3.454 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^5 (d+e x)} \, dx\)

Optimal. Leaf size=295 \[ -\frac {\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{5/2} d^{7/2} e^{5/2}}+\frac {\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 x^3} \]

[Out]

-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x^4-1/24*(3*c/a/e-5*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)/x^3-1/128*(-a*e^2+c*d^2)^3*(5*a*e^2+3*c*d^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^(5/2)/d^(7/2)/e^(5/2)+1/64*(-a*e^2+c*d^2)*(5*a*e^2+3*c*d^2)*(2*a*d
*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d^3/e^2/x^2

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Rubi [A]  time = 0.39, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {849, 834, 806, 720, 724, 206} \[ \frac {\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (5 a e^2+3 c d^2\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{5/2} d^{7/2} e^{5/2}}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 x^3}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 d x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^5*(d + e*x)),x]

[Out]

((c*d^2 - a*e^2)*(3*c*d^2 + 5*a*e^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]
)/(64*a^2*d^3*e^2*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(4*d*x^4) - (((3*c)/(a*e) - (5*e)/d^2)*
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*x^3) - ((c*d^2 - a*e^2)^3*(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])])/(128*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 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5 (d+e x)} \, dx &=\int \frac {(a e+c d x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^5} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (3 c d^2-5 a e^2\right )+a c d e^2 x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4} \, dx}{4 a d e}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}-\frac {\left (\frac {3 c^2 d^2}{a}+2 c e^2-\frac {5 a e^4}{d^2}\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{16 e}\\ &=\frac {\left (c d^2-a e^2\right ) \left (3 c d^2+5 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\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^2 d^3 e^2}\\ &=\frac {\left (c d^2-a e^2\right ) \left (3 c d^2+5 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (3 c d^2+5 a e^2\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^2 d^3 e^2}\\ &=\frac {\left (c d^2-a e^2\right ) \left (3 c d^2+5 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a^2 d^3 e^2 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 d x^4}-\frac {\left (\frac {3 c}{a e}-\frac {5 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 x^3}-\frac {\left (c d^2-a e^2\right )^3 \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 )}{128 a^{5/2} d^{7/2} e^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 253, normalized size = 0.86 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {x \left (5 a e^2+3 c d^2\right ) \left (\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {d+e x} \sqrt {a e+c d x} \left (a^2 e^2 \left (8 d^2+2 d e x-3 e^2 x^2\right )+2 a c d^2 e x (7 d+4 e x)+3 c^2 d^4 x^2\right )-3 x^3 \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )\right )}{a^{3/2} d^{5/2} e^{3/2} \sqrt {d+e x} \sqrt {a e+c d x}}-48 (d+e x) (a e+c d x)^2\right )}{192 a d e x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^5*(d + e*x)),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-48*(a*e + c*d*x)^2*(d + e*x) + ((3*c*d^2 + 5*a*e^2)*x*(Sqrt[a]*Sqrt[d]*Sqrt[e
]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(3*c^2*d^4*x^2 + 2*a*c*d^2*e*x*(7*d + 4*e*x) + a^2*e^2*(8*d^2 + 2*d*e*x - 3*
e^2*x^2)) - 3*(c*d^2 - a*e^2)^3*x^3*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^
(3/2)*d^(5/2)*e^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(192*a*d*e*x^4)

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fricas [A]  time = 17.38, size = 704, normalized size = 2.39 \[ \left [-\frac {3 \, {\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {a d e} x^{4} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (48 \, a^{4} d^{4} e^{4} - {\left (9 \, a c^{3} d^{7} e - 9 \, a^{2} c^{2} d^{5} e^{3} + 31 \, a^{3} c d^{3} e^{5} - 15 \, a^{4} d e^{7}\right )} x^{3} + 2 \, {\left (3 \, a^{2} c^{2} d^{6} e^{2} + 10 \, a^{3} c d^{4} e^{4} - 5 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (9 \, a^{3} c d^{5} e^{3} + a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, a^{3} d^{4} e^{3} x^{4}}, \frac {3 \, {\left (3 \, c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} - 6 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 5 \, a^{4} e^{8}\right )} \sqrt {-a d e} x^{4} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (48 \, a^{4} d^{4} e^{4} - {\left (9 \, a c^{3} d^{7} e - 9 \, a^{2} c^{2} d^{5} e^{3} + 31 \, a^{3} c d^{3} e^{5} - 15 \, a^{4} d e^{7}\right )} x^{3} + 2 \, {\left (3 \, a^{2} c^{2} d^{6} e^{2} + 10 \, a^{3} c d^{4} e^{4} - 5 \, a^{4} d^{2} e^{6}\right )} x^{2} + 8 \, {\left (9 \, a^{3} c d^{5} e^{3} + a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, a^{3} d^{4} e^{3} x^{4}}\right ] \]

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)^(3/2)/x^5/(e*x+d),x, algorithm="fricas")

[Out]

[-1/768*(3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 - 5*a^4*e^8)*sqrt(a*d*e)*x^4*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*(48*a^4*d^4*e^4 - (9*a*c^3*d^7*
e - 9*a^2*c^2*d^5*e^3 + 31*a^3*c*d^3*e^5 - 15*a^4*d*e^7)*x^3 + 2*(3*a^2*c^2*d^6*e^2 + 10*a^3*c*d^4*e^4 - 5*a^4
*d^2*e^6)*x^2 + 8*(9*a^3*c*d^5*e^3 + a^4*d^3*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^4*e^3
*x^4), 1/384*(3*(3*c^4*d^8 - 4*a*c^3*d^6*e^2 - 6*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 - 5*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 - (9*a*c^3*d^7*e - 9*a^2*c^2*d^5*e^3
+ 31*a^3*c*d^3*e^5 - 15*a^4*d*e^7)*x^3 + 2*(3*a^2*c^2*d^6*e^2 + 10*a^3*c*d^4*e^4 - 5*a^4*d^2*e^6)*x^2 + 8*(9*a
^3*c*d^5*e^3 + a^4*d^3*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^4*e^3*x^4)]

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

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)^(3/2)/x^5/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((-2*exp(1)^3*a^2*exp(2)^2+4*exp(1)^5*
a^2*exp(2)-2*exp(1)^7*a^2)/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)))+(3*a
^4*exp(2)^4+8*exp(1)^2*a^4*exp(2)^3+48*exp(1)^4*a^4*exp(2)^2-192*exp(1)^6*a^4*exp(2)+128*exp(1)^8*a^4+12*c*d^2
*a^3*exp(2)^3+18*c^2*d^4*a^2*exp(2)^2-24*c^2*d^4*exp(1)^2*a^2*exp(2)+12*c^3*d^6*a*exp(2)-16*c^3*d^6*exp(1)^2*a
+3*c^4*d^8)/64/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))-s
qrt(c*d*exp(1))*x)/sqrt(-a*d*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)^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*ex
p(2)^3-240*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)+36*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+54*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)^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)+36*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)^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+9*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)^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
)^6*a^4*exp(2)^2+768*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))-sq
rt(c*d*exp(1))*x)^6*a^4+384*c^2*d^5*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)^6*a^2-33*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+40*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+624*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)-132*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)^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^4*exp(2)^2+768*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^4-198*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)^2-888*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)-132*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)-464*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^2-33*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)^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)^4*a^5*exp(2)^3+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)^2-1920*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*e
xp(1)+a*x*exp(2)+c*d^2*x+c*d*x^2*exp(1))-sqrt(c*d*exp(1))*x)^4*a^5+1152*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)^4*a^4*exp(2)^2+768*c*d^4*exp(1)^4*sqrt(c*d*e
xp(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+1152*c^2*d^6*e
xp(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)^4*a^3*exp(2)+
384*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+384*c^3*d^8*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)^4*a^2-33*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)^4-88*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)^3-52
8*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)-132*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)^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^5*exp(2)+768*c*d^4*exp(1)^8*(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^5-198*c^2*d^6*exp(1)^2*(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)^3*a^4*exp(2)^2-1272*c^2*d^6*exp(1)^4*(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^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^4-132*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)-592*c^3*d^8*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^3-33*c^4*d^10*exp(1)^2*(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)^3*a^2+1536*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+768*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*exp(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+768*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+9*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*e
xp(1))-sqrt(c*d*exp(1))*x)*a^7*exp(2)^3+144*d^3*exp(1)^7*(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^7*exp(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)+36*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)^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^6+
54*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)^2-72*c^
2*d^7*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^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+36*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)-48*c^3*d^9*exp(1)^5*(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)*a^4+9*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)^8*sqrt(c*d*exp(1))*a^7*exp(2)+384*d^4*exp(1)^
10*sqrt(c*d*exp(1))*a^7+128*c*d^6*exp(1)^8*sqrt(c*d*exp(1))*a^6)/384/d^3/exp(1)^2/a^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)^2-d*exp(1)*a)^4)

________________________________________________________________________________________

maple [B]  time = 0.03, size = 2427, normalized size = 8.23 \[ \text {result too large to display} \]

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)^(3/2)/x^5/(e*x+d),x)

[Out]

-3/64/d*e^2/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^2-3/32/d*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)*c-133/192/d^4*e^3*c/a*(c*d*e*x^2+a*d*e+(a*e^2
+c*d^2)*x)^(3/2)*x-3/16/d^2*e^5*a*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+3/64*d^3/a^3/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^4+1/8/d/a^2/e^2/x^3*(
c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c+1/64*d^2/a^4/e^3*c^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-1/64*d
/a^4/e^4/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^3-3/128*d^5/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^4+1/16/d^6*e^9*a^3/c*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)+3/16/d^2*e^5*a*c*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)-1/16/d^6*e^9*a^3/c*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)-13/48/e/d^2/a^2/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c+19/192/e^2/d/a^3/x*(c*d*e*x^2+a*d*
e+(a*e^2+c*d^2)*x)^(5/2)*c^2-91/192/d^2*e/a^2*c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+1/8/d^2*e^3*c*((x+
d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/16*e^3*c^2*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)+11/24/d^3/a/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
5/2)-1/32*e/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^2-29/96/d/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*
c^2+1/16*e^3*c^2*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)-5/64/d^4*e^5*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-3/32/d^2*e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)
^(1/2)*c-1/3/d^5*e^4*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)-23/64/d^5*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(3/2)+3/16/d^4*e^7*a^2*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)+133/192/d^5*e^2/a/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)+3/64*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^2-1/32/a^3/e^3/x^2*(c*d*e
*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2+3/64*d^4/a^3/e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^4+1/64*d^3/a^
4/e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^4-1/4/d^2/a/e/x^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)-53/9
6/d^3*e^2/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c-21/64/d^3*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*
c+5/128/d^3*e^6*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)-59/96/d^4*e/a/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)+91/192/d^3/a^2/x*(c*d*e*x^2+a*d*e+(a*e^2
+c*d^2)*x)^(5/2)*c+1/4/d^5*e^6*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+1/8/d^6*e^7*a^2/c*(c*d*e*x^2+a*d*e+
(a*e^2+c*d^2)*x)^(1/2)-1/4/d^5*e^6*a*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-1/8/d^6*e^7*a^2/c*((x+d/e
)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-3/16/d^4*e^7*a^2*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)+1/4/d^3*e^4*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/
e))^(1/2)*x+1/32/e*d^2/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^3-5/96/e^2*d/a^3*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(3/2)*c^3-19/192/e/a^3*c^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+1/32*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)*c^3+5/64*d/a^2*(c*d*e*x^2+a*
d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} x^{5}}\,{d x} \]

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)^(3/2)/x^5/(e*x+d),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^5), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^5\,\left (d+e\,x\right )} \,d x \]

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)^(3/2)/(x^5*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^5*(d + e*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}{x^{5} \left (d + e x\right )}\, dx \]

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)**(3/2)/x**5/(e*x+d),x)

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(3/2)/(x**5*(d + e*x)), x)

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