3.3.78 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^4 (d+e x)} \, dx\)

Optimal. Leaf size=286 \[ \frac {\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 x^2}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d x^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.40, antiderivative size = 286, 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 = {849, 834, 806, 724, 206} \begin {gather*} -\frac {\left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\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 )}{16 a^{5/2} d^{7/2} e^{5/2}}+\frac {\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 x^2}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^4*(d + e*x)),x]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*d*x^3) - ((c/(a*e) - (5*e)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(12*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*d^2 + 3*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(24*a^2*d^3*e^2*x) - ((c*d^2 - a*e^2)*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*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])])/(16*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 834

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])

Rule 849

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rubi steps

\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx &=\int \frac {a e+c d x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\int \frac {-\frac {1}{2} a e \left (c d^2-5 a e^2\right )+2 a c d e^2 x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a d e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\int \frac {-\frac {1}{4} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right )-\frac {1}{2} a c d e^2 \left (c d^2-5 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}{6 a^2 d^2 e^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \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}}{24 a^2 d^3 e^2 x}+\frac {\left (\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 a^2 d^3 e^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \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}}{24 a^2 d^3 e^2 x}-\frac {\left (\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\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 )}{8 a^2 d^3 e^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac {\left (\frac {c}{a e}-\frac {5 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac {\left (3 c d^2-5 a e^2\right ) \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}}{24 a^2 d^3 e^2 x}-\frac {\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\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 )}{16 a^{5/2} d^{7/2} e^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 210, normalized size = 0.73 \begin {gather*} \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (a^2 e^2 \left (-8 d^2+10 d e x-15 e^2 x^2\right )-2 a c d^2 e x (d-2 e x)+3 c^2 d^4 x^2\right )}{x^3}-\frac {3 \left (-5 a^3 e^6+3 a^2 c d^2 e^4+a c^2 d^4 e^2+c^3 d^6\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {d+e x} \sqrt {a e+c d x}}\right )}{24 a^{5/2} d^{7/2} e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^4*(d + e*x)),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(3*c^2*d^4*x^2 - 2*a*c*d^2*e*x*(d - 2*e*x) + a^2*e^2*
(-8*d^2 + 10*d*e*x - 15*e^2*x^2)))/x^3 - (3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*ArcTanh[(S
qrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(24*a^(5/2)*d^
(7/2)*e^(5/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.23, size = 228, normalized size = 0.80 \begin {gather*} \frac {\sqrt {a d e+a e^2 x+c d^2 x+c d e x^2} \left (-8 a^2 d^2 e^2+10 a^2 d e^3 x-15 a^2 e^4 x^2-2 a c d^3 e x+4 a c d^2 e^2 x^2+3 c^2 d^4 x^2\right )}{24 a^2 d^3 e^2 x^3}+\frac {\left (-5 a^3 e^6+3 a^2 c d^2 e^4+a c^2 d^4 e^2+c^3 d^6\right ) \tanh ^{-1}\left (\frac {x \sqrt {c d e}-\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {a} \sqrt {d} \sqrt {e}}\right )}{8 a^{5/2} d^{7/2} e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^4*(d + e*x)),x]

[Out]

(Sqrt[a*d*e + c*d^2*x + a*e^2*x + c*d*e*x^2]*(-8*a^2*d^2*e^2 - 2*a*c*d^3*e*x + 10*a^2*d*e^3*x + 3*c^2*d^4*x^2
+ 4*a*c*d^2*e^2*x^2 - 15*a^2*e^4*x^2))/(24*a^2*d^3*e^2*x^3) + ((c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*
a^3*e^6)*ArcTanh[(Sqrt[c*d*e]*x - Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[a]*Sqrt[d]*Sqrt[e])])/(8*
a^(5/2)*d^(7/2)*e^(5/2))

________________________________________________________________________________________

fricas [A]  time = 2.12, size = 558, normalized size = 1.95 \begin {gather*} \left [-\frac {3 \, {\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \sqrt {a d e} x^{3} \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 (8 \, a^{3} d^{3} e^{3} - {\left (3 \, a c^{2} d^{5} e + 4 \, a^{2} c d^{3} e^{3} - 15 \, a^{3} d e^{5}\right )} x^{2} + 2 \, {\left (a^{2} c d^{4} e^{2} - 5 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, a^{3} d^{4} e^{3} x^{3}}, \frac {3 \, {\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \sqrt {-a d e} x^{3} \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 (8 \, a^{3} d^{3} e^{3} - {\left (3 \, a c^{2} d^{5} e + 4 \, a^{2} c d^{3} e^{3} - 15 \, a^{3} d e^{5}\right )} x^{2} + 2 \, {\left (a^{2} c d^{4} e^{2} - 5 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, a^{3} d^{4} e^{3} x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x, algorithm="fricas")

[Out]

[-1/96*(3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(a*d*e)*x^3*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*(8*a^3*d^3*e^3 - (3*a*c^2*d^5*e + 4*a^2*c*d^3*e^3 - 15*a^
3*d*e^5)*x^2 + 2*(a^2*c*d^4*e^2 - 5*a^3*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^4*e^3*
x^3), 1/48*(3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(-a*d*e)*x^3*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*(8*a^3*d^3*e^3 - (3*a*c^2*d^5*e + 4*a^2*c*d^3*e^3 - 15*a^3*d*e^5)*x^2 + 2*(a^2*c
*d^4*e^2 - 5*a^3*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^4*e^3*x^3)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-2*exp(1)^3*a*exp(2)+2*exp(1)^5*a)/2
/d^3/sqrt(-a*d*exp(1)^3+a*d*exp(1)*exp(2))*atan((-d*sqrt(c*d*exp(1))+(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x
^2*exp(1))-sqrt(c*d*exp(1))*x)*exp(1))/sqrt(-a*d*exp(1)^3+a*d*exp(1)*exp(2)))-(-a^3*exp(2)^3-2*exp(1)^2*a^3*ex
p(2)^2-8*exp(1)^4*a^3*exp(2)+16*exp(1)^6*a^3-3*c*d^2*a^2*exp(2)^2-3*c^2*d^4*a*exp(2)+2*c^2*d^4*exp(1)^2*a-c^3*
d^6)/8/d^3/exp(1)^2/a^2/2/sqrt(-a*d*exp(1))*atan((sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*
exp(1))*x)/sqrt(-a*d*exp(1)))+(3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^3
*exp(2)^3+6*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^3*exp(2)^2-24
*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^3*exp(2)+9*c*d^2*(sqrt(a
*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a^2*exp(2)^2+9*c^2*d^4*(sqrt(a*d*exp(1)+a*x
*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a*exp(2)-6*c^2*d^4*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+
c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^5*a+3*c^3*d^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-
sqrt(c*d*exp(1))*x)^5-48*d*exp(1)^3*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(
c*d*exp(1))*x)^4*a^3*exp(2)+48*d*exp(1)^5*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))
-sqrt(c*d*exp(1))*x)^4*a^3-8*d*exp(1)*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^
3*a^4*exp(2)^3+48*d*exp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^4*exp
(2)-24*c*d^3*exp(1)*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^3*exp(2)^2-48*
c*d^3*exp(1)^3*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^3*exp(2)+48*c*d^3*e
xp(1)^5*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^3-24*c^2*d^5*exp(1)*(sqrt(
a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^2*exp(2)-48*c^2*d^5*exp(1)^3*(sqrt(a*d*e
xp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^2-8*c^3*d^7*exp(1)*(sqrt(a*d*exp(1)+a*x*exp(2
)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a+48*d^2*exp(1)^2*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)
+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^4*exp(2)^2+48*d^2*exp(1)^4*sqrt(c*d*exp(1))*(sqrt(a*d*exp(1)+
a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^4*exp(2)-96*d^2*exp(1)^6*sqrt(c*d*exp(1))*(sqrt(a*d
*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^4+96*c*d^4*exp(1)^2*sqrt(c*d*exp(1))*(sqrt(
a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^3*exp(2)+48*c*d^4*exp(1)^4*sqrt(c*d*exp(
1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^3+48*c^2*d^6*exp(1)^2*sqrt(c*d
*exp(1))*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2*a^2-3*d^2*exp(1)^2*(sqrt(a*
d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^5*exp(2)^3-6*d^2*exp(1)^4*(sqrt(a*d*exp(1)+a
*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^5*exp(2)^2-24*d^2*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)
+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^5*exp(2)-9*c*d^4*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c
*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^4*exp(2)^2-48*c*d^4*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*
exp(1))-sqrt(c*d*exp(1))*x)*a^4*exp(2)-48*c*d^4*exp(1)^6*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-s
qrt(c*d*exp(1))*x)*a^4-9*c^2*d^6*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))
*x)*a^3*exp(2)-42*c^2*d^6*exp(1)^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^3
-3*c^3*d^8*exp(1)^2*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^2+48*d^3*exp(1)^
7*sqrt(c*d*exp(1))*a^5+16*c*d^5*exp(1)^5*sqrt(c*d*exp(1))*a^4)/48/d^3/exp(1)^2/a^2/((sqrt(a*d*exp(1)+a*x*exp(2
)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^2-d*exp(1)*a)^3)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 1165, normalized size = 4.07

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)/x^4/(e*x+d),x)

[Out]

3/4/d^3/a/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+3/8/e/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^2+1/
4/d/a^2/e^2/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c-1/16*d^3/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^3+1/8*d/a^3/e^2*c^3*(c*d*e*x^2+a*d*e+(a*e
^2+c*d^2)*x)^(1/2)*x-1/2/e/d^2/a^2/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c+11/8/d^3*e^2*c/a*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(1/2)*x+1/d^4*e^3*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3/8/d^4*e^3*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(1/2)-1/16*d/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)*c^2+1/2/d/a^2*c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x-1/3/d^2/a/e/x^3*(c*d*e*
x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)-1/8/a^3/e^3/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^2+1/8*d^2/a^3/e^3*(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^3-1/2/d^4*e^5*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2
+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a+1/2/d^4*e^5*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2
)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)*a-1/2/d^2*e^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c
*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)*c-11/8/d^4*e/a/x*(c*d*e*x^2+a
*d*e+(a*e^2+c*d^2)*x)^(3/2)+9/8/d^2*e/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c+1/2/d^2*e^3*ln((c*d*e*x+1/2*
a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*c+5/16/d^3*e^4*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)-3/16/d*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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((e*x + d)*x^4), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^4\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(x^4*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(x^4*(d + e*x)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{x^{4} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**4/(e*x+d),x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(x**4*(d + e*x)), x)

________________________________________________________________________________________