3.475 \(\int \frac{1}{x^2 (d+e x) \sqrt{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=229 \[ \frac{\left (3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left (c d^2-3 a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d^2 e x \left (c d^2-a e^2\right )}-\frac{2 e (a e+c d x)}{d x \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[Out]

(-2*e*(a*e + c*d*x))/(d*(c*d^2 - a*e^2)*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((c*d^2 - 3*a*e^2)*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*d^2*e*(c*d^2 - a*e^2)*x) + ((c*d^2 + 3*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^(3/2)*d^(5/
2)*e^(3/2))

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Rubi [A]  time = 0.292443, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {851, 822, 806, 724, 206} \[ \frac{\left (3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left (c d^2-3 a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d^2 e x \left (c d^2-a e^2\right )}-\frac{2 e (a e+c d x)}{d x \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e*(a*e + c*d*x))/(d*(c*d^2 - a*e^2)*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((c*d^2 - 3*a*e^2)*Sq
rt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a*d^2*e*(c*d^2 - a*e^2)*x) + ((c*d^2 + 3*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^(3/2)*d^(5/
2)*e^(3/2))

Rule 851

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])

Rule 822

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 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((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[Sim
plify[m + 2*p + 3], 0]

Rule 724

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 206

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

Rubi steps

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

Mathematica [A]  time = 0.129399, size = 201, normalized size = 0.88 \[ \frac{\sqrt{a} \sqrt{d} \sqrt{e} \left (a^2 e^3 (d+3 e x)-a c d e \left (d^2-3 e^2 x^2\right )-c^2 d^3 x (d+e x)\right )+x \sqrt{d+e x} \left (-3 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )}{a^{3/2} d^{5/2} e^{3/2} x \left (c d^2-a e^2\right ) \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*Sqrt[d]*Sqrt[e]*(-(c^2*d^3*x*(d + e*x)) + a^2*e^3*(d + 3*e*x) - a*c*d*e*(d^2 - 3*e^2*x^2)) + (c^2*d^4
+ 2*a*c*d^2*e^2 - 3*a^2*e^4)*x*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*S
qrt[e]*Sqrt[d + e*x])])/(a^(3/2)*d^(5/2)*e^(3/2)*(c*d^2 - a*e^2)*x*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.064, size = 253, normalized size = 1.1 \begin{align*}{\frac{3\,e}{2\,{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}}-2\,{\frac{e}{{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cde \left ({\frac{d}{e}}+x \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) } \left ({\frac{d}{e}}+x \right ) ^{-1}}-{\frac{1}{a{d}^{2}ex}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{c}{2\,ae}\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}} \end{align*}

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)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (e x + d\right )} x^{2}}\,{d x} \end{align*}

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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 11.0505, size = 1245, normalized size = 5.44 \begin{align*} \left [\frac{\sqrt{a d e}{\left ({\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x^{2} +{\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \log \left (\frac{8 \, a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{a d e} + 8 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) - 4 \,{\left (a c d^{4} e - a^{2} d^{2} e^{3} +{\left (a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{4 \,{\left ({\left (a^{2} c d^{5} e^{3} - a^{3} d^{3} e^{5}\right )} x^{2} +{\left (a^{2} c d^{6} e^{2} - a^{3} d^{4} e^{4}\right )} x\right )}}, -\frac{\sqrt{-a d e}{\left ({\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} - 3 \, a^{2} e^{5}\right )} x^{2} +{\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-a d e}}{2 \,{\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} +{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \,{\left (a c d^{4} e - a^{2} d^{2} e^{3} +{\left (a c d^{3} e^{2} - 3 \, a^{2} d e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{2 \,{\left ({\left (a^{2} c d^{5} e^{3} - a^{3} d^{3} e^{5}\right )} x^{2} +{\left (a^{2} c d^{6} e^{2} - a^{3} d^{4} e^{4}\right )} x\right )}}\right ] \end{align*}

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)^(1/2),x, algorithm="fricas")

[Out]

[1/4*(sqrt(a*d*e)*((c^2*d^4*e + 2*a*c*d^2*e^3 - 3*a^2*e^5)*x^2 + (c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x)*lo
g((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*
a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) - 4*(a*c*d^4*e - a^2*d^2*e^3 + (a*c
*d^3*e^2 - 3*a^2*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^2*c*d^5*e^3 - a^3*d^3*e^5)*x^2 + (
a^2*c*d^6*e^2 - a^3*d^4*e^4)*x), -1/2*(sqrt(-a*d*e)*((c^2*d^4*e + 2*a*c*d^2*e^3 - 3*a^2*e^5)*x^2 + (c^2*d^5 +
2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e
^2)*x)*sqrt(-a*d*e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)) + 2*(a*c*d^4*e - a^2*d^2*e^3
+ (a*c*d^3*e^2 - 3*a^2*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^2*c*d^5*e^3 - a^3*d^3*e^5)*x
^2 + (a^2*c*d^6*e^2 - a^3*d^4*e^4)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**(1/2),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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)^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError