3.3.89 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^6 (d+e x)} \, dx\)

Optimal. Leaf size=395 \[ \frac {\left (-35 a^2 e^4+12 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{7/2} d^{9/2} e^{7/2}}-\frac {\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^3 d^4 e^3 x^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 c}{a e}-\frac {7 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 x^4} \]

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Rubi [A]  time = 0.51, antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -\frac {\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\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}}{128 a^3 d^4 e^3 x^2}+\frac {\left (-35 a^2 e^4+12 a c d^2 e^2+15 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\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 )}{256 a^{7/2} d^{9/2} e^{7/2}}-\frac {\left (\frac {3 c}{a e}-\frac {7 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}}{40 x^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 d x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]

[Out]

-((c*d^2 - a*e^2)*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*(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])/(128*a^3*d^4*e^3*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(5*d*x^5) - (((3*
c)/(a*e) - (7*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(40*x^4) + ((15*c^2*d^4 + 12*a*c*d^2*e^2
- 35*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(240*a^2*d^3*e^2*x^3) + ((c*d^2 - a*e^2)^3*(3*c^2
*d^4 + 6*a*c*d^2*e^2 + 7*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])])/(256*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 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^6 (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^6} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (3 c d^2-7 a e^2\right )+2 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^5} \, dx}{5 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}}{5 d x^5}-\frac {\left (\frac {3 c}{a e}-\frac {7 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}}{40 x^4}+\frac {\int \frac {\left (-\frac {1}{4} a e \left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right )-\frac {1}{2} a c d e^2 \left (3 c d^2-7 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4} \, dx}{20 a^2 d^2 e^2}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 c}{a e}-\frac {7 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}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{32 a^2 d^3 e^2}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}}{128 a^3 d^4 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 c}{a e}-\frac {7 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}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 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}{256 a^3 d^4 e^3}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}}{128 a^3 d^4 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 c}{a e}-\frac {7 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}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 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 )}{128 a^3 d^4 e^3}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\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}}{128 a^3 d^4 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 c}{a e}-\frac {7 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}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 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 )}{256 a^{7/2} d^{9/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.49, size = 310, normalized size = 0.78 \begin {gather*} \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {5 x^2 \left (7 a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\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^{5/2} d^{7/2} e^{5/2} \sqrt {d+e x} \sqrt {a e+c d x}}+\frac {48 x (d+e x) \left (7 a e^2+5 c d^2\right ) (a e+c d x)^2}{a d e}-384 (d+e x) (a e+c d x)^2\right )}{1920 a d e x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-384*(a*e + c*d*x)^2*(d + e*x) + (48*(5*c*d^2 + 7*a*e^2)*x*(a*e + c*d*x)^2*(d
+ e*x))/(a*d*e) + (5*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 7*a^2*e^4)*x^2*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sq
rt[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^(5/2)*d^(7/2)*e^(5/2
)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(1920*a*d*e*x^5)

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IntegrateAlgebraic [F]  time = 180.11, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]

[Out]

$Aborted

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fricas [A]  time = 20.24, size = 872, normalized size = 2.21 \begin {gather*} \left [-\frac {15 \, {\left (3 \, c^{5} d^{10} - 3 \, a c^{4} d^{8} e^{2} - 2 \, a^{2} c^{3} d^{6} e^{4} - 6 \, a^{3} c^{2} d^{4} e^{6} + 15 \, a^{4} c d^{2} e^{8} - 7 \, a^{5} e^{10}\right )} \sqrt {a d e} x^{5} \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 (384 \, a^{5} d^{5} e^{5} + {\left (45 \, a c^{4} d^{9} e - 30 \, a^{2} c^{3} d^{7} e^{3} - 36 \, a^{3} c^{2} d^{5} e^{5} + 190 \, a^{4} c d^{3} e^{7} - 105 \, a^{5} d e^{9}\right )} x^{4} - 2 \, {\left (15 \, a^{2} c^{3} d^{8} e^{2} - 9 \, a^{3} c^{2} d^{6} e^{4} + 61 \, a^{4} c d^{4} e^{6} - 35 \, a^{5} d^{2} e^{8}\right )} x^{3} + 8 \, {\left (3 \, a^{3} c^{2} d^{7} e^{3} + 12 \, a^{4} c d^{5} e^{5} - 7 \, a^{5} d^{3} e^{7}\right )} x^{2} + 48 \, {\left (11 \, a^{4} c d^{6} e^{4} + a^{5} d^{4} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{7680 \, a^{4} d^{5} e^{4} x^{5}}, -\frac {15 \, {\left (3 \, c^{5} d^{10} - 3 \, a c^{4} d^{8} e^{2} - 2 \, a^{2} c^{3} d^{6} e^{4} - 6 \, a^{3} c^{2} d^{4} e^{6} + 15 \, a^{4} c d^{2} e^{8} - 7 \, a^{5} e^{10}\right )} \sqrt {-a d e} x^{5} \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 (384 \, a^{5} d^{5} e^{5} + {\left (45 \, a c^{4} d^{9} e - 30 \, a^{2} c^{3} d^{7} e^{3} - 36 \, a^{3} c^{2} d^{5} e^{5} + 190 \, a^{4} c d^{3} e^{7} - 105 \, a^{5} d e^{9}\right )} x^{4} - 2 \, {\left (15 \, a^{2} c^{3} d^{8} e^{2} - 9 \, a^{3} c^{2} d^{6} e^{4} + 61 \, a^{4} c d^{4} e^{6} - 35 \, a^{5} d^{2} e^{8}\right )} x^{3} + 8 \, {\left (3 \, a^{3} c^{2} d^{7} e^{3} + 12 \, a^{4} c d^{5} e^{5} - 7 \, a^{5} d^{3} e^{7}\right )} x^{2} + 48 \, {\left (11 \, a^{4} c d^{6} e^{4} + a^{5} d^{4} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3840 \, a^{4} d^{5} e^{4} x^{5}}\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)^(3/2)/x^6/(e*x+d),x, algorithm="fricas")

[Out]

[-1/7680*(15*(3*c^5*d^10 - 3*a*c^4*d^8*e^2 - 2*a^2*c^3*d^6*e^4 - 6*a^3*c^2*d^4*e^6 + 15*a^4*c*d^2*e^8 - 7*a^5*
e^10)*sqrt(a*d*e)*x^5*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*(384*a^
5*d^5*e^5 + (45*a*c^4*d^9*e - 30*a^2*c^3*d^7*e^3 - 36*a^3*c^2*d^5*e^5 + 190*a^4*c*d^3*e^7 - 105*a^5*d*e^9)*x^4
- 2*(15*a^2*c^3*d^8*e^2 - 9*a^3*c^2*d^6*e^4 + 61*a^4*c*d^4*e^6 - 35*a^5*d^2*e^8)*x^3 + 8*(3*a^3*c^2*d^7*e^3 +
12*a^4*c*d^5*e^5 - 7*a^5*d^3*e^7)*x^2 + 48*(11*a^4*c*d^6*e^4 + a^5*d^4*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^
2 + a*e^2)*x))/(a^4*d^5*e^4*x^5), -1/3840*(15*(3*c^5*d^10 - 3*a*c^4*d^8*e^2 - 2*a^2*c^3*d^6*e^4 - 6*a^3*c^2*d^
4*e^6 + 15*a^4*c*d^2*e^8 - 7*a^5*e^10)*sqrt(-a*d*e)*x^5*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*(
384*a^5*d^5*e^5 + (45*a*c^4*d^9*e - 30*a^2*c^3*d^7*e^3 - 36*a^3*c^2*d^5*e^5 + 190*a^4*c*d^3*e^7 - 105*a^5*d*e^
9)*x^4 - 2*(15*a^2*c^3*d^8*e^2 - 9*a^3*c^2*d^6*e^4 + 61*a^4*c*d^4*e^6 - 35*a^5*d^2*e^8)*x^3 + 8*(3*a^3*c^2*d^7
*e^3 + 12*a^4*c*d^5*e^5 - 7*a^5*d^3*e^7)*x^2 + 48*(11*a^4*c*d^6*e^4 + a^5*d^4*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e +
(c*d^2 + a*e^2)*x))/(a^4*d^5*e^4*x^5)]

________________________________________________________________________________________

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)^(3/2)/x^6/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((2*exp(1)^4*a^2*exp(2)^2-4*exp(1)^6*a
^2*exp(2)+2*exp(1)^8*a^2)/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)))-(3*a^
5*exp(2)^5+6*exp(1)^2*a^5*exp(2)^4+16*exp(1)^4*a^5*exp(2)^3+96*exp(1)^6*a^5*exp(2)^2-384*exp(1)^8*a^5*exp(2)+2
56*exp(1)^10*a^5+15*c*d^2*a^4*exp(2)^4+30*c^2*d^4*a^3*exp(2)^3-36*c^2*d^4*exp(1)^2*a^3*exp(2)^2+30*c^3*d^6*a^2
*exp(2)^2-48*c^3*d^6*exp(1)^2*a^2*exp(2)+16*c^3*d^6*exp(1)^4*a^2+15*c^4*d^8*a*exp(2)-18*c^4*d^8*exp(1)^2*a+3*c
^5*d^10)/128/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))-sqr
t(c*d*exp(1))*x)/sqrt(-a*d*exp(1)))-(45*(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
)^9*a^5*exp(2)^5+90*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)^9*a^5*exp
(2)^4+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)^9*a^5*exp(2)^3-2400
*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)^9*a^5*exp(2)^2+1920*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)^9*a^5*exp(2)+225*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)^9*a^4*exp(2)^4+450*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)^9*a^3*exp(2)^3-540*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)^9*a^3*exp(2)^2+450*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)^9*a^2*exp(2)^2-720*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)^9*a^2*exp(2)+240*c^3*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)^9*a^2+225*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)^9*a*exp(2)-270*c^4*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
)^9*a+45*c^5*d^10*(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)^9-3840*d*exp(1)^5*sq
rt(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)^8*a^5*exp(2)^2+7680*d*e
xp(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)^8*a^5*exp(2)-
3840*d*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)^8*a^5
-210*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)^7*a^6*exp(2)^5-420*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)^7*a^6*exp(2)^4+160*d*exp(1)^5*(s
qrt(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^6*exp(2)^3+8640*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)^7*a^6*exp(2)^2-7680*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^6*exp(2)-1050*c*d^3*exp(1)*(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)^7*a^5*exp(2)^4-3840*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)^7*a^5*exp(2)^2+7680*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)^7*a^5*exp(2)-3840*c*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)^7*a^5-2100*c^2*d^5*exp(1)*(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^4*exp(2)^3+2520*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)^7*a^4*exp(2)^2-2100*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)^7*a^3*exp(2)^2+3360*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)^7*a^3*exp(2)+4000*c^3*d^7*exp(1)^5*(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)^7*a^3-1050*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)^7*a^2*exp(2)+1260*c^4*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)^7*a^2-210*c^5*d^11*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)^7*a+3840*
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)^6*a^6*ex
p(2)^3+7680*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)^6*a^6*exp(2)^2-26880*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))-sqr
t(c*d*exp(1))*x)^6*a^6*exp(2)+15360*d^2*exp(1)^10*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^6-3840*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)^6*a^5*exp(2)^2+7680*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)^6*a^5*exp(2)-3840*c*d^4*exp(1)^8*sqrt(c*d*exp(1))*(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)^6*a^5-19200*c^2*d^6*exp(1)^4*sqrt(c*d*exp(1))*(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)^6*a^4*exp(2)-11520*c^3*d^8*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)^6*a^3+384*d^2*exp(1)^2*(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)^5*a^7*exp(2)^5-1280*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)^5*a^7*exp(2)^3-11520*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)^5*a^7*exp(2)^2+11520*d^2*exp(1)^10*(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^7*exp(2)+1920*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)^5*a^6*exp(2)^4+7680*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)^5*a^6*exp(2)^3-3840*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)^5*a^6*exp(2)^2-15360*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)^5*a^6*exp(2)+11520*c*d^4*exp(1)^10*(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^6+3840*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)^5*a^5*exp(2)^3+23040*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)^5*a^5*exp(2)^2+7680*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)^5*a^5*exp(2)-3840*c^2*d^6*exp(1)^8*(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)^5*a^5+3840*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)^5*a^4*exp(2)^2+23040*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)^5*a^4*exp(2)+10240*c^3*d^8*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)^5*a^4+1920*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))-sqrt(c*d*exp(1)
)*x)^5*a^3*exp(2)+7680*c^4*d^10*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+384*c^5*d^12*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^2-3
840*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)^4*a^
7*exp(2)^4-3840*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)^4*a^7*exp(2)^3-3840*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)^4*a^7*exp(2)^2+34560*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)^4*a^7*exp(2)-23040*d^3*exp(1)^11*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^7-15360*c*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)^4*a^6*exp(2)^3-19200*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)^4*a^6*exp(2)^2+6400*c*d^5*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)^4*a^6-23040*c^2*d^7*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)^4*a^5*exp(2)^2-26880*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)^4*a^5
*exp(2)-3840*c^2*d^7*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*ex
p(1))*x)^4*a^5-15360*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))-sqr
t(c*d*exp(1))*x)^4*a^4*exp(2)-11520*c^3*d^9*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^4-3840*c^4*d^11*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+210*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^8*exp(2)^5+420*d^3*exp(1)^5*(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^8*exp(2)^4+1120*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)^3*a^8*exp(2)^3+6720*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*ex
p(1))*x)^3*a^8*exp(2)^2-7680*d^3*exp(1)^11*(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^8*exp(2)+1050*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)^
3*a^7*exp(2)^4+7680*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)^3*a
^7*exp(2)^3+7680*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)^3*a^7*
exp(2)^2+7680*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)^3*a^7*exp
(2)-11520*c*d^5*exp(1)^11*(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^7+2100*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)^3*a^6*exp(2)^3+20520*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)^3*a^6*exp(2)^2+19200*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)^3*a^6*exp(2)+2100*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)^3*a^5*exp(2)^2+19680*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)^3*a^5*exp(2)+12640*c^3*d^
9*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)^3*a^5+1050*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)^3*a^4*exp(2)+6420*c^4*d^11*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+210*c^5*d^13*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^3-19200*d^4*exp(1)^10*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^8*exp(2)+15360*d^4*exp(1)^12*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)^2*a^8-7680*c*d^6*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^7*exp(2)^2-7680*c*d^6*
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*exp(1))*x)^2*a^7*exp(2)
-1280*c*d^6*exp(1)^10*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^7-15360*c^2*d^8*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*ex
p(1))*x)^2*a^6*exp(2)-7680*c^2*d^8*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*exp(1))*x)^2*a^6-7680*c^3*d^10*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^5-45*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^9*exp(2)^5-90*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)*a^9*exp(2)^4-240*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)*a^
9*exp(2)^3-1440*d^4*exp(1)^10*(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^9*exp(
2)^2+1920*d^4*exp(1)^12*(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^9*exp(2)-225
*c*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^8*exp(2)^4+3840*c*d^
6*exp(1)^12*(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^8-450*c^2*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)*a^7*exp(2)^3+540*c^2*d^8*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^7*exp(2)^2+3840*c^2*d^8*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)*a^7*exp(2)+3840*c^2*d^8*exp(1)^10*(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^7-450*c^3*d^10*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^6*exp(2)^2+720*c^3*d^10*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^6*exp(2)+3600*c^3*d^10*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)*a^6-225*c^4*d^12*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)+270*c^4*d^12*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^5-45*c^5*d^14*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+3840*d^5*exp(1)^11*sqrt(c*d*exp(1))*a^9*exp(2)-3840*d^5*exp(1)^13*sqrt(c*d*exp(1))*a^9-1280*c*d
^7*exp(1)^11*sqrt(c*d*exp(1))*a^8-768*c^2*d^9*exp(1)^9*sqrt(c*d*exp(1))*a^7)/3840/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)^5)

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maple [B]  time = 0.03, size = 2888, normalized size = 7.31 \begin {gather*} \text {output too large to display} \end {gather*}

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^6/(e*x+d),x)

[Out]

7/128/d^5*e^6*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-1/32*d/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^3
-3/128*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^2-1/8/d^3*e^4*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3/8/d^3/a/x^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(5/2)+7/48/e/a^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^3+15/128/d^3*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(1/2)*c+1/3/d^6*e^5*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)+45/128/d^6*e^5*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(3/2)+121/192/d^5*e^2/a/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)+19/48/d^2*e/a^2*(c*d*e*x^2+a*d*e
+(a*e^2+c*d^2)*x)^(3/2)*c^2+1/32/d*e^2/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^2+1/4/d^6*e^7*a*((x+d/e)^2*
c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+1/8/d^7*e^8*a^2/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3/16/d^5*
e^8*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^4*e^5*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+1/16/d*e^4*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)-1/16/a^3/e^3/x^3*
(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2-3/64/a^4/e^3/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^3-1/5/d^2
/a/e/x^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)-3/128*d^3/a^3/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^4
+3/128*d^2/a^4/e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^4-1/128*d^4/a^5/e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(3/2)*c^5-3/128*d^5/a^4/e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^5-3/16/d^5*e^8*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)-1/16/d*e^4*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)-263/384/d^6*e^3/a
/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)+227/384/d^4*e^3/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c+39/128/
d^4*e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c-7/256/d^4*e^7*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)-1/32*e/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/
2)*x*c^3+23/96/d/a^3*c^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+73/192/d^3/a^2/x^2*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(5/2)*c-25/48/d^4*e/a/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)-1/4/d^6*e^7*a*(c*d*e*x^2+a*d*e+(a*
e^2+c*d^2)*x)^(1/2)*x-1/8/d^7*e^8*a^2/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+1/8/d/a^2/e^2/x^4*(c*d*e*x^2+a
*d*e+(a*e^2+c*d^2)*x)^(5/2)*c-1/4/d^2/a^2/e/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c-1/128*d^2*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^3-23/96/d^2/
e/a^3/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2+1/16/d^7*e^10*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)+103/192/d^3*e^2/a^2*c^2*(c*d*e*x^2+a*d*e+(a*e^
2+c*d^2)*x)^(3/2)*x-103/192/d^4*e/a^2/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c+3/64/d^2*e^3/a*(c*d*e*x^2+a*
d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^2+263/384/d^5*e^4*c/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+3/16/d^3*e^6*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+15/256
/d^2*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)*c-3/64/e*d^2/a^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^4-1/16/d^7*e^10*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^3*e^6*a*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)
-3/256*d^4/a^2/e/(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+3/256*d^6/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^5-3/128*d^4/a^4/e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^5+3/64*d/a^4/e^2*c^4*
(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x-1/128*d^3/a^5/e^4*c^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+9/64
/d/a^3/e^2/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^2+1/64*d/a^4/e^4/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x
)^(5/2)*c^3+1/128*d^2/a^5/e^5/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^4

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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^{6}}\,{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)^(3/2)/x^6/(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^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^6\,\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)^(3/2)/(x^6*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(3/2)/x**6/(e*x+d),x)

[Out]

Timed out

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