3.5.52 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^3 (d+e x)} \, dx\) [452]

Optimal. Leaf size=256 \[ -\frac {\left (2 a d e+\left (5 c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d x^2}+c^{3/2} d^{3/2} \sqrt {e} \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )-\frac {\left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {a} d^{3/2} \sqrt {e}} \]

[Out]

-1/8*(-a^2*e^4+6*a*c*d^2*e^2+3*c^2*d^4)*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))/d^(3/2)/a^(1/2)/e^(1/2)+c^(3/2)*d^(3/2)*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)/c
^(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))*e^(1/2)-1/4*(2*a*d*e+(a*e^2+5*c*d^2)*x)*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/d/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 256, 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 = {863, 824, 857, 635, 212, 738} \begin {gather*} -\frac {\left (-a^2 e^4+6 a c d^2 e^2+3 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 )}{8 \sqrt {a} d^{3/2} \sqrt {e}}+c^{3/2} d^{3/2} \sqrt {e} \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac {\left (x \left (a e^2+5 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}}{4 d x^2} \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^3*(d + e*x)),x]

[Out]

-1/4*((2*a*d*e + (5*c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d*x^2) + c^(3/2)*d^(3/2)*S
qrt[e]*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2])] - ((3*c^2*d^4 + 6*a*c*d^2*e^2 - 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])])/(8*Sqrt[a]*d^(3/2)*Sqrt[e])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 738

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 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2)
)*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d -
b*e)*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*
x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m +
1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m +
1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
0]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rule 863

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

________________________________________________________________________________________

Mathematica [A]
time = 0.57, size = 243, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (5 c d^2 x+a e (2 d+e x)\right )-8 \sqrt {a} c^{3/2} d^3 e x^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )+\left (3 c^2 d^4+6 a c d^2 e^2-a^2 e^4\right ) x^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )\right )}{4 \sqrt {a} d^{3/2} \sqrt {e} x^2 \sqrt {(a e+c d x) (d+e x)}} \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^3*(d + e*x)),x]

[Out]

-1/4*(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(5*c*d^2*x + a*
e*(2*d + e*x)) - 8*Sqrt[a]*c^(3/2)*d^3*e*x^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x
])] + (3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*x^2*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*
d*x])]))/(Sqrt[a]*d^(3/2)*Sqrt[e]*x^2*Sqrt[(a*e + c*d*x)*(d + e*x)])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2437\) vs. \(2(218)=436\).
time = 0.09, size = 2438, normalized size = 9.52

method	result	size
default	\(\text {Expression too large to display}\)	\(2438\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^3/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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^3/(e*x+d),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 5.26, size = 1379, normalized size = 5.39 \begin {gather*} \left [\frac {{\left (8 \, \sqrt {c d} a c d^{3} x^{2} e^{\frac {3}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) - {\left (3 \, c^{2} d^{4} x^{2} + 6 \, a c d^{2} x^{2} e^{2} - a^{2} x^{2} e^{4}\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} + 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) - 4 \, {\left (5 \, a c d^{3} x e + a^{2} d x e^{3} + 2 \, a^{2} d^{2} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-1\right )}}{16 \, a d^{2} x^{2}}, -\frac {{\left (16 \, \sqrt {-c d e} a c d^{3} x^{2} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) e + {\left (3 \, c^{2} d^{4} x^{2} + 6 \, a c d^{2} x^{2} e^{2} - a^{2} x^{2} e^{4}\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} + 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) + 4 \, {\left (5 \, a c d^{3} x e + a^{2} d x e^{3} + 2 \, a^{2} d^{2} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-1\right )}}{16 \, a d^{2} x^{2}}, \frac {{\left (4 \, \sqrt {c d} a c d^{3} x^{2} e^{\frac {3}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + {\left (3 \, c^{2} d^{4} x^{2} + 6 \, a c d^{2} x^{2} e^{2} - a^{2} x^{2} e^{4}\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (5 \, a c d^{3} x e + a^{2} d x e^{3} + 2 \, a^{2} d^{2} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-1\right )}}{8 \, a d^{2} x^{2}}, -\frac {{\left (8 \, \sqrt {-c d e} a c d^{3} x^{2} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) e - {\left (3 \, c^{2} d^{4} x^{2} + 6 \, a c d^{2} x^{2} e^{2} - a^{2} x^{2} e^{4}\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (5 \, a c d^{3} x e + a^{2} d x e^{3} + 2 \, a^{2} d^{2} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-1\right )}}{8 \, a d^{2} x^{2}}\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^3/(e*x+d),x, algorithm="fricas")

[Out]

[1/16*(8*sqrt(c*d)*a*c*d^3*x^2*e^(3/2)*log(8*c^2*d^3*x*e + c^2*d^4 + 8*a*c*d*x*e^3 + a^2*e^4 + 4*sqrt(c*d^2*x
+ a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(c*d)*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e
^2) - (3*c^2*d^4*x^2 + 6*a*c*d^2*x^2*e^2 - a^2*x^2*e^4)*sqrt(a*d)*e^(1/2)*log((c^2*d^4*x^2 + 8*a*c*d^3*x*e + a
^2*x^2*e^4 + 8*a^2*d*x*e^3 + 4*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(
a*d)*e^(1/2) + 2*(3*a*c*d^2*x^2 + 4*a^2*d^2)*e^2)/x^2) - 4*(5*a*c*d^3*x*e + a^2*d*x*e^3 + 2*a^2*d^2*e^2)*sqrt(
c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-1)/(a*d^2*x^2), -1/16*(16*sqrt(-c*d*e)*a*c*d^3*x^2*arctan(1/2*sqrt
(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^3*x*e + a*c*d*x*e^3 +
(c^2*d^2*x^2 + a*c*d^2)*e^2))*e + (3*c^2*d^4*x^2 + 6*a*c*d^2*x^2*e^2 - a^2*x^2*e^4)*sqrt(a*d)*e^(1/2)*log((c^2
*d^4*x^2 + 8*a*c*d^3*x*e + a^2*x^2*e^4 + 8*a^2*d*x*e^3 + 4*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^
2 + (c*d*x^2 + a*d)*e)*sqrt(a*d)*e^(1/2) + 2*(3*a*c*d^2*x^2 + 4*a^2*d^2)*e^2)/x^2) + 4*(5*a*c*d^3*x*e + a^2*d*
x*e^3 + 2*a^2*d^2*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-1)/(a*d^2*x^2), 1/8*(4*sqrt(c*d)*a*c*d
^3*x^2*e^(3/2)*log(8*c^2*d^3*x*e + c^2*d^4 + 8*a*c*d*x*e^3 + a^2*e^4 + 4*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a
*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(c*d)*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) + (3*c^2*d^4*x^2 + 6
*a*c*d^2*x^2*e^2 - a^2*x^2*e^4)*sqrt(-a*d*e)*arctan(1/2*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 +
(c*d*x^2 + a*d)*e)*sqrt(-a*d*e)/(a*c*d^3*x*e + a^2*d*x*e^3 + (a*c*d^2*x^2 + a^2*d^2)*e^2)) - 2*(5*a*c*d^3*x*e
+ a^2*d*x*e^3 + 2*a^2*d^2*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-1)/(a*d^2*x^2), -1/8*(8*sqrt(
-c*d*e)*a*c*d^3*x^2*arctan(1/2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-c
*d*e)/(c^2*d^3*x*e + a*c*d*x*e^3 + (c^2*d^2*x^2 + a*c*d^2)*e^2))*e - (3*c^2*d^4*x^2 + 6*a*c*d^2*x^2*e^2 - a^2*
x^2*e^4)*sqrt(-a*d*e)*arctan(1/2*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqr
t(-a*d*e)/(a*c*d^3*x*e + a^2*d*x*e^3 + (a*c*d^2*x^2 + a^2*d^2)*e^2)) + 2*(5*a*c*d^3*x*e + a^2*d*x*e^3 + 2*a^2*
d^2*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-1)/(a*d^2*x^2)]

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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**3/(e*x+d),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (215) = 430\).
time = 1.81, size = 647, normalized size = 2.53 \begin {gather*} -\sqrt {c d} c d e^{\frac {1}{2}} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right ) + \frac {{\left (3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \arctan \left (-\frac {\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}}{\sqrt {-a d e}}\right )}{4 \, \sqrt {-a d e} d} - \frac {3 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a c^{2} d^{5} e - 5 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} c^{2} d^{4} + 8 \, \sqrt {c d} a^{2} c d^{4} e^{\frac {5}{2}} - 16 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} \sqrt {c d} a c d^{3} e^{\frac {3}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{2} c d^{3} e^{3} - 10 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a c d^{2} e^{2} - 8 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} \sqrt {c d} a^{2} d e^{\frac {7}{2}} - {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} a^{3} d e^{5} - {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{3} a^{2} e^{4}}{4 \, {\left (a d e - {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2}\right )}^{2} d} \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^3/(e*x+d),x, algorithm="giac")

[Out]

-sqrt(c*d)*c*d*e^(1/2)*log(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x +
a*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2))) + 1/4*(3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*arctan(-(sqrt(c*d)
*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))/sqrt(-a*d*e))/(sqrt(-a*d*e)*d) - 1/4*(3*(sqrt(c*d)*x
*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a*c^2*d^5*e - 5*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e
+ c*d^2*x + a*x*e^2 + a*d*e))^3*c^2*d^4 + 8*sqrt(c*d)*a^2*c*d^4*e^(5/2) - 16*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x
^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a*c*d^3*e^(3/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d
^2*x + a*x*e^2 + a*d*e))*a^2*c*d^3*e^3 - 10*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)
)^3*a*c*d^2*e^2 - 8*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a^2*d*e^(7
/2) - (sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^3*d*e^5 - (sqrt(c*d)*x*e^(1/2) - s
qrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^2*e^4)/((a*d*e - (sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2
*x + a*x*e^2 + a*d*e))^2)^2*d)

________________________________________________________________________________________

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^3\,\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^3*(d + e*x)),x)

[Out]

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

________________________________________________________________________________________