3.5.84 \(\int \frac {1}{x^2 (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [484]

Optimal. Leaf size=394 \[ -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}+\frac {\left (3 c d^2+5 a e^2\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 a^{5/2} d^{7/2} e^{5/2}} \]

[Out]

-2/3*e*(c*d*x+a*e)/d/(-a*e^2+c*d^2)/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/2*(5*a*e^2+3*c*d^2)*arctanh(1/
2*(2*a*d*e+(a*e^2+c*d^2)*x)/a^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/a^(5/2)/d^(7/2)/e
^(5/2)+2/3*(3*c^3*d^6+a*c^2*d^4*e^2+9*a^2*c*d^2*e^4-5*a^3*e^6+c*d*e*(-5*a^2*e^4+10*a*c*d^2*e^2+3*c^2*d^4)*x)/a
/d^2/e/(-a*e^2+c*d^2)^3/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/3*(-15*a^3*e^6+31*a^2*c*d^2*e^4-9*a*c^2*d^
4*e^2+9*c^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^3/e^2/(-a*e^2+c*d^2)^3/x

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Rubi [A]
time = 0.36, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {865, 836, 820, 738, 212} \begin {gather*} \frac {\left (5 a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{5/2} d^{7/2} e^{5/2}}+\frac {2 \left (-5 a^3 e^6+c d e x \left (-5 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+9 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-15 a^3 e^6+31 a^2 c d^2 e^4-9 a c^2 d^4 e^2+9 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x \left (c d^2-a e^2\right )^3}-\frac {2 e (a e+c d x)}{3 d x \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) + (2*(3*c^3*d^6 + a
*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 5*a^3*e^6 + c*d*e*(3*c^2*d^4 + 10*a*c*d^2*e^2 - 5*a^2*e^4)*x))/(3*a*d^2*e*(c*
d^2 - a*e^2)^3*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((9*c^3*d^6 - 9*a*c^2*d^4*e^2 + 31*a^2*c*d^2*e
^4 - 15*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x) + ((3*c*d^2
+ 5*a*e^2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2])])/(2*a^(5/2)*d^(7/2)*e^(5/2))

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 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 865

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + p))/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] &&
NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[m, 0] && In
tegerQ[n] && (LtQ[n, 0] || GtQ[p, 0])

Rubi steps

\begin {align*} \int \frac {1}{x^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\int \frac {a e+c d x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 \int \frac {-\frac {1}{2} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2-a e^2\right )+3 a c d e^2 \left (c d^2-a e^2\right ) x}{x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 a d e \left (c d^2-a e^2\right )^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 \int \frac {\frac {1}{4} a e \left (c d^2-a e^2\right ) \left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right )+\frac {1}{2} a c d e^2 \left (c d^2-a e^2\right ) \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a^2 d^2 e^2 \left (c d^2-a e^2\right )^4}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}-\frac {\left (3 c d^2+5 a e^2\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 a^2 d^3 e^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}+\frac {\left (3 c d^2+5 a e^2\right ) \text {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^2 d^3 e^2}\\ &=-\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+9 a^2 c d^2 e^4-5 a^3 e^6+c d e \left (3 c^2 d^4+10 a c d^2 e^2-5 a^2 e^4\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (9 c^3 d^6-9 a c^2 d^4 e^2+31 a^2 c d^2 e^4-15 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 d^3 e^2 \left (c d^2-a e^2\right )^3 x}+\frac {\left (3 c d^2+5 a e^2\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 a^{5/2} d^{7/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.55, size = 303, normalized size = 0.77 \begin {gather*} \frac {-\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (-9 c^4 d^7 x (d+e x)^2-3 a c^3 d^5 e (d-3 e x) (d+e x)^2+a^4 e^7 \left (3 d^2+20 d e x+15 e^2 x^2\right )+a^2 c^2 d^3 e^3 \left (9 d^3+9 d^2 e x-33 d e^2 x^2-31 e^3 x^3\right )-a^3 c d e^5 \left (9 d^3+39 d^2 e x+11 d e^2 x^2-15 e^3 x^3\right )\right )}{\left (-c d^2+a e^2\right )^3 x}+3 \left (3 c d^2+5 a e^2\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{3 a^{5/2} d^{7/2} e^{5/2} ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(-9*c^4*d^7*x*(d + e*x)^2 - 3*a*c^3*d^5*e*(d - 3*e*x)*(d + e*x)^2 +
a^4*e^7*(3*d^2 + 20*d*e*x + 15*e^2*x^2) + a^2*c^2*d^3*e^3*(9*d^3 + 9*d^2*e*x - 33*d*e^2*x^2 - 31*e^3*x^3) - a^
3*c*d*e^5*(9*d^3 + 39*d^2*e*x + 11*d*e^2*x^2 - 15*e^3*x^3)))/((-(c*d^2) + a*e^2)^3*x)) + 3*(3*c*d^2 + 5*a*e^2)
*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d*x])])/(3*
a^(5/2)*d^(7/2)*e^(5/2)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [A]
time = 0.10, size = 716, normalized size = 1.82

method	result	size
default	\(\frac {e \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{d^{2}}+\frac {-\frac {1}{a d e x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}\right )}{2 a d e}-\frac {4 c \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}}{d}-\frac {e \left (\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}\right )}{d^{2}}\)	\(716\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e/d^2*(-2/3/(a*e^2-c*d^2)/(x+d/e)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*c*d*e/(a*e^2-c*d^2)^3*(2*c
*d*e*(x+d/e)+a*e^2-c*d^2)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))+1/d*(-1/a/d/e/x/(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)-3/2*(a*e^2+c*d^2)/a/d/e*(1/a/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/a/d/
e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/a/d/e/(a*d
*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x))-4*c/a*(2*c*
d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))-e/d^2*(1/a/d/e/(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/a/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^
2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/a/d/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (355) = 710\).
time = 32.92, size = 1889, normalized size = 4.79 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/12*(3*(3*c^5*d^11*x^2 - 6*a^2*c^3*d^5*x^4*e^6 - 5*a^5*x^3*e^11 - 5*(a^4*c*d*x^4 + 2*a^5*d*x^2)*e^10 + (2*a^
4*c*d^2*x^3 - 5*a^5*d^2*x)*e^9 + (12*a^3*c^2*d^3*x^4 + 19*a^4*c*d^3*x^2)*e^8 + 6*(3*a^3*c^2*d^4*x^3 + 2*a^4*c*
d^4*x)*e^7 - 2*(8*a^2*c^3*d^6*x^3 + 3*a^3*c^2*d^6*x)*e^5 - 2*(2*a*c^4*d^7*x^4 + 7*a^2*c^3*d^7*x^2)*e^4 - (5*a*
c^4*d^8*x^3 + 4*a^2*c^3*d^8*x)*e^3 + (3*c^5*d^9*x^4 + 2*a*c^4*d^9*x^2)*e^2 + 3*(2*c^5*d^10*x^3 + a*c^4*d^10*x)
*e)*sqrt(a*d)*e^(1/2)*log((c^2*d^4*x^2 + 8*a*c*d^3*x*e + a^2*x^2*e^4 + 8*a^2*d*x*e^3 + 4*(c*d^2*x + a*x*e^2 +
2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(a*d)*e^(1/2) + 2*(3*a*c*d^2*x^2 + 4*a^2*d^2)*e^2)/x^
2) - 4*(9*a*c^4*d^10*x*e - 15*a^5*d*x^2*e^10 - 5*(3*a^4*c*d^2*x^3 + 4*a^5*d^2*x)*e^9 + (11*a^4*c*d^3*x^2 - 3*a
^5*d^3)*e^8 + (31*a^3*c^2*d^4*x^3 + 39*a^4*c*d^4*x)*e^7 + 3*(11*a^3*c^2*d^5*x^2 + 3*a^4*c*d^5)*e^6 - 9*(a^2*c^
3*d^6*x^3 + a^3*c^2*d^6*x)*e^5 - 3*(5*a^2*c^3*d^7*x^2 + 3*a^3*c^2*d^7)*e^4 + 3*(3*a*c^4*d^8*x^3 - a^2*c^3*d^8*
x)*e^3 + 3*(6*a*c^4*d^9*x^2 + a^2*c^3*d^9)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(a^3*c^4*d^13*x^2
*e^3 - a^7*d^4*x^3*e^12 - (a^6*c*d^5*x^4 + 2*a^7*d^5*x^2)*e^11 + (a^6*c*d^6*x^3 - a^7*d^6*x)*e^10 + (3*a^5*c^2
*d^7*x^4 + 5*a^6*c*d^7*x^2)*e^9 + 3*(a^5*c^2*d^8*x^3 + a^6*c*d^8*x)*e^8 - 3*(a^4*c^3*d^9*x^4 + a^5*c^2*d^9*x^2
)*e^7 - (5*a^4*c^3*d^10*x^3 + 3*a^5*c^2*d^10*x)*e^6 + (a^3*c^4*d^11*x^4 - a^4*c^3*d^11*x^2)*e^5 + (2*a^3*c^4*d
^12*x^3 + a^4*c^3*d^12*x)*e^4), -1/6*(3*(3*c^5*d^11*x^2 - 6*a^2*c^3*d^5*x^4*e^6 - 5*a^5*x^3*e^11 - 5*(a^4*c*d*
x^4 + 2*a^5*d*x^2)*e^10 + (2*a^4*c*d^2*x^3 - 5*a^5*d^2*x)*e^9 + (12*a^3*c^2*d^3*x^4 + 19*a^4*c*d^3*x^2)*e^8 +
6*(3*a^3*c^2*d^4*x^3 + 2*a^4*c*d^4*x)*e^7 - 2*(8*a^2*c^3*d^6*x^3 + 3*a^3*c^2*d^6*x)*e^5 - 2*(2*a*c^4*d^7*x^4 +
7*a^2*c^3*d^7*x^2)*e^4 - (5*a*c^4*d^8*x^3 + 4*a^2*c^3*d^8*x)*e^3 + (3*c^5*d^9*x^4 + 2*a*c^4*d^9*x^2)*e^2 + 3*
(2*c^5*d^10*x^3 + a*c^4*d^10*x)*e)*sqrt(-a*d*e)*arctan(1/2*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^
2 + (c*d*x^2 + a*d)*e)*sqrt(-a*d*e)/(a*c*d^3*x*e + a^2*d*x*e^3 + (a*c*d^2*x^2 + a^2*d^2)*e^2)) + 2*(9*a*c^4*d^
10*x*e - 15*a^5*d*x^2*e^10 - 5*(3*a^4*c*d^2*x^3 + 4*a^5*d^2*x)*e^9 + (11*a^4*c*d^3*x^2 - 3*a^5*d^3)*e^8 + (31*
a^3*c^2*d^4*x^3 + 39*a^4*c*d^4*x)*e^7 + 3*(11*a^3*c^2*d^5*x^2 + 3*a^4*c*d^5)*e^6 - 9*(a^2*c^3*d^6*x^3 + a^3*c^
2*d^6*x)*e^5 - 3*(5*a^2*c^3*d^7*x^2 + 3*a^3*c^2*d^7)*e^4 + 3*(3*a*c^4*d^8*x^3 - a^2*c^3*d^8*x)*e^3 + 3*(6*a*c^
4*d^9*x^2 + a^2*c^3*d^9)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(a^3*c^4*d^13*x^2*e^3 - a^7*d^4*x^3
*e^12 - (a^6*c*d^5*x^4 + 2*a^7*d^5*x^2)*e^11 + (a^6*c*d^6*x^3 - a^7*d^6*x)*e^10 + (3*a^5*c^2*d^7*x^4 + 5*a^6*c
*d^7*x^2)*e^9 + 3*(a^5*c^2*d^8*x^3 + a^6*c*d^8*x)*e^8 - 3*(a^4*c^3*d^9*x^4 + a^5*c^2*d^9*x^2)*e^7 - (5*a^4*c^3
*d^10*x^3 + 3*a^5*c^2*d^10*x)*e^6 + (a^3*c^4*d^11*x^4 - a^4*c^3*d^11*x^2)*e^5 + (2*a^3*c^4*d^12*x^3 + a^4*c^3*
d^12*x)*e^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x^2\,\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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