3.3.79 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^5 (d+e x)} \, dx\)

Optimal. Leaf size=389 \[ \frac {\left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (-105 a^3 e^6+25 a^2 c d^2 e^4+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (c d^2-a e^2\right ) \left (35 a^3 e^6+15 a^2 c d^2 e^4+9 a c^2 d^4 e^2+5 c^3 d^6\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 x^3} \]

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Rubi [A]  time = 0.59, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 7, 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 (25 a^2 c d^2 e^4-105 a^3 e^6+17 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (-35 a^2 e^4+6 a c d^2 e^2+5 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 a^2 d^3 e^2 x^2}+\frac {\left (c d^2-a e^2\right ) \left (15 a^2 c d^2 e^4+35 a^3 e^6+9 a c^2 d^4 e^2+5 c^3 d^6\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 )}{128 a^{7/2} d^{9/2} e^{7/2}}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 x^3}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^4} \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^5*(d + e*x)),x]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(4*d*x^4) - ((c/(a*e) - (7*e)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(24*x^3) + ((5*c^2*d^4 + 6*a*c*d^2*e^2 - 35*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
])/(96*a^2*d^3*e^2*x^2) - ((15*c^3*d^6 + 17*a*c^2*d^4*e^2 + 25*a^2*c*d^2*e^4 - 105*a^3*e^6)*Sqrt[a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2])/(192*a^3*d^4*e^3*x) + ((c*d^2 - a*e^2)*(5*c^3*d^6 + 9*a*c^2*d^4*e^2 + 15*a^2*c*d^2*
e^4 + 35*a^3*e^6)*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])])/(128*a^(7/2)*d^(9/2)*e^(7/2))

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^5 (d+e x)} \, dx &=\int \frac {a e+c d x}{x^5 \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}}{4 d x^4}-\frac {\int \frac {-\frac {1}{2} a e \left (c d^2-7 a e^2\right )+3 a c d e^2 x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 a d e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\int \frac {-\frac {1}{4} a e \left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right )-a c d e^2 \left (c d^2-7 a e^2\right ) x}{x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{12 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}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\int \frac {-\frac {1}{8} a e \left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right )-\frac {1}{4} a c d e^2 \left (5 c^2 d^4+6 a c d^2 e^2-35 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}{24 a^3 d^3 e^3}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}-\frac {\left (\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\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}{128 a^3 d^4 e^3}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\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 )}{64 a^3 d^4 e^3}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^4}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 x^3}+\frac {\left (5 c^2 d^4+6 a c d^2 e^2-35 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 a^2 d^3 e^2 x^2}-\frac {\left (15 c^3 d^6+17 a c^2 d^4 e^2+25 a^2 c d^2 e^4-105 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 a^3 d^4 e^3 x}+\frac {\left (c d^2-a e^2\right ) \left (5 c^3 d^6+9 a c^2 d^4 e^2+15 a^2 c d^2 e^4+35 a^3 e^6\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 )}{128 a^{7/2} d^{9/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 273, normalized size = 0.70 \begin {gather*} \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (a^3 e^3 \left (-48 d^3+56 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+a^2 c d^2 e^2 x \left (-8 d^2+12 d e x-25 e^2 x^2\right )+a c^2 d^4 e x^2 (10 d-17 e x)-15 c^3 d^6 x^3\right )}{x^4}+\frac {3 \left (-35 a^4 e^8+20 a^3 c d^2 e^6+6 a^2 c^2 d^4 e^4+4 a c^3 d^6 e^2+5 c^4 d^8\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 )}{192 a^{7/2} d^{9/2} e^{7/2}} \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^5*(d + e*x)),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-15*c^3*d^6*x^3 + a*c^2*d^4*e*x^2*(10*d - 17*e*x) +
a^2*c*d^2*e^2*x*(-8*d^2 + 12*d*e*x - 25*e^2*x^2) + a^3*e^3*(-48*d^3 + 56*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)
))/x^4 + (3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTanh[(Sqrt[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])))/(192*a^(7/2)*d^(9/2)
*e^(7/2))

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IntegrateAlgebraic [A]  time = 1.74, size = 307, normalized size = 0.79 \begin {gather*} \frac {\sqrt {a d e+a e^2 x+c d^2 x+c d e x^2} \left (-48 a^3 d^3 e^3+56 a^3 d^2 e^4 x-70 a^3 d e^5 x^2+105 a^3 e^6 x^3-8 a^2 c d^4 e^2 x+12 a^2 c d^3 e^3 x^2-25 a^2 c d^2 e^4 x^3+10 a c^2 d^5 e x^2-17 a c^2 d^4 e^2 x^3-15 c^3 d^6 x^3\right )}{192 a^3 d^4 e^3 x^4}+\frac {\left (35 a^4 e^8-20 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4-4 a c^3 d^6 e^2-5 c^4 d^8\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 )}{64 a^{7/2} d^{9/2} e^{7/2}} \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^5*(d + e*x)),x]

[Out]

(Sqrt[a*d*e + c*d^2*x + a*e^2*x + c*d*e*x^2]*(-48*a^3*d^3*e^3 - 8*a^2*c*d^4*e^2*x + 56*a^3*d^2*e^4*x + 10*a*c^
2*d^5*e*x^2 + 12*a^2*c*d^3*e^3*x^2 - 70*a^3*d*e^5*x^2 - 15*c^3*d^6*x^3 - 17*a*c^2*d^4*e^2*x^3 - 25*a^2*c*d^2*e
^4*x^3 + 105*a^3*e^6*x^3))/(192*a^3*d^4*e^3*x^4) + ((-5*c^4*d^8 - 4*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 - 20*a^3
*c*d^2*e^6 + 35*a^4*e^8)*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])])/(64*a^(7/2)*d^(9/2)*e^(7/2))

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fricas [A]  time = 9.70, size = 702, normalized size = 1.80 \begin {gather*} \left [-\frac {3 \, {\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} \sqrt {a d e} x^{4} \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 (48 \, a^{4} d^{4} e^{4} + {\left (15 \, a c^{3} d^{7} e + 17 \, a^{2} c^{2} d^{5} e^{3} + 25 \, a^{3} c d^{3} e^{5} - 105 \, a^{4} d e^{7}\right )} x^{3} - 2 \, {\left (5 \, a^{2} c^{2} d^{6} e^{2} + 6 \, a^{3} c d^{4} e^{4} - 35 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (a^{3} c d^{5} e^{3} - 7 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, a^{4} d^{5} e^{4} x^{4}}, -\frac {3 \, {\left (5 \, c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} \sqrt {-a d e} x^{4} \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 (48 \, a^{4} d^{4} e^{4} + {\left (15 \, a c^{3} d^{7} e + 17 \, a^{2} c^{2} d^{5} e^{3} + 25 \, a^{3} c d^{3} e^{5} - 105 \, a^{4} d e^{7}\right )} x^{3} - 2 \, {\left (5 \, a^{2} c^{2} d^{6} e^{2} + 6 \, a^{3} c d^{4} e^{4} - 35 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (a^{3} c d^{5} e^{3} - 7 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, a^{4} d^{5} e^{4} x^{4}}\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^5/(e*x+d),x, algorithm="fricas")

[Out]

[-1/768*(3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*sqrt(a*d*e)*x^4*l
og((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*(48*a^4*d^4*e^4 + (15*a*c^3*d^
7*e + 17*a^2*c^2*d^5*e^3 + 25*a^3*c*d^3*e^5 - 105*a^4*d*e^7)*x^3 - 2*(5*a^2*c^2*d^6*e^2 + 6*a^3*c*d^4*e^4 - 35
*a^4*d^2*e^6)*x^2 + 8*(a^3*c*d^5*e^3 - 7*a^4*d^3*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^5
*e^4*x^4), -1/384*(3*(5*c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 35*a^4*e^8)*sqrt(-a
*d*e)*x^4*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*(48*a^4*d^4*e^4 + (15*a*c^3*d^7*e + 17*a^2*c^2*
d^5*e^3 + 25*a^3*c*d^3*e^5 - 105*a^4*d*e^7)*x^3 - 2*(5*a^2*c^2*d^6*e^2 + 6*a^3*c*d^4*e^4 - 35*a^4*d^2*e^6)*x^2
+ 8*(a^3*c*d^5*e^3 - 7*a^4*d^3*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^5*e^4*x^4)]

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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^5/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((2*exp(1)^4*a*exp(2)-2*exp(1)^6*a)/2/
d^4/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)))+(-5*a^4*exp(2)^4-8*exp(1)^2*a^4*e
xp(2)^3-16*exp(1)^4*a^4*exp(2)^2-64*exp(1)^6*a^4*exp(2)+128*exp(1)^8*a^4-20*c*d^2*a^3*exp(2)^3-30*c^2*d^4*a^2*
exp(2)^2+24*c^2*d^4*exp(1)^2*a^2*exp(2)-20*c^3*d^6*a*exp(2)+16*c^3*d^6*exp(1)^2*a-5*c^4*d^8)/64/d^4/exp(1)^3/a
^3/2/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)))+(-15*(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)^7*a^4*exp(2)^4-24*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)^7*a^4*exp(2)^3-48*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)^7*a^4*exp(2)^2+192*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)^7*a^4*exp(2)-60*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)^7*a^3*exp(2)^3-90*c^2*d^4*(sqrt(a*d*exp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*ex
p(1))-sqrt(c*d*exp(1))*x)^7*a^2*exp(2)^2+72*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)^7*a^2*exp(2)-60*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*ex
p(1))*x)^7*a*exp(2)+48*c^3*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
)^7*a-15*c^4*d^8*(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)^7+384*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)^6*a^4*exp(2)-384*d*exp(1)
^7*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)^6*a^4+55*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)^5*a^5*exp(2)^4+88*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^5*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)^5*a^5*exp(2)^2-576*d*exp(1)^7*(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^5*exp(2)+220*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)^5*a^4*exp(2)^3+384*c*d^3*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)^5*a^4*exp(2)-384*c*d^3*exp(1)^7*(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^4+330*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)^5*a^3*exp(2)^2-264*c^2*d^5*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)+220*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)^5*a^2*exp(2)-176*c^3*d^7*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^2+55*c^4*d^9*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)^5*a-384*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)^4*a^5*exp(2
)^2-768*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)^
4*a^5*exp(2)+1152*d^2*exp(1)^8*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*e
xp(1))*x)^4*a^5+384*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)^4*a^4*exp(2)-384*c*d^4*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)^4*a^4+768*c^2*d^6*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)^4*a^3-73*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)^3*a^6*exp(2)^4-40*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)^3*a^6*exp(2)^3+48*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)
^3*a^6*exp(2)^2+576*d^2*exp(1)^8*(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^6
*exp(2)-292*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)^3*a^5*exp(2
)^3-768*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)^3*a^5*exp(2)^2+
768*c*d^4*exp(1)^8*(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^5-438*c^2*d^6*e
xp(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)^3*a^4*exp(2)^2-1416*c^2*d^6*ex
p(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)^3*a^4*exp(2)-384*c^2*d^6*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)^3*a^4-292*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)^3*a^3*exp(2)-688*c^3*d^8*exp(1)^4*(sqrt(a*d*ex
p(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^3-73*c^4*d^10*exp(1)^2*(sqrt(a*d*exp(1)+a*x*ex
p(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^3*a^2+384*d^3*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)^2*a^6*exp(2)^3+384*d^3*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)^2*a^6*exp(2)^2+384*d^3*exp(1)^7*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^6*exp(2)-1152*d^3*exp(1)^9*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^6+1152*c*d^5*exp(1)^3*s
qrt(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^5*exp(2)^2+1024*c*
d^5*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)^2*a^5*ex
p(2)+256*c*d^5*exp(1)^7*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^5+1152*c^2*d^7*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*e
xp(1))*x)^2*a^4*exp(2)+640*c^2*d^7*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)^2*a^4+384*c^3*d^9*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)^2*a^3-15*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)*a^7*exp(2)^4-24*d^3*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)*a^7*exp(2)^3-48*d^3*exp(1)^7*(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^7*e
xp(2)^2-192*d^3*exp(1)^9*(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^7*exp(2)-60
*c*d^5*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)*a^6*exp(2)^3-384*c*d^5
*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)*a^6*exp(2)^2-384*c*d^5*exp(1
)^7*(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^6*exp(2)-384*c*d^5*exp(1)^9*(sqr
t(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^6-90*c^2*d^7*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)*a^5*exp(2)^2-696*c^2*d^7*exp(1)^5*(sqrt(a*d*exp(1)+a*x*e
xp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)*a^5*exp(2)-384*c^2*d^7*exp(1)^7*(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^5-60*c^3*d^9*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)*a^4*exp(2)-336*c^3*d^9*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)*a^4-15*c^4*d^11*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)*a^3+384*d^4*exp(1)^10*sqrt(c*d*exp(1))*a^7+128*c*d^6*exp(1)^6*sqrt(c*d*exp(1))*a^6*exp(2)+128*c*d^6
*exp(1)^8*sqrt(c*d*exp(1))*a^6+128*c^2*d^8*exp(1)^6*sqrt(c*d*exp(1))*a^5)/384/d^4/exp(1)^3/a^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)^2-d*exp(1)*a)^4)

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maple [B]  time = 0.02, size = 1494, normalized size = 3.84

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^5/(e*x+d),x)

[Out]

19/64/e^2/d/a^3/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^2+1/32/e*d^2/a^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^3-43/64/d^2*e/a^2*c^2*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(1/2)*x-93/64/d^4*e^3*c/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+13/24/d^3/a/x^3*(c*d*e*x^
2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)-17/32/d/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^2+5/128*d^4/a^3/e^3/(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^4-5/64*d^2/
a^4/e^3*c^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+5/64*d/a^4/e^4/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)
*c^3+5/24/d/a^2/e^2/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c-7/16/e/d^2/a^2/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c
*d^2)*x)^(3/2)*c-1/d^5*e^4*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-29/64/d^5*e^4*(c*d*e*x^2+a*d*e+(a*e^2
+c*d^2)*x)^(1/2)-1/4/d^2/a/e/x^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+1/2/d^5*e^6*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^5*e^6*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^3*e^4*l
n((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+93/64/d^5*e^2/a/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)-39/32/d^3*e^2/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*
x)^(1/2)*c-1/2/d^3*e^4*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-35/128/d^4*e^5*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)+5/32/d^2*e^3/(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-29/32/d^4*e/a/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+43/64/d^3/a^2/x*(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c+3/64*e/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^2-7/32/e^2*d/a^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^3-19/64/e
/a^3*c^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x-5/32/a^3/e^3/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^
2-5/64*d^3/a^4/e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^4

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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^{5}}\,{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^5/(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^5), x)

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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^5\,\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^5*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(x^5*(d + e*x)), x)

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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^{5} \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**5/(e*x+d),x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(x**5*(d + e*x)), x)

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