\(\int \frac {1}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [1961]

Optimal result

Integrand size = 37, antiderivative size = 181 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{5 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 627} \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {16 c^2 d^2 \left (a e^2+c d^2+2 c d e x\right )}{5 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {4 c d}{5 (d+e x) \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

[In]

Int[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

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

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 672

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

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \left (a^3 e^6-a^2 c d e^4 (5 d+2 e x)+a c^2 d^2 e^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )+c^3 d^3 \left (5 d^3+30 d^2 e x+40 d e^2 x^2+16 e^3 x^3\right )\right )}{5 \left (c d^2-a e^2\right )^4 (d+e x)^2 \sqrt {(a e+c d x) (d+e x)}} \]

[In]

Integrate[1/((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*(a^3*e^6 - a^2*c*d*e^4*(5*d + 2*e*x) + a*c^2*d^2*e^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + c^3*d^3*(5*d^3 + 30
*d^2*e*x + 40*d*e^2*x^2 + 16*e^3*x^3)))/(5*(c*d^2 - a*e^2)^4*(d + e*x)^2*Sqrt[(a*e + c*d*x)*(d + e*x)])

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.19

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (16 x^{3} c^{3} d^{3} e^{3}+8 x^{2} a \,c^{2} d^{2} e^{4}+40 x^{2} c^{3} d^{4} e^{2}-2 x \,a^{2} c d \,e^{5}+20 x a \,c^{2} d^{3} e^{3}+30 x \,c^{3} d^{5} e +e^{6} a^{3}-5 d^{2} e^{4} a^{2} c +15 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}\right )}{5 \left (e x +d \right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(216\)
trager	\(-\frac {2 \left (16 x^{3} c^{3} d^{3} e^{3}+8 x^{2} a \,c^{2} d^{2} e^{4}+40 x^{2} c^{3} d^{4} e^{2}-2 x \,a^{2} c d \,e^{5}+20 x a \,c^{2} d^{3} e^{3}+30 x \,c^{3} d^{5} e +e^{6} a^{3}-5 d^{2} e^{4} a^{2} c +15 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{5 \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right ) \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \left (e x +d \right )^{3}}\)	\(219\)
default	\(\frac {-\frac {2}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 c d e \left (-\frac {2}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right )}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (e^{2} a -c \,d^{2}\right )}}{e^{2}}\)	\(227\)

[In]

int(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*(c*d*x+a*e)*(16*c^3*d^3*e^3*x^3+8*a*c^2*d^2*e^4*x^2+40*c^3*d^4*e^2*x^2-2*a^2*c*d*e^5*x+20*a*c^2*d^3*e^3*x
+30*c^3*d^5*e*x+a^3*e^6-5*a^2*c*d^2*e^4+15*a*c^2*d^4*e^2+5*c^3*d^6)/(e*x+d)/(a^4*e^8-4*a^3*c*d^2*e^6+6*a^2*c^2
*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (169) = 338\).

Time = 3.70 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.73 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (16 \, c^{3} d^{3} e^{3} x^{3} + 5 \, c^{3} d^{6} + 15 \, a c^{2} d^{4} e^{2} - 5 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + 8 \, {\left (5 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (15 \, c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{5 \, {\left (a c^{4} d^{11} e - 4 \, a^{2} c^{3} d^{9} e^{3} + 6 \, a^{3} c^{2} d^{7} e^{5} - 4 \, a^{4} c d^{5} e^{7} + a^{5} d^{3} e^{9} + {\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x^{4} + {\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )} x^{3} + 3 \, {\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )} x^{2} + {\left (c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}\right )} x\right )}} \]

[In]

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

-2/5*(16*c^3*d^3*e^3*x^3 + 5*c^3*d^6 + 15*a*c^2*d^4*e^2 - 5*a^2*c*d^2*e^4 + a^3*e^6 + 8*(5*c^3*d^4*e^2 + a*c^2
*d^2*e^4)*x^2 + 2*(15*c^3*d^5*e + 10*a*c^2*d^3*e^3 - a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*
x)/(a*c^4*d^11*e - 4*a^2*c^3*d^9*e^3 + 6*a^3*c^2*d^7*e^5 - 4*a^4*c*d^5*e^7 + a^5*d^3*e^9 + (c^5*d^9*e^3 - 4*a*
c^4*d^7*e^5 + 6*a^2*c^3*d^5*e^7 - 4*a^3*c^2*d^3*e^9 + a^4*c*d*e^11)*x^4 + (3*c^5*d^10*e^2 - 11*a*c^4*d^8*e^4 +
14*a^2*c^3*d^6*e^6 - 6*a^3*c^2*d^4*e^8 - a^4*c*d^2*e^10 + a^5*e^12)*x^3 + 3*(c^5*d^11*e - 3*a*c^4*d^9*e^3 + 2
*a^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5*e^7 - 3*a^4*c*d^3*e^9 + a^5*d*e^11)*x^2 + (c^5*d^12 - a*c^4*d^10*e^2 - 6*a^2*
c^3*d^8*e^4 + 14*a^3*c^2*d^6*e^6 - 11*a^4*c*d^4*e^8 + 3*a^5*d^2*e^10)*x)

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]

[In]

integrate(1/(e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Integral(1/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assu
me?` for mor

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3930 vs. \(2 (169) = 338\).

Time = 0.45 (sec) , antiderivative size = 3930, normalized size of antiderivative = 21.71 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

2/5*(16*c^3*d^3*e*abs(e)*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d*e)*c^4*d^8 - 4*sqrt(c*d*e)*a*c^3*d^6*e^2 + 6*sqrt(c
*d*e)*a^2*c^2*d^4*e^4 - 4*sqrt(c*d*e)*a^3*c*d^2*e^6 + sqrt(c*d*e)*a^4*e^8) - (5*c^3*d^3/((c^4*d^8*e^2*sgn(1/(e
*x + d))*sgn(e) - 4*a*c^3*d^6*e^4*sgn(1/(e*x + d))*sgn(e) + 6*a^2*c^2*d^4*e^6*sgn(1/(e*x + d))*sgn(e) - 4*a^3*
c*d^2*e^8*sgn(1/(e*x + d))*sgn(e) + a^4*e^10*sgn(1/(e*x + d))*sgn(e))*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(
e*x + d))) + (15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^18*d^34*e^22*sgn(1/(e*x + d))^4*sgn(e)^4
- 240*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^17*d^32*e^24*sgn(1/(e*x + d))^4*sgn(e)^4 + 1800*sq
rt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^2*c^16*d^30*e^26*sgn(1/(e*x + d))^4*sgn(e)^4 - 8400*sqrt(c*d
*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^3*c^15*d^28*e^28*sgn(1/(e*x + d))^4*sgn(e)^4 + 27300*sqrt(c*d*e -
c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^4*c^14*d^26*e^30*sgn(1/(e*x + d))^4*sgn(e)^4 - 65520*sqrt(c*d*e - c*d^2
*e/(e*x + d) + a*e^3/(e*x + d))*a^5*c^13*d^24*e^32*sgn(1/(e*x + d))^4*sgn(e)^4 + 120120*sqrt(c*d*e - c*d^2*e/(
e*x + d) + a*e^3/(e*x + d))*a^6*c^12*d^22*e^34*sgn(1/(e*x + d))^4*sgn(e)^4 - 171600*sqrt(c*d*e - c*d^2*e/(e*x
+ d) + a*e^3/(e*x + d))*a^7*c^11*d^20*e^36*sgn(1/(e*x + d))^4*sgn(e)^4 + 193050*sqrt(c*d*e - c*d^2*e/(e*x + d)
+ a*e^3/(e*x + d))*a^8*c^10*d^18*e^38*sgn(1/(e*x + d))^4*sgn(e)^4 - 171600*sqrt(c*d*e - c*d^2*e/(e*x + d) + a
*e^3/(e*x + d))*a^9*c^9*d^16*e^40*sgn(1/(e*x + d))^4*sgn(e)^4 + 120120*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/
(e*x + d))*a^10*c^8*d^14*e^42*sgn(1/(e*x + d))^4*sgn(e)^4 - 65520*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x
+ d))*a^11*c^7*d^12*e^44*sgn(1/(e*x + d))^4*sgn(e)^4 + 27300*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))
*a^12*c^6*d^10*e^46*sgn(1/(e*x + d))^4*sgn(e)^4 - 8400*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^13*
c^5*d^8*e^48*sgn(1/(e*x + d))^4*sgn(e)^4 + 1800*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^14*c^4*d^6
*e^50*sgn(1/(e*x + d))^4*sgn(e)^4 - 240*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^15*c^3*d^4*e^52*sg
n(1/(e*x + d))^4*sgn(e)^4 + 15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^16*c^2*d^2*e^54*sgn(1/(e*x
+ d))^4*sgn(e)^4 - 5*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c^17*d^33*e^21*sgn(1/(e*x + d))^4*sgn
(e)^4 + 80*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a*c^16*d^31*e^23*sgn(1/(e*x + d))^4*sgn(e)^4 -
600*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^2*c^15*d^29*e^25*sgn(1/(e*x + d))^4*sgn(e)^4 + 2800*
(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^3*c^14*d^27*e^27*sgn(1/(e*x + d))^4*sgn(e)^4 - 9100*(c*d
*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^4*c^13*d^25*e^29*sgn(1/(e*x + d))^4*sgn(e)^4 + 21840*(c*d*e
- c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^5*c^12*d^23*e^31*sgn(1/(e*x + d))^4*sgn(e)^4 - 40040*(c*d*e - c
*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^6*c^11*d^21*e^33*sgn(1/(e*x + d))^4*sgn(e)^4 + 57200*(c*d*e - c*d^
2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^7*c^10*d^19*e^35*sgn(1/(e*x + d))^4*sgn(e)^4 - 64350*(c*d*e - c*d^2*e
/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^8*c^9*d^17*e^37*sgn(1/(e*x + d))^4*sgn(e)^4 + 57200*(c*d*e - c*d^2*e/(e*
x + d) + a*e^3/(e*x + d))^(3/2)*a^9*c^8*d^15*e^39*sgn(1/(e*x + d))^4*sgn(e)^4 - 40040*(c*d*e - c*d^2*e/(e*x +
d) + a*e^3/(e*x + d))^(3/2)*a^10*c^7*d^13*e^41*sgn(1/(e*x + d))^4*sgn(e)^4 + 21840*(c*d*e - c*d^2*e/(e*x + d)
+ a*e^3/(e*x + d))^(3/2)*a^11*c^6*d^11*e^43*sgn(1/(e*x + d))^4*sgn(e)^4 - 9100*(c*d*e - c*d^2*e/(e*x + d) + a*
e^3/(e*x + d))^(3/2)*a^12*c^5*d^9*e^45*sgn(1/(e*x + d))^4*sgn(e)^4 + 2800*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(
e*x + d))^(3/2)*a^13*c^4*d^7*e^47*sgn(1/(e*x + d))^4*sgn(e)^4 - 600*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x +
d))^(3/2)*a^14*c^3*d^5*e^49*sgn(1/(e*x + d))^4*sgn(e)^4 + 80*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/
2)*a^15*c^2*d^3*e^51*sgn(1/(e*x + d))^4*sgn(e)^4 - 5*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^16*
c*d*e^53*sgn(1/(e*x + d))^4*sgn(e)^4 + (c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*c^16*d^32*e^20*sgn(
1/(e*x + d))^4*sgn(e)^4 - 16*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a*c^15*d^30*e^22*sgn(1/(e*x +
d))^4*sgn(e)^4 + 120*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^2*c^14*d^28*e^24*sgn(1/(e*x + d))^
4*sgn(e)^4 - 560*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^3*c^13*d^26*e^26*sgn(1/(e*x + d))^4*sgn
(e)^4 + 1820*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^4*c^12*d^24*e^28*sgn(1/(e*x + d))^4*sgn(e)^
4 - 4368*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^5*c^11*d^22*e^30*sgn(1/(e*x + d))^4*sgn(e)^4 +
8008*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^6*c^10*d^20*e^32*sgn(1/(e*x + d))^4*sgn(e)^4 - 1144
0*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^7*c^9*d^18*e^34*sgn(1/(e*x + d))^4*sgn(e)^4 + 12870*(c
*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^8*c^8*d^16*e^36*sgn(1/(e*x + d))^4*sgn(e)^4 - 11440*(c*d*e
- c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^9*c^7*d^14*e^38*sgn(1/(e*x + d))^4*sgn(e)^4 + 8008*(c*d*e - c*
d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^10*c^6*d^12*e^40*sgn(1/(e*x + d))^4*sgn(e)^4 - 4368*(c*d*e - c*d^2*
e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*a^11*c^5*d^10*e^42*sgn(1/(e*x + d))^4*sgn(e)^4 + 1820*(c*d*e - c*d^2*e/(e
*x + d) + a*e^3/(e*x + d))^(5/2)*a^12*c^4*d^8*e^44*sgn(1/(e*x + d))^4*sgn(e)^4 - 560*(c*d*e - c*d^2*e/(e*x + d
) + a*e^3/(e*x + d))^(5/2)*a^13*c^3*d^6*e^46*sgn(1/(e*x + d))^4*sgn(e)^4 + 120*(c*d*e - c*d^2*e/(e*x + d) + a*
e^3/(e*x + d))^(5/2)*a^14*c^2*d^4*e^48*sgn(1/(e*x + d))^4*sgn(e)^4 - 16*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*
x + d))^(5/2)*a^15*c*d^2*e^50*sgn(1/(e*x + d))^4*sgn(e)^4 + (c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2
)*a^16*e^52*sgn(1/(e*x + d))^4*sgn(e)^4)/(c^20*d^40*e^25*sgn(1/(e*x + d))^5*sgn(e)^5 - 20*a*c^19*d^38*e^27*sgn
(1/(e*x + d))^5*sgn(e)^5 + 190*a^2*c^18*d^36*e^29*sgn(1/(e*x + d))^5*sgn(e)^5 - 1140*a^3*c^17*d^34*e^31*sgn(1/
(e*x + d))^5*sgn(e)^5 + 4845*a^4*c^16*d^32*e^33*sgn(1/(e*x + d))^5*sgn(e)^5 - 15504*a^5*c^15*d^30*e^35*sgn(1/(
e*x + d))^5*sgn(e)^5 + 38760*a^6*c^14*d^28*e^37*sgn(1/(e*x + d))^5*sgn(e)^5 - 77520*a^7*c^13*d^26*e^39*sgn(1/(
e*x + d))^5*sgn(e)^5 + 125970*a^8*c^12*d^24*e^41*sgn(1/(e*x + d))^5*sgn(e)^5 - 167960*a^9*c^11*d^22*e^43*sgn(1
/(e*x + d))^5*sgn(e)^5 + 184756*a^10*c^10*d^20*e^45*sgn(1/(e*x + d))^5*sgn(e)^5 - 167960*a^11*c^9*d^18*e^47*sg
n(1/(e*x + d))^5*sgn(e)^5 + 125970*a^12*c^8*d^16*e^49*sgn(1/(e*x + d))^5*sgn(e)^5 - 77520*a^13*c^7*d^14*e^51*s
gn(1/(e*x + d))^5*sgn(e)^5 + 38760*a^14*c^6*d^12*e^53*sgn(1/(e*x + d))^5*sgn(e)^5 - 15504*a^15*c^5*d^10*e^55*s
gn(1/(e*x + d))^5*sgn(e)^5 + 4845*a^16*c^4*d^8*e^57*sgn(1/(e*x + d))^5*sgn(e)^5 - 1140*a^17*c^3*d^6*e^59*sgn(1
/(e*x + d))^5*sgn(e)^5 + 190*a^18*c^2*d^4*e^61*sgn(1/(e*x + d))^5*sgn(e)^5 - 20*a^19*c*d^2*e^63*sgn(1/(e*x + d
))^5*sgn(e)^5 + a^20*e^65*sgn(1/(e*x + d))^5*sgn(e)^5))*e^3*abs(e))/abs(e)

Mupad [B] (verification not implemented)

Time = 10.82 (sec) , antiderivative size = 1005, normalized size of antiderivative = 5.55 \[ \int \frac {1}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\left (\frac {16\,c^3\,d^4\,e}{15\,{\left (a\,e^2-c\,d^2\right )}^5}-\frac {8\,c^2\,d^2\,e\,\left (c\,d^2+a\,e^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^5}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {e^2\,\left (10\,c^2\,d^3-18\,a\,c\,d\,e^2\right )}{5\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )}+\frac {8\,c^2\,d^3\,e^2}{5\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a^2\,e^5-6\,a\,c\,d^2\,e^3+3\,c^2\,d^4\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {2\,e^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3\,\left (5\,a^2\,e^5-10\,a\,c\,d^2\,e^3+5\,c^2\,d^4\,e\right )}-\frac {\left (x\,\left (\frac {32\,a\,c^5\,d^6\,e^4}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}-\frac {\left (c\,d^2+a\,e^2\right )\,\left (\frac {16\,c^5\,d^5\,e^3\,\left (c\,d^2+a\,e^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}+\frac {8\,c^5\,d^5\,e^3\,\left (3\,a\,e^2-11\,c\,d^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}\right )}{c\,d\,e}+\frac {2\,c^2\,d^2\,e^2\,\left (58\,a^2\,c^2\,d^2\,e^4-104\,a\,c^3\,d^4\,e^2+30\,c^4\,d^6\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}+\frac {4\,c^4\,d^4\,e^2\,\left (c\,d^2+a\,e^2\right )\,\left (3\,a\,e^2-11\,c\,d^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}\right )-\frac {a\,\left (\frac {16\,c^5\,d^5\,e^3\,\left (c\,d^2+a\,e^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}+\frac {8\,c^5\,d^5\,e^3\,\left (3\,a\,e^2-11\,c\,d^2\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}\right )}{c}+\frac {c\,d\,e\,\left (c\,d^2+a\,e^2\right )\,\left (58\,a^2\,c^2\,d^2\,e^4-104\,a\,c^3\,d^4\,e^2+30\,c^4\,d^6\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^4\,\left (a^2\,c\,d\,e^5-2\,a\,c^2\,d^3\,e^3+c^3\,d^5\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\left (a\,e+c\,d\,x\right )\,\left (d+e\,x\right )} \]

[In]

int(1/((d + e*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

(((16*c^3*d^4*e)/(15*(a*e^2 - c*d^2)^5) - (8*c^2*d^2*e*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^5))*(x*(a*e^2 + c*
d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((e^2*(10*c^2*d^3 - 18*a*c*d*e^2))/(5*(a*e^2 - c*d^2)^2*(3*a^2*e
^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)) + (8*c^2*d^3*e^2)/(5*(a*e^2 - c*d^2)^2*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2
*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (2*e^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e
*x^2)^(1/2))/((d + e*x)^3*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)) - ((x*((32*a*c^5*d^6*e^4)/(15*(a*e^2 - c
*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - ((a*e^2 + c*d^2)*((16*c^5*d^5*e^3*(a*e^2 + c*d^2))/(15*
(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^5*d^5*e^3*(3*a*e^2 - 11*c*d^2))/(15*(a*e
^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(30*c^4*d^6 - 104*a*c^3*
d^4*e^2 + 58*a^2*c^2*d^2*e^4))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (4*c^4*d^4
*e^2*(a*e^2 + c*d^2)*(3*a*e^2 - 11*c*d^2))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))
- (a*((16*c^5*d^5*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (
8*c^5*d^5*e^3*(3*a*e^2 - 11*c*d^2))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (
c*d*e*(a*e^2 + c*d^2)*(30*c^4*d^6 - 104*a*c^3*d^4*e^2 + 58*a^2*c^2*d^2*e^4))/(15*(a*e^2 - c*d^2)^4*(c^3*d^5*e
- 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((a*e + c*d*x)*(d + e*x))