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

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

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Rubi [A]  time = 0.59, 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 = {851, 822, 806, 724, 206} \begin {gather*} -\frac {\left (31 a^2 c d^2 e^4-15 a^3 e^6-9 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 \left (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^4-5 a^3 e^6+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 (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 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 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])

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 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 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 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])

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 ) \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 )}{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 = 0.57, size = 370, normalized size = 0.94 \begin {gather*} \frac {(a e+c d x) \left (3 a^{3/2} d^{5/2} e^{3/2} \left (a e^2-c d^2\right )^3+\sqrt {a} d^{3/2} \sqrt {e} x \left (a e^2-c d^2\right ) \left (5 a^2 e^5-6 a c d^2 e^3+9 c^2 d^4 e\right ) (a e+c d x)+x (d+e x) \sqrt {a e+c d x} \left (\sqrt {a} \sqrt {d} \sqrt {e} \left (15 a^3 e^7-31 a^2 c d^2 e^5+9 a c^2 d^4 e^3-9 c^3 d^6 e\right ) \sqrt {a e+c d x}+3 \sqrt {d+e x} \left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )\right )+3 \sqrt {a} c d^{7/2} \sqrt {e} x \left (c d^2-a e^2\right )^2 \left (a e^2-3 c d^2\right )\right )}{3 a^{5/2} d^{7/2} e^{5/2} x \left (c d^2-a e^2\right )^3 ((d+e x) (a e+c d 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]

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

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IntegrateAlgebraic [F]  time = 180.04, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[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]

$Aborted

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fricas [B]  time = 19.59, size = 1812, normalized size = 4.60

result too large to display

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^9*e^2 - 4*a*c^4*d^7*e^4 - 6*a^2*c^3*d^5*e^6 + 12*a^3*c^2*d^3*e^8 - 5*a^4*c*d*e^10)*x^4 + (6
*c^5*d^10*e - 5*a*c^4*d^8*e^3 - 16*a^2*c^3*d^6*e^5 + 18*a^3*c^2*d^4*e^7 + 2*a^4*c*d^2*e^9 - 5*a^5*e^11)*x^3 +
(3*c^5*d^11 + 2*a*c^4*d^9*e^2 - 14*a^2*c^3*d^7*e^4 + 19*a^4*c*d^3*e^8 - 10*a^5*d*e^10)*x^2 + (3*a*c^4*d^10*e -
4*a^2*c^3*d^8*e^3 - 6*a^3*c^2*d^6*e^5 + 12*a^4*c*d^4*e^7 - 5*a^5*d^2*e^9)*x)*sqrt(a*d*e)*log((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*(3*a^2*c^3*d^9*e^2 - 9*a^3*c^2*d^7*e^4 + 9*a^4*c*
d^5*e^6 - 3*a^5*d^3*e^8 + (9*a*c^4*d^8*e^3 - 9*a^2*c^3*d^6*e^5 + 31*a^3*c^2*d^4*e^7 - 15*a^4*c*d^2*e^9)*x^3 +
(18*a*c^4*d^9*e^2 - 15*a^2*c^3*d^7*e^4 + 33*a^3*c^2*d^5*e^6 + 11*a^4*c*d^3*e^8 - 15*a^5*d*e^10)*x^2 + (9*a*c^4
*d^10*e - 3*a^2*c^3*d^8*e^3 - 9*a^3*c^2*d^6*e^5 + 39*a^4*c*d^4*e^7 - 20*a^5*d^2*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x))/((a^3*c^4*d^11*e^5 - 3*a^4*c^3*d^9*e^7 + 3*a^5*c^2*d^7*e^9 - a^6*c*d^5*e^11)*x^4 + (2*a
^3*c^4*d^12*e^4 - 5*a^4*c^3*d^10*e^6 + 3*a^5*c^2*d^8*e^8 + a^6*c*d^6*e^10 - a^7*d^4*e^12)*x^3 + (a^3*c^4*d^13*
e^3 - a^4*c^3*d^11*e^5 - 3*a^5*c^2*d^9*e^7 + 5*a^6*c*d^7*e^9 - 2*a^7*d^5*e^11)*x^2 + (a^4*c^3*d^12*e^4 - 3*a^5
*c^2*d^10*e^6 + 3*a^6*c*d^8*e^8 - a^7*d^6*e^10)*x), -1/6*(3*((3*c^5*d^9*e^2 - 4*a*c^4*d^7*e^4 - 6*a^2*c^3*d^5*
e^6 + 12*a^3*c^2*d^3*e^8 - 5*a^4*c*d*e^10)*x^4 + (6*c^5*d^10*e - 5*a*c^4*d^8*e^3 - 16*a^2*c^3*d^6*e^5 + 18*a^3
*c^2*d^4*e^7 + 2*a^4*c*d^2*e^9 - 5*a^5*e^11)*x^3 + (3*c^5*d^11 + 2*a*c^4*d^9*e^2 - 14*a^2*c^3*d^7*e^4 + 19*a^4
*c*d^3*e^8 - 10*a^5*d*e^10)*x^2 + (3*a*c^4*d^10*e - 4*a^2*c^3*d^8*e^3 - 6*a^3*c^2*d^6*e^5 + 12*a^4*c*d^4*e^7 -
5*a^5*d^2*e^9)*x)*sqrt(-a*d*e)*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*(3*a^2*c^3*d^9*e^2 - 9*a^
3*c^2*d^7*e^4 + 9*a^4*c*d^5*e^6 - 3*a^5*d^3*e^8 + (9*a*c^4*d^8*e^3 - 9*a^2*c^3*d^6*e^5 + 31*a^3*c^2*d^4*e^7 -
15*a^4*c*d^2*e^9)*x^3 + (18*a*c^4*d^9*e^2 - 15*a^2*c^3*d^7*e^4 + 33*a^3*c^2*d^5*e^6 + 11*a^4*c*d^3*e^8 - 15*a^
5*d*e^10)*x^2 + (9*a*c^4*d^10*e - 3*a^2*c^3*d^8*e^3 - 9*a^3*c^2*d^6*e^5 + 39*a^4*c*d^4*e^7 - 20*a^5*d^2*e^9)*x
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/((a^3*c^4*d^11*e^5 - 3*a^4*c^3*d^9*e^7 + 3*a^5*c^2*d^7*e^9 - a^
6*c*d^5*e^11)*x^4 + (2*a^3*c^4*d^12*e^4 - 5*a^4*c^3*d^10*e^6 + 3*a^5*c^2*d^8*e^8 + a^6*c*d^6*e^10 - a^7*d^4*e^
12)*x^3 + (a^3*c^4*d^13*e^3 - a^4*c^3*d^11*e^5 - 3*a^5*c^2*d^9*e^7 + 5*a^6*c*d^7*e^9 - 2*a^7*d^5*e^11)*x^2 + (
a^4*c^3*d^12*e^4 - 3*a^5*c^2*d^10*e^6 + 3*a^6*c*d^8*e^8 - a^7*d^6*e^10)*x)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.41Unable to transpose Error: Bad Argument Value

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maple [B]  time = 0.02, size = 912, normalized size = 2.31 \begin {gather*} \frac {16 c^{2} e^{3} x}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 a c \,e^{4}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, d}+\frac {8 c^{2} d \,e^{2}}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {3 c^{3} d^{2} x}{\left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{2} e}+\frac {5 c \,e^{3} x}{\left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d^{2}}+\frac {5 a \,e^{4}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d^{3}}+\frac {3 c^{2} d}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a}+\frac {3 c^{3} d^{3}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{2} e^{2}}+\frac {5 c \,e^{2}}{2 \left (-a^{2} e^{4}+2 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d}-\frac {2 e}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c d e +\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\, d^{2}}+\frac {5 \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{2 \sqrt {a d e}\, a \,d^{3}}+\frac {3 c \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}}{x}\right )}{2 \sqrt {a d e}\, a^{2} d \,e^{2}}-\frac {5}{2 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a \,d^{3}}-\frac {3 c}{2 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{2} d \,e^{2}}-\frac {1}{\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a \,d^{2} e x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d^2/a/e/x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-5/2/d^3/a/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-3/2/d/a
^2/e^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c+5/d^2*e^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(
a*e^2+c*d^2)*x)^(1/2)*x*c+3*d^2/a^2/e/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)
*x*c^3+5/2/d^3*a*e^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+5/2/d*e^2/(-a^2*
e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c+3/2*d/a/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/
(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^2+3/2*d^3/a^2/e^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(1/2)*c^3+5/2/d^3/a/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*
e+(a*e^2+c*d^2)*x)^(1/2))/x)+3/2/d/a^2/e^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^
2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c-2/3*e/d^2/(a*e^2-c*d^2)/(x+d/e)/((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(
1/2)+16/3*e^3*c^2/(a*e^2-c*d^2)^3/((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+8/3*e^4/d*c/(a*e^2-c*d^2)^3/
((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*a+8/3*e^2*d*c^2/(a*e^2-c*d^2)^3/((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(
x+d/e))^(1/2)

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