3.5.83 \(\int \frac {1}{x (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [483]

Optimal. Leaf size=271 \[ -\frac {2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \left (3 c^3 d^6+a c^2 d^4 e^2+7 a^2 c d^2 e^4-3 a^3 e^6+c d e \left (3 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right ) x\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{3/2} d^{5/2} e^{3/2}} \]

[Out]

-2/3*e*(c*d*x+a*e)/d/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-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^(3/2)/d^(5/2)/e^(3/2)+2/3*(3*c^3*d^6+a*
c^2*d^4*e^2+7*a^2*c*d^2*e^4-3*a^3*e^6+c*d*e*(-a*e^2+3*c*d^2)*(3*a*e^2+c*d^2)*x)/a/d^2/e/(-a*e^2+c*d^2)^3/(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 271, 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, 12, 738, 212} \begin {gather*} -\frac {\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 )}{a^{3/2} d^{5/2} e^{3/2}}+\frac {2 \left (-3 a^3 e^6+7 a^2 c d^2 e^4+a c^2 d^4 e^2+c d e x \left (3 c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )+3 c^3 d^6\right )}{3 a d^2 e \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 {2 e (a e+c d x)}{3 d \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*(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)*(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 + 7*a^2*c*d^2*e^4 - 3*a^3*e^6 + c*d*e*(3*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2)*x))/(3*a*d^2*e*(c*d^2 - a
*e^2)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^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])]/(a^(3/2)*d^(5/2)*e^(3/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.35, size = 217, normalized size = 0.80 \begin {gather*} \frac {2 \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} (a e+c d x) \left (-3 c^3 d^5 (d+e x)^2+a^3 e^6 (4 d+3 e x)-a c^2 d^3 e^3 x (9 d+8 e x)+a^2 c d e^4 \left (-9 d^2-4 d e x+3 e^2 x^2\right )\right )}{\left (-c d^2+a e^2\right )^3}-3 (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 )\right )}{3 a^{3/2} d^{5/2} e^{3/2} ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(-3*c^3*d^5*(d + e*x)^2 + a^3*e^6*(4*d + 3*e*x) - a*c^2*d^3*e^3*x*(
9*d + 8*e*x) + a^2*c*d*e^4*(-9*d^2 - 4*d*e*x + 3*e^2*x^2)))/(-(c*d^2) + a*e^2)^3 - 3*(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^(3/2)*d^(5/2)*e^(3/2)*(
(a*e + c*d*x)*(d + e*x))^(3/2))

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 360, normalized size = 1.33

method	result	size
default	\(-\frac {-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}}{d}+\frac {\frac {1}{a d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{a d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{x}\right )}{a d e \sqrt {a d e}}}{d}\)	\(360\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*(-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/(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))

________________________________________________________________________________________

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/(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), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (243) = 486\).
time = 8.95, size = 1483, normalized size = 5.47 \begin {gather*} \left [\frac {3 \, {\left (c^{4} d^{9} x - a^{4} x^{2} e^{9} - {\left (a^{3} c d x^{3} + 2 \, a^{4} d x\right )} e^{8} + {\left (a^{3} c d^{2} x^{2} - a^{4} d^{2}\right )} e^{7} + {\left (3 \, a^{2} c^{2} d^{3} x^{3} + 5 \, a^{3} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{5} - 3 \, {\left (a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e^{4} - {\left (5 \, a c^{3} d^{6} x^{2} + 3 \, a^{2} c^{2} d^{6}\right )} e^{3} + {\left (c^{4} d^{7} x^{3} - a c^{3} d^{7} x\right )} e^{2} + {\left (2 \, c^{4} d^{8} x^{2} + a c^{3} d^{8}\right )} e\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {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 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) + 4 \, {\left (3 \, a c^{3} d^{6} x^{2} e^{3} + 6 \, a c^{3} d^{7} x e^{2} + 3 \, a c^{3} d^{8} e + 9 \, a^{2} c^{2} d^{5} x e^{4} + 4 \, a^{3} c d^{3} x e^{6} - 3 \, a^{4} d x e^{8} - {\left (3 \, a^{3} c d^{2} x^{2} + 4 \, a^{4} d^{2}\right )} e^{7} + {\left (8 \, a^{2} c^{2} d^{4} x^{2} + 9 \, a^{3} c d^{4}\right )} e^{5}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{6 \, {\left (a^{2} c^{4} d^{12} x e^{2} - a^{6} d^{3} x^{2} e^{11} - {\left (a^{5} c d^{4} x^{3} + 2 \, a^{6} d^{4} x\right )} e^{10} + {\left (a^{5} c d^{5} x^{2} - a^{6} d^{5}\right )} e^{9} + {\left (3 \, a^{4} c^{2} d^{6} x^{3} + 5 \, a^{5} c d^{6} x\right )} e^{8} + 3 \, {\left (a^{4} c^{2} d^{7} x^{2} + a^{5} c d^{7}\right )} e^{7} - 3 \, {\left (a^{3} c^{3} d^{8} x^{3} + a^{4} c^{2} d^{8} x\right )} e^{6} - {\left (5 \, a^{3} c^{3} d^{9} x^{2} + 3 \, a^{4} c^{2} d^{9}\right )} e^{5} + {\left (a^{2} c^{4} d^{10} x^{3} - a^{3} c^{3} d^{10} x\right )} e^{4} + {\left (2 \, a^{2} c^{4} d^{11} x^{2} + a^{3} c^{3} d^{11}\right )} e^{3}\right )}}, \frac {3 \, {\left (c^{4} d^{9} x - a^{4} x^{2} e^{9} - {\left (a^{3} c d x^{3} + 2 \, a^{4} d x\right )} e^{8} + {\left (a^{3} c d^{2} x^{2} - a^{4} d^{2}\right )} e^{7} + {\left (3 \, a^{2} c^{2} d^{3} x^{3} + 5 \, a^{3} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{5} - 3 \, {\left (a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e^{4} - {\left (5 \, a c^{3} d^{6} x^{2} + 3 \, a^{2} c^{2} d^{6}\right )} e^{3} + {\left (c^{4} d^{7} x^{3} - a c^{3} d^{7} x\right )} e^{2} + {\left (2 \, c^{4} d^{8} x^{2} + a c^{3} d^{8}\right )} e\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (3 \, a c^{3} d^{6} x^{2} e^{3} + 6 \, a c^{3} d^{7} x e^{2} + 3 \, a c^{3} d^{8} e + 9 \, a^{2} c^{2} d^{5} x e^{4} + 4 \, a^{3} c d^{3} x e^{6} - 3 \, a^{4} d x e^{8} - {\left (3 \, a^{3} c d^{2} x^{2} + 4 \, a^{4} d^{2}\right )} e^{7} + {\left (8 \, a^{2} c^{2} d^{4} x^{2} + 9 \, a^{3} c d^{4}\right )} e^{5}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{3 \, {\left (a^{2} c^{4} d^{12} x e^{2} - a^{6} d^{3} x^{2} e^{11} - {\left (a^{5} c d^{4} x^{3} + 2 \, a^{6} d^{4} x\right )} e^{10} + {\left (a^{5} c d^{5} x^{2} - a^{6} d^{5}\right )} e^{9} + {\left (3 \, a^{4} c^{2} d^{6} x^{3} + 5 \, a^{5} c d^{6} x\right )} e^{8} + 3 \, {\left (a^{4} c^{2} d^{7} x^{2} + a^{5} c d^{7}\right )} e^{7} - 3 \, {\left (a^{3} c^{3} d^{8} x^{3} + a^{4} c^{2} d^{8} x\right )} e^{6} - {\left (5 \, a^{3} c^{3} d^{9} x^{2} + 3 \, a^{4} c^{2} d^{9}\right )} e^{5} + {\left (a^{2} c^{4} d^{10} x^{3} - a^{3} c^{3} d^{10} x\right )} e^{4} + {\left (2 \, a^{2} c^{4} d^{11} x^{2} + a^{3} c^{3} d^{11}\right )} e^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(c^4*d^9*x - a^4*x^2*e^9 - (a^3*c*d*x^3 + 2*a^4*d*x)*e^8 + (a^3*c*d^2*x^2 - a^4*d^2)*e^7 + (3*a^2*c^2*
d^3*x^3 + 5*a^3*c*d^3*x)*e^6 + 3*(a^2*c^2*d^4*x^2 + a^3*c*d^4)*e^5 - 3*(a*c^3*d^5*x^3 + a^2*c^2*d^5*x)*e^4 - (
5*a*c^3*d^6*x^2 + 3*a^2*c^2*d^6)*e^3 + (c^4*d^7*x^3 - a*c^3*d^7*x)*e^2 + (2*c^4*d^8*x^2 + a*c^3*d^8)*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)*s
qrt(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*(3*
a*c^3*d^6*x^2*e^3 + 6*a*c^3*d^7*x*e^2 + 3*a*c^3*d^8*e + 9*a^2*c^2*d^5*x*e^4 + 4*a^3*c*d^3*x*e^6 - 3*a^4*d*x*e^
8 - (3*a^3*c*d^2*x^2 + 4*a^4*d^2)*e^7 + (8*a^2*c^2*d^4*x^2 + 9*a^3*c*d^4)*e^5)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x
^2 + a*d)*e))/(a^2*c^4*d^12*x*e^2 - a^6*d^3*x^2*e^11 - (a^5*c*d^4*x^3 + 2*a^6*d^4*x)*e^10 + (a^5*c*d^5*x^2 - a
^6*d^5)*e^9 + (3*a^4*c^2*d^6*x^3 + 5*a^5*c*d^6*x)*e^8 + 3*(a^4*c^2*d^7*x^2 + a^5*c*d^7)*e^7 - 3*(a^3*c^3*d^8*x
^3 + a^4*c^2*d^8*x)*e^6 - (5*a^3*c^3*d^9*x^2 + 3*a^4*c^2*d^9)*e^5 + (a^2*c^4*d^10*x^3 - a^3*c^3*d^10*x)*e^4 +
(2*a^2*c^4*d^11*x^2 + a^3*c^3*d^11)*e^3), 1/3*(3*(c^4*d^9*x - a^4*x^2*e^9 - (a^3*c*d*x^3 + 2*a^4*d*x)*e^8 + (a
^3*c*d^2*x^2 - a^4*d^2)*e^7 + (3*a^2*c^2*d^3*x^3 + 5*a^3*c*d^3*x)*e^6 + 3*(a^2*c^2*d^4*x^2 + a^3*c*d^4)*e^5 -
3*(a*c^3*d^5*x^3 + a^2*c^2*d^5*x)*e^4 - (5*a*c^3*d^6*x^2 + 3*a^2*c^2*d^6)*e^3 + (c^4*d^7*x^3 - a*c^3*d^7*x)*e^
2 + (2*c^4*d^8*x^2 + a*c^3*d^8)*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*(3*a*c^3*d^
6*x^2*e^3 + 6*a*c^3*d^7*x*e^2 + 3*a*c^3*d^8*e + 9*a^2*c^2*d^5*x*e^4 + 4*a^3*c*d^3*x*e^6 - 3*a^4*d*x*e^8 - (3*a
^3*c*d^2*x^2 + 4*a^4*d^2)*e^7 + (8*a^2*c^2*d^4*x^2 + 9*a^3*c*d^4)*e^5)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d
)*e))/(a^2*c^4*d^12*x*e^2 - a^6*d^3*x^2*e^11 - (a^5*c*d^4*x^3 + 2*a^6*d^4*x)*e^10 + (a^5*c*d^5*x^2 - a^6*d^5)*
e^9 + (3*a^4*c^2*d^6*x^3 + 5*a^5*c*d^6*x)*e^8 + 3*(a^4*c^2*d^7*x^2 + a^5*c*d^7)*e^7 - 3*(a^3*c^3*d^8*x^3 + a^4
*c^2*d^8*x)*e^6 - (5*a^3*c^3*d^9*x^2 + 3*a^4*c^2*d^9)*e^5 + (a^2*c^4*d^10*x^3 - a^3*c^3*d^10*x)*e^4 + (2*a^2*c
^4*d^11*x^2 + a^3*c^3*d^11)*e^3)]

________________________________________________________________________________________

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

________________________________________________________________________________________

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/(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), x)

________________________________________________________________________________________

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

________________________________________________________________________________________