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

Optimal. Leaf size=181 \[ \frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {4 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {16 c^2 d^2 \left (c d^2+a e^2+2 c d e x\right )}{5 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[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)

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Rubi [A]
time = 0.05, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 627} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {(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*}

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Mathematica [A]
time = 0.16, size = 123, normalized size = 0.68 \begin {gather*} -\frac {2 (a e+c d x)^5 \left (e^3-\frac {5 c d e^2 (d+e x)}{a e+c d x}+\frac {15 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {5 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )}{5 \left (c d^2-a e^2\right )^4 ((a e+c d x) (d+e x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[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*e + c*d*x)^5*(e^3 - (5*c*d*e^2*(d + e*x))/(a*e + c*d*x) + (15*c^2*d^2*e*(d + e*x)^2)/(a*e + c*d*x)^2 +
(5*c^3*d^3*(d + e*x)^3)/(a*e + c*d*x)^3))/(5*(c*d^2 - a*e^2)^4*((a*e + c*d*x)*(d + e*x))^(5/2))

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Maple [A]
time = 0.72, size = 227, normalized size = 1.25

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

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (171) = 342\).
time = 17.33, size = 503, normalized size = 2.78 \begin {gather*} -\frac {2 \, {\left (30 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} - 2 \, a^{2} c d x e^{5} + a^{3} e^{6} + {\left (8 \, a c^{2} d^{2} x^{2} - 5 \, a^{2} c d^{2}\right )} e^{4} + 4 \, {\left (4 \, c^{3} d^{3} x^{3} + 5 \, a c^{2} d^{3} x\right )} e^{3} + 5 \, {\left (8 \, c^{3} d^{4} x^{2} + 3 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{5 \, {\left (c^{5} d^{12} x + a^{5} x^{3} e^{12} + {\left (a^{4} c d x^{4} + 3 \, a^{5} d x^{2}\right )} e^{11} - {\left (a^{4} c d^{2} x^{3} - 3 \, a^{5} d^{2} x\right )} e^{10} - {\left (4 \, a^{3} c^{2} d^{3} x^{4} + 9 \, a^{4} c d^{3} x^{2} - a^{5} d^{3}\right )} e^{9} - {\left (6 \, a^{3} c^{2} d^{4} x^{3} + 11 \, a^{4} c d^{4} x\right )} e^{8} + 2 \, {\left (3 \, a^{2} c^{3} d^{5} x^{4} + 3 \, a^{3} c^{2} d^{5} x^{2} - 2 \, a^{4} c d^{5}\right )} e^{7} + 14 \, {\left (a^{2} c^{3} d^{6} x^{3} + a^{3} c^{2} d^{6} x\right )} e^{6} - 2 \, {\left (2 \, a c^{4} d^{7} x^{4} - 3 \, a^{2} c^{3} d^{7} x^{2} - 3 \, a^{3} c^{2} d^{7}\right )} e^{5} - {\left (11 \, a c^{4} d^{8} x^{3} + 6 \, a^{2} c^{3} d^{8} x\right )} e^{4} + {\left (c^{5} d^{9} x^{4} - 9 \, a c^{4} d^{9} x^{2} - 4 \, a^{2} c^{3} d^{9}\right )} e^{3} + {\left (3 \, c^{5} d^{10} x^{3} - a c^{4} d^{10} x\right )} e^{2} + {\left (3 \, c^{5} d^{11} x^{2} + a c^{4} d^{11}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3774 vs. \(2 (171) = 342\).
time = 2.17, size = 3774, normalized size = 20.85 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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^2*sgn(1/(x*e + d))/(sqrt(c*d)*c^4*d^8*e^(1/2) - 4*sqrt(c*d)*a*c^3*d^6*e^(5/2) + 6*sqrt(c*d)*
a^2*c^2*d^4*e^(9/2) - 4*sqrt(c*d)*a^3*c*d^2*e^(13/2) + sqrt(c*d)*a^4*e^(17/2)) - (5*c^3*d^3/((c^4*d^8*e^2*sgn(
1/(x*e + d)) - 4*a*c^3*d^6*e^4*sgn(1/(x*e + d)) + 6*a^2*c^2*d^4*e^6*sgn(1/(x*e + d)) - 4*a^3*c*d^2*e^8*sgn(1/(
x*e + d)) + a^4*e^10*sgn(1/(x*e + d)))*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))) + (15*sqrt(c*d*e - c
*d^2*e/(x*e + d) + a*e^3/(x*e + d))*c^18*d^34*e^22*sgn(1/(x*e + d))^4 - 5*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(
x*e + d))^(3/2)*c^17*d^33*e^21*sgn(1/(x*e + d))^4 - 240*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a*c^
17*d^32*e^24*sgn(1/(x*e + d))^4 + (c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*c^16*d^32*e^20*sgn(1/(x*
e + d))^4 + 80*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a*c^16*d^31*e^23*sgn(1/(x*e + d))^4 + 1800*
sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^2*c^16*d^30*e^26*sgn(1/(x*e + d))^4 - 16*(c*d*e - c*d^2*e/
(x*e + d) + a*e^3/(x*e + d))^(5/2)*a*c^15*d^30*e^22*sgn(1/(x*e + d))^4 - 600*(c*d*e - c*d^2*e/(x*e + d) + a*e^
3/(x*e + d))^(3/2)*a^2*c^15*d^29*e^25*sgn(1/(x*e + d))^4 - 8400*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e +
d))*a^3*c^15*d^28*e^28*sgn(1/(x*e + d))^4 + 120*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^2*c^14*d
^28*e^24*sgn(1/(x*e + d))^4 + 2800*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^3*c^14*d^27*e^27*sgn(
1/(x*e + d))^4 + 27300*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^4*c^14*d^26*e^30*sgn(1/(x*e + d))^4
- 560*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^3*c^13*d^26*e^26*sgn(1/(x*e + d))^4 - 9100*(c*d*e
- c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^4*c^13*d^25*e^29*sgn(1/(x*e + d))^4 - 65520*sqrt(c*d*e - c*d^2
*e/(x*e + d) + a*e^3/(x*e + d))*a^5*c^13*d^24*e^32*sgn(1/(x*e + d))^4 + 1820*(c*d*e - c*d^2*e/(x*e + d) + a*e^
3/(x*e + d))^(5/2)*a^4*c^12*d^24*e^28*sgn(1/(x*e + d))^4 + 21840*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))
^(3/2)*a^5*c^12*d^23*e^31*sgn(1/(x*e + d))^4 + 120120*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^6*c^
12*d^22*e^34*sgn(1/(x*e + d))^4 - 4368*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^5*c^11*d^22*e^30*
sgn(1/(x*e + d))^4 - 40040*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^6*c^11*d^21*e^33*sgn(1/(x*e +
d))^4 - 171600*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^7*c^11*d^20*e^36*sgn(1/(x*e + d))^4 + 8008
*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^6*c^10*d^20*e^32*sgn(1/(x*e + d))^4 + 57200*(c*d*e - c*
d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^7*c^10*d^19*e^35*sgn(1/(x*e + d))^4 + 193050*sqrt(c*d*e - c*d^2*e/(
x*e + d) + a*e^3/(x*e + d))*a^8*c^10*d^18*e^38*sgn(1/(x*e + d))^4 - 11440*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(
x*e + d))^(5/2)*a^7*c^9*d^18*e^34*sgn(1/(x*e + d))^4 - 64350*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/
2)*a^8*c^9*d^17*e^37*sgn(1/(x*e + d))^4 - 171600*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^9*c^9*d^1
6*e^40*sgn(1/(x*e + d))^4 + 12870*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^8*c^8*d^16*e^36*sgn(1/
(x*e + d))^4 + 57200*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^9*c^8*d^15*e^39*sgn(1/(x*e + d))^4
+ 120120*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^10*c^8*d^14*e^42*sgn(1/(x*e + d))^4 - 11440*(c*d*
e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^9*c^7*d^14*e^38*sgn(1/(x*e + d))^4 - 40040*(c*d*e - c*d^2*e/(
x*e + d) + a*e^3/(x*e + d))^(3/2)*a^10*c^7*d^13*e^41*sgn(1/(x*e + d))^4 - 65520*sqrt(c*d*e - c*d^2*e/(x*e + d)
+ a*e^3/(x*e + d))*a^11*c^7*d^12*e^44*sgn(1/(x*e + d))^4 + 8008*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))
^(5/2)*a^10*c^6*d^12*e^40*sgn(1/(x*e + d))^4 + 21840*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^11*
c^6*d^11*e^43*sgn(1/(x*e + d))^4 + 27300*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^12*c^6*d^10*e^46*
sgn(1/(x*e + d))^4 - 4368*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^11*c^5*d^10*e^42*sgn(1/(x*e +
d))^4 - 9100*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^12*c^5*d^9*e^45*sgn(1/(x*e + d))^4 - 8400*s
qrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^13*c^5*d^8*e^48*sgn(1/(x*e + d))^4 + 1820*(c*d*e - c*d^2*e/
(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^12*c^4*d^8*e^44*sgn(1/(x*e + d))^4 + 2800*(c*d*e - c*d^2*e/(x*e + d) + a*
e^3/(x*e + d))^(3/2)*a^13*c^4*d^7*e^47*sgn(1/(x*e + d))^4 + 1800*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e +
d))*a^14*c^4*d^6*e^50*sgn(1/(x*e + d))^4 - 560*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^13*c^3*d
^6*e^46*sgn(1/(x*e + d))^4 - 600*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^14*c^3*d^5*e^49*sgn(1/(
x*e + d))^4 - 240*sqrt(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))*a^15*c^3*d^4*e^52*sgn(1/(x*e + d))^4 + 120
*(c*d*e - c*d^2*e/(x*e + d) + a*e^3/(x*e + d))^(5/2)*a^14*c^2*d^4*e^48*sgn(1/(x*e + d))^4 + 80*(c*d*e - c*d^2*
e/(x*e + d) + a*e^3/(x*e + d))^(3/2)*a^15*c^2*d^3*e^51*sgn(1/(x*e + d))^4 + 15*sqrt(c*d*e - c*d^2*e/(x*e + d)
+ a*e^3/(x*e + d))*a^16*c^2*d^2*e^54*sgn(1/(x*e...

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Mupad [B]
time = 1.52, size = 1005, normalized size = 5.55 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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