*(((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^4 + (5*b^5 - 24*a*b^3*c + 16*a
^2*b*c^2)*x^3 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^2)*e^6 - 2*(8*(b^3*c^2 - 4*a*b*c^3)*d*x^4 + (3*b^4*c -
8*a*b^2*c^2 - 16*a^2*c^3)*d*x^3 - (5*b^5 - 32*a*b^3*c + 48*a^2*b*c^2)*d*x^2 - (5*a*b^4 - 24*a^2*b^2*c + 16*a^
3*c^2)*d*x)*e^5 + (16*(b^2*c^3 - 4*a*c^4)*d^2*x^4 - 16*(b^3*c^2 - 4*a*b*c^3)*d^2*x^3 - 3*(9*b^4*c - 40*a*b^2*c
^2 + 16*a^2*c^3)*d^2*x^2 + (5*b^5 - 56*a*b^3*c + 144*a^2*b*c^2)*d^2*x + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*
d^2)*e^4 + 16*(2*(b^2*c^3 - 4*a*c^4)*d^3*x^3 + (b^3*c^2 - 4*a*b*c^3)*d^3*x^2 - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^
3)*d^3*x - (a*b^3*c - 4*a^2*b*c^2)*d^3)*e^3 + 16*((b^2*c^3 - 4*a*c^4)*d^4*x^2 + (b^3*c^2 - 4*a*b*c^3)*d^4*x +
(a*b^2*c^2 - 4*a^2*c^3)*d^4)*e^2)*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*(16*c^5*d^7*x + 8*b*c^4*d^7
+ (2*a^3*b^2 - 8*a^4*c - (15*a*b^3*c - 52*a^2*b*c^2)*x^3 - (15*a*b^4 - 62*a^2*b^2*c + 24*a^3*c^2)*x^2 - 5*(a^2
*b^3 - 4*a^3*b*c)*x)*e^7 + ((15*b^4*c - 14*a*b^2*c^2 - 104*a^2*c^3)*d*x^3 + (15*b^5 - 49*a*b^3*c - 20*a^2*b*c^
2)*d*x^2 - 2*(10*a*b^4 - 47*a^2*b^2*c + 44*a^3*c^2)*d*x - 11*(a^2*b^3 - 4*a^3*b*c)*d)*e^6 - ((53*b^3*c^2 - 132
*a*b*c^3)*d^2*x^3 + 2*(14*b^4*c - 73*a*b^2*c^2 + 68*a^2*c^3)*d^2*x^2 - (25*b^5 - 63*a*b^3*c - 76*a^2*b*c^2)*d^
2*x - (a*b^4 + 14*a^2*b^2*c - 88*a^3*c^2)*d^2)*e^5 + (2*(31*b^2*c^3 - 44*a*c^4)*d^3*x^3 - (27*b^3*c^2 - 28*a*b
*c^3)*d^3*x^2 - (81*b^4*c - 274*a*b^2*c^2 + 40*a^2*c^3)*d^3*x + (8*b^5 - 33*a*b^3*c + 44*a^2*b*c^2)*d^3)*e^4 -
8*(5*b*c^4*d^4*x^3 - 10*(b^2*c^3 - a*c^4)*d^4*x^2 - 10*(b^3*c^2 - 3*a*b*c^3)*d^4*x + 4*(b^4*c - 3*a*b^2*c^2 +
a^2*c^3)*d^4)*e^3 + 8*(2*c^5*d^5*x^3 - 9*b*c^4*d^5*x^2 - 2*(b^2*c^3 - 4*a*c^4)*d^5*x + 2*(3*b^3*c^2 - 7*a*b*c
^3)*d^5)*e^2 + 8*(4*c^5*d^6*x^2 - 3*b*c^4*d^6*x - 2*(2*b^2*c^3 - 3*a*c^4)*d^6)*e)*sqrt(c*x^2 + b*x + a))/((b^2
*c^5 - 4*a*c^6)*d^10*x^2 + (b^3*c^4 - 4*a*b*c^5)*d^10*x + (a*b^2*c^4 - 4*a^2*c^5)*d^10 + ((a^4*b^2*c - 4*a^5*c
^2)*x^4 + (a^4*b^3 - 4*a^5*b*c)*x^3 + (a^5*b^2 - 4*a^6*c)*x^2)*e^10 - 2*(2*(a^3*b^3*c - 4*a^4*b*c^2)*d*x^4 + (
2*a^3*b^4 - 9*a^4*b^2*c + 4*a^5*c^2)*d*x^3 + (a^4*b^3 - 4*a^5*b*c)*d*x^2 - (a^5*b^2 - 4*a^6*c)*d*x)*e^9 + (2*(
3*a^2*b^4*c - 10*a^3*b^2*c^2 - 8*a^4*c^3)*d^2*x^4 + 2*(3*a^2*b^5 - 14*a^3*b^3*c + 8*a^4*b*c^2)*d^2*x^3 - (2*a^
3*b^4 - 13*a^4*b^2*c + 20*a^5*c^2)*d^2*x^2 - 7*(a^4*b^3 - 4*a^5*b*c)*d^2*x + (a^5*b^2 - 4*a^6*c)*d^2)*e^8 - 4*
((a*b^5*c - a^2*b^3*c^2 - 12*a^3*b*c^3)*d^3*x^4 + (a*b^6 - 4*a^2*b^4*c - 2*a^3*b^2*c^2 + 8*a^4*c^3)*d^3*x^3 -
2*(a^2*b^5 - 5*a^3*b^3*c + 4*a^4*b*c^2)*d^3*x^2 - 2*(a^3*b^4 - 3*a^4*b^2*c - 4*a^5*c^2)*d^3*x + (a^4*b^3 - 4*a
^5*b*c)*d^3)*e^7 + ((b^6*c + 8*a*b^4*c^2 - 42*a^2*b^2*c^3 - 24*a^3*c^4)*d^4*x^4 + (b^7 - 34*a^2*b^3*c^2 + 72*a
^3*b*c^3)*d^4*x^3 - (7*a*b^6 - 22*a^2*b^4*c - 34*a^3*b^2*c^2 + 40*a^4*c^3)*d^4*x^2 - 2*(a^2*b^5 + 6*a^3*b^3*c
- 40*a^4*b*c^2)*d^4*x + 2*(3*a^3*b^4 - 10*a^4*b^2*c - 8*a^5*c^2)*d^4)*e^6 - 2*(2*(b^5*c^2 - a*b^3*c^3 - 12*a^2
*b*c^4)*d^5*x^4 + (b^6*c - 10*a*b^4*c^2 + 18*a^2*b^2*c^3 + 24*a^3*c^4)*d^5*x^3 - (b^7 + 4*a*b^5*c - 38*a^2*b^3
*c^2 + 24*a^3*b*c^3)*d^5*x^2 + (a*b^6 - 10*a^2*b^4*c + 18*a^3*b^2*c^2 + 24*a^4*c^3)*d^5*x + 2*(a^2*b^5 - a^3*b
^3*c - 12*a^4*b*c^2)*d^5)*e^5 + (2*(3*b^4*c^3 - 10*a*b^2*c^4 - 8*a^2*c^5)*d^6*x^4 - 2*(b^5*c^2 + 6*a*b^3*c^3 -
40*a^2*b*c^4)*d^6*x^3 - (7*b^6*c - 22*a*b^4*c^2 - 34*a^2*b^2*c^3 + 40*a^3*c^4)*d^6*x^2 + (b^7 - 34*a^2*b^3*c^
2 + 72*a^3*b*c^3)*d^6*x + (a*b^6 + 8*a^2*b^4*c - 42*a^3*b^2*c^2 - 24*a^4*c^3)*d^6)*e^4 - 4*((b^3*c^4 - 4*a*b*c
^5)*d^7*x^4 - 2*(b^4*c^3 - 3*a*b^2*c^4 - 4*a^2*c^5)*d^7*x^3 - 2*(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*d^7*x^2
+ (b^6*c - 4*a*b^4*c^2 - 2*a^2*b^2*c^3 + 8*a^3*c^4)*d^7*x + (a*b^5*c - a^2*b^3*c^2 - 12*a^3*b*c^3)*d^7)*e^3 +
((b^2*c^5 - 4*a*c^6)*d^8*x^4 - 7*(b^3*c^4 - 4*a*b*c^5)*d^8*x^3 - (2*b^4*c^3 - 13*a*b^2*c^4 + 20*a^2*c^5)*d^8*x
^2 + 2*(3*b^5*c^2 - 14*a*b^3*c^3 + 8*a^2*b*c^4)*d^8*x + 2*(3*a*b^4*c^2 - 10*a^2*b^2*c^3 - 8*a^3*c^4)*d^8)*e^2
+ 2*((b^2*c^5 - 4*a*c^6)*d^9*x^3 - (b^3*c^4 - 4*a*b*c^5)*d^9*x^2 - (2*b^4*c^3 - 9*a*b^2*c^4 + 4*a^2*c^5)*d^9*x
- 2*(a*b^3*c^3 - 4*a^2*b*c^4)*d^9)*e), 1/8*(3*sqrt(-c*d^2 + b*d*e - a*e^2)*(((5*b^4*c - 24*a*b^2*c^2 + 16*a^2
*c^3)*x^4 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*x^3 + (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*x^2)*e^6 - 2*(8*(b
^3*c^2 - 4*a*b*c^3)*d*x^4 + (3*b^4*c - 8*a*b^2*c^2 - 16*a^2*c^3)*d*x^3 - (5*b^5 - 32*a*b^3*c + 48*a^2*b*c^2)*d
*x^2 - (5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*d*x)*e^5 + (16*(b^2*c^3 - 4*a*c^4)*d^2*x^4 - 16*(b^3*c^2 - 4*a*b*
c^3)*d^2*x^3 - 3*(9*b^4*c - 40*a*b^2*c^2 + 16*a^2*c^3)*d^2*x^2 + (5*b^5 - 56*a*b^3*c + 144*a^2*b*c^2)*d^2*x +
(5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2)*d^2)*e^4 + 16*(2*(b^2*c^3 - 4*a*c^4)*d^3*x^3 + (b^3*c^2 - 4*a*b*c^3)*d^3
*x^2 - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d^3*x ...

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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 {3}{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)**(3/2),x)

[Out]

Integral(1/((d + e*x)**3*(a + b*x + c*x**2)**(3/2)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2574 vs. \(2 (364) = 728\).
time = 0.99, size = 2574, normalized size = 6.94 \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)^(3/2),x, algorithm="giac")

[Out]

-2*((2*c^7*d^9 - 9*b*c^6*d^8*e + 18*b^2*c^5*d^7*e^2 - 21*b^3*c^4*d^6*e^3 + 15*b^4*c^3*d^5*e^4 + 6*a*b^2*c^4*d^
5*e^4 - 12*a^2*c^5*d^5*e^4 - 6*b^5*c^2*d^4*e^5 - 15*a*b^3*c^3*d^4*e^5 + 30*a^2*b*c^4*d^4*e^5 + b^6*c*d^3*e^6 +
12*a*b^4*c^2*d^3*e^6 - 18*a^2*b^2*c^3*d^3*e^6 - 16*a^3*c^4*d^3*e^6 - 3*a*b^5*c*d^2*e^7 - 3*a^2*b^3*c^2*d^2*e^
7 + 24*a^3*b*c^3*d^2*e^7 + 3*a^2*b^4*c*d*e^8 - 6*a^3*b^2*c^2*d*e^8 - 6*a^4*c^3*d*e^8 - a^3*b^3*c*e^9 + 3*a^4*b
*c^2*e^9)*x/(b^2*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^11*e + 24*a*b*c^6*d^11*e + 15*b^4*c^4*d^10*e^2 - 54*a*b
^2*c^5*d^10*e^2 - 24*a^2*c^6*d^10*e^2 - 20*b^5*c^3*d^9*e^3 + 50*a*b^3*c^4*d^9*e^3 + 120*a^2*b*c^5*d^9*e^3 + 15
*b^6*c^2*d^8*e^4 - 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5*d^8*e^4 - 6*b^7*c*d^7*e^5 - 36*a*b^5*c^2*d^7*e^5 + 180
*a^2*b^3*c^3*d^7*e^5 + 240*a^3*b*c^4*d^7*e^5 + b^8*d^6*e^6 + 26*a*b^6*c*d^6*e^6 - 30*a^2*b^4*c^2*d^6*e^6 - 340
*a^3*b^2*c^3*d^6*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5*e^7 - 36*a^2*b^5*c*d^5*e^7 + 180*a^3*b^3*c^2*d^5*e^7 +
240*a^4*b*c^3*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 225*a^4*b^2*c^2*d^4*e^8 - 60*a^5*c^3*d^4*e^8 - 20*a^3*b^5*d^3*e^
9 + 50*a^4*b^3*c*d^3*e^9 + 120*a^5*b*c^2*d^3*e^9 + 15*a^4*b^4*d^2*e^10 - 54*a^5*b^2*c*d^2*e^10 - 24*a^6*c^2*d^
2*e^10 - 6*a^5*b^3*d*e^11 + 24*a^6*b*c*d*e^11 + a^6*b^2*e^12 - 4*a^7*c*e^12) + (b*c^6*d^9 - 6*b^2*c^5*d^8*e +
6*a*c^6*d^8*e + 15*b^3*c^4*d^7*e^2 - 24*a*b*c^5*d^7*e^2 - 20*b^4*c^3*d^6*e^3 + 34*a*b^2*c^4*d^6*e^3 + 16*a^2*c
^5*d^6*e^3 + 15*b^5*c^2*d^5*e^4 - 15*a*b^3*c^3*d^5*e^4 - 54*a^2*b*c^4*d^5*e^4 - 6*b^6*c*d^4*e^5 - 9*a*b^4*c^2*
d^4*e^5 + 66*a^2*b^2*c^3*d^4*e^5 + 12*a^3*c^4*d^4*e^5 + b^7*d^3*e^6 + 11*a*b^5*c*d^3*e^6 - 31*a^2*b^3*c^2*d^3*
e^6 - 32*a^3*b*c^3*d^3*e^6 - 3*a*b^6*d^2*e^7 + 30*a^3*b^2*c^2*d^2*e^7 + 3*a^2*b^5*d*e^8 - 9*a^3*b^3*c*d*e^8 -
3*a^4*b*c^2*d*e^8 - a^3*b^4*e^9 + 4*a^4*b^2*c*e^9 - 2*a^5*c^2*e^9)/(b^2*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^
11*e + 24*a*b*c^6*d^11*e + 15*b^4*c^4*d^10*e^2 - 54*a*b^2*c^5*d^10*e^2 - 24*a^2*c^6*d^10*e^2 - 20*b^5*c^3*d^9*
e^3 + 50*a*b^3*c^4*d^9*e^3 + 120*a^2*b*c^5*d^9*e^3 + 15*b^6*c^2*d^8*e^4 - 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5
*d^8*e^4 - 6*b^7*c*d^7*e^5 - 36*a*b^5*c^2*d^7*e^5 + 180*a^2*b^3*c^3*d^7*e^5 + 240*a^3*b*c^4*d^7*e^5 + b^8*d^6*
e^6 + 26*a*b^6*c*d^6*e^6 - 30*a^2*b^4*c^2*d^6*e^6 - 340*a^3*b^2*c^3*d^6*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5
*e^7 - 36*a^2*b^5*c*d^5*e^7 + 180*a^3*b^3*c^2*d^5*e^7 + 240*a^4*b*c^3*d^5*e^7 + 15*a^2*b^6*d^4*e^8 - 225*a^4*b
^2*c^2*d^4*e^8 - 60*a^5*c^3*d^4*e^8 - 20*a^3*b^5*d^3*e^9 + 50*a^4*b^3*c*d^3*e^9 + 120*a^5*b*c^2*d^3*e^9 + 15*a
^4*b^4*d^2*e^10 - 54*a^5*b^2*c*d^2*e^10 - 24*a^6*c^2*d^2*e^10 - 6*a^5*b^3*d*e^11 + 24*a^6*b*c*d*e^11 + a^6*b^2
*e^12 - 4*a^7*c*e^12))/sqrt(c*x^2 + b*x + a) + 3/4*(16*c^2*d^2*e^2 - 16*b*c*d*e^3 + 5*b^2*e^4 - 4*a*c*e^4)*arc
tan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*b*c^2*d^5
*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3
*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) - 1/4*(56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*
d^3*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e^3 + 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*
d^3*e^2 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*d^2*e^3 + 14*b^2*c^(3/2)*d^3*e^2 - 24*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*b*c*d*e^4 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c*d^2*e^3 - 88*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))*a*c^2*d^2*e^3 + 13*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*d*e^4 - 28*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*d*e^4 - 7*b^3*sqrt(c)*d^2*e^3 - 44*a*b*c^(3/2)*d^2*e^3 + 7*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^3*b^2*e^5 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*e^5 + 9*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*b^3*d*e^4 + 60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c*d*e^4 + 8*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^2*a*b*sqrt(c)*e^5 + 23*a*b^2*sqrt(c)*d*e^4 + 28*a^2*c^(3/2)*d*e^4 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*a*b^2*e^5 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*e^5 - 16*a^2*b*sqrt(c)*e^5)/((c^3*d^6 - 3*b*c^2*d
^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 -
3*a^2*b*d*e^5 + a^3*e^6)*((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqr
t(c)*d + b*d - a*e)^2)

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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 )}^{3/2}} \,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)^(3/2)),x)

[Out]

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

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