3.5.58 \(\int \frac {1}{(d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}})^2} \, dx\) [458]

Optimal. Leaf size=151 \[ -\frac {a f^2}{2 d^2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {1+\frac {a f^2}{d^2}}{2 e \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {a f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{d^3 e}+\frac {a f^2 \log \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{d^3 e} \]

[Out]

-a*f^2*ln(e*x+f*(a+e^2*x^2/f^2)^(1/2))/d^3/e+a*f^2*ln(d+e*x+f*(a+e^2*x^2/f^2)^(1/2))/d^3/e-1/2*a*f^2/d^2/e/(e*
x+f*(a+e^2*x^2/f^2)^(1/2))+1/2*(-1-a*f^2/d^2)/e/(d+e*x+f*(a+e^2*x^2/f^2)^(1/2))

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Rubi [A]
time = 0.08, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2142, 907} \begin {gather*} -\frac {a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}{d^3 e}+\frac {a f^2 \log \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )}{d^3 e}-\frac {a f^2}{2 d^2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+e x\right )}-\frac {\frac {a f^2}{d^2}+1}{2 e \left (f \sqrt {a+\frac {e^2 x^2}{f^2}}+d+e x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-2),x]

[Out]

-1/2*(a*f^2)/(d^2*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) - (1 + (a*f^2)/d^2)/(2*e*(d + e*x + f*Sqrt[a + (e^2*x^2
)/f^2])) - (a*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/(d^3*e) + (a*f^2*Log[d + e*x + f*Sqrt[a + (e^2*x^2)/f^
2]])/(d^3*e)

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2142

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[(g + h*x^n)^p*((d^2 + a*f^2 - 2*d*x + x^2)/(d - x)^2), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {d^2+a f^2-2 d x+x^2}{(d-x)^2 x^2} \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a f^2}{d^2 (d-x)^2}+\frac {2 a f^2}{d^3 (d-x)}+\frac {d^2+a f^2}{d^2 x^2}+\frac {2 a f^2}{d^3 x}\right ) \, dx,x,d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{2 e}\\ &=-\frac {a f^2}{2 d^2 e \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {1+\frac {a f^2}{d^2}}{2 e \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}-\frac {a f^2 \log \left (e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{d^3 e}+\frac {a f^2 \log \left (d+e x+f \sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{d^3 e}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(378\) vs. \(2(151)=302\).
time = 1.45, size = 378, normalized size = 2.50 \begin {gather*} \frac {\frac {2 d x \left (d^4+a^2 f^4+d^3 e x-a d e f^2 x\right )}{\left (d^2-a f^2\right ) \left (d^2-a f^2+2 d e x\right )}+\frac {2 d \left (a f^3-d e f x\right ) \sqrt {a+\frac {e^2 x^2}{f^2}}}{e \left (d^2-a f^2+2 d e x\right )}+\frac {a f^2 \left (-e+\sqrt {\frac {e^2}{f^2}} f\right ) \log \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )}{e^2}+\frac {a f \log \left (a f+d \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )}{\sqrt {\frac {e^2}{f^2}}}+\frac {a f^2 \log \left (d^2 e \left (a f+d \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )\right )}{e}-\frac {a f \log \left (d+f \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )}{\sqrt {\frac {e^2}{f^2}}}+\frac {a f^2 \log \left (d^3 e \left (d+f \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+\frac {e^2 x^2}{f^2}}\right )\right )\right )}{e}}{2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^(-2),x]

[Out]

((2*d*x*(d^4 + a^2*f^4 + d^3*e*x - a*d*e*f^2*x))/((d^2 - a*f^2)*(d^2 - a*f^2 + 2*d*e*x)) + (2*d*(a*f^3 - d*e*f
*x)*Sqrt[a + (e^2*x^2)/f^2])/(e*(d^2 - a*f^2 + 2*d*e*x)) + (a*f^2*(-e + Sqrt[e^2/f^2]*f)*Log[-(Sqrt[e^2/f^2]*x
) + Sqrt[a + (e^2*x^2)/f^2]])/e^2 + (a*f*Log[a*f + d*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + (e^2*x^2)/f^2])])/Sqrt[e^2
/f^2] + (a*f^2*Log[d^2*e*(a*f + d*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + (e^2*x^2)/f^2]))])/e - (a*f*Log[d + f*(-(Sqrt
[e^2/f^2]*x) + Sqrt[a + (e^2*x^2)/f^2])])/Sqrt[e^2/f^2] + (a*f^2*Log[d^3*e*(d + f*(-(Sqrt[e^2/f^2]*x) + Sqrt[a
+ (e^2*x^2)/f^2]))])/e)/(2*d^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4160\) vs. \(2(140)=280\).
time = 0.04, size = 4161, normalized size = 27.56

method	result	size
default	\(\text {Expression too large to display}\)	\(4161\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/e*f^7/d^4/(a^2*f^4+2*a*d^2*f^2+d^4)/((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*ln((1/2*(a^2*f^4+2*a*d^2*f^2
+d^4)/d^2/f^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/2*((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*(4/f^
2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/
f^2)^(1/2))/(x+1/2*(-a*f^2+d^2)/d/e))*a^4+1/2*a*f^2/(a*f^2-2*d*e*x-d^2)/d/e-1/4/e/d^3/(-a*f^2+2*d*e*x+d^2)*a^2
*f^4-f^3/d/(a^2*f^4+2*a*d^2*f^2+d^4)*(1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+
d^2)/d/e)+1/4*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*x*a+1/4/e*f^3/d^4/((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(
1/2)*ln((1/2*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/2*((a^2*f^4+2*a*
d^2*f^2+d^4)/d^2/f^2)^(1/2)*(4/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/
e)+(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2))/(x+1/2*(-a*f^2+d^2)/d/e))*a^2+1/2/e*f/d^2/((a^2*f^4+2*a*d^2*f^2+d
^4)/d^2/f^2)^(1/2)*ln((1/2*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/2*
((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*(4/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+1/2*(
-a*f^2+d^2)/d/e)+(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2))/(x+1/2*(-a*f^2+d^2)/d/e))*a+1/4/e*d/(-a*f^2+2*d*e*x
+d^2)+1/2/d^2*f^5/e/(a^2*f^4+2*a*d^2*f^2+d^4)/((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*ln((1/2*(a^2*f^4+2*a*d
^2*f^2+d^4)/d^2/f^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/2*((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)
*(4/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+(a^2*f^4+2*a*d^2*f^2+d^4
)/d^2/f^2)^(1/2))/(x+1/2*(-a*f^2+d^2)/d/e))*a^3+1/2/e/d^3*ln(-a*f^2+2*d*e*x+d^2)*a*f^2+1/2/d^2*x+1/2*d/(a*f^2-
2*d*e*x-d^2)/e+1/e^2*f^5/d/(a^2*f^4+2*a*d^2*f^2+d^4)/(x-1/2/e/d*a*f^2+1/2*d/e)*(1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/
d/e)^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/4*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(3/2)*a-1/2*d^2*f/e
/(a^2*f^4+2*a*d^2*f^2+d^4)/((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*ln((1/2*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2
+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/2*((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*(4/f^2*e^2*(x+1/2*
(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2))/(
x+1/2*(-a*f^2+d^2)/d/e))*a-1/4*f/d^3*ln((1/2*e*(a*f^2-d^2)/d/f^2+1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e))/(1/f^2*e^
2)^(1/2)+(1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/4*(a^2*f^4+2*a*d
^2*f^2+d^4)/d^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)*a-1/4/e*f^5/d^2/(a^2*f^4+2*a*d^2*f^2+d^4)*(4/f^2*e^2*(x+1/2*(-a*
f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*a^2-1/
4*f^5/d^3/(a^2*f^4+2*a*d^2*f^2+d^4)*ln((1/2*e*(a*f^2-d^2)/d/f^2+1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e))/(1/f^2*e^2
)^(1/2)+(1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/4*(a^2*f^4+2*a*d^
2*f^2+d^4)/d^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)*a^3+d*f^3/e^2/(a^2*f^4+2*a*d^2*f^2+d^4)/(x-1/2/e/d*a*f^2+1/2*d/e)
*(1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/4*(a^2*f^4+2*a*d^2*f^2+d
^4)/d^2/f^2)^(3/2)-3/4/d*f^3/(a^2*f^4+2*a*d^2*f^2+d^4)*ln((1/2*e*(a*f^2-d^2)/d/f^2+1/f^2*e^2*(x+1/2*(-a*f^2+d^
2)/d/e))/(1/f^2*e^2)^(1/2)+(1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+
1/4*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)*a^2-3/4*d*f/(a^2*f^4+2*a*d^2*f^2+d^4)*ln((1/2*
e*(a*f^2-d^2)/d/f^2+1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e))/(1/f^2*e^2)^(1/2)+(1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^
2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/4*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)
*a-1/4*d^4/f/e/(a^2*f^4+2*a*d^2*f^2+d^4)/((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*ln((1/2*(a^2*f^4+2*a*d^2*f^
2+d^4)/d^2/f^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+1/2*((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*(4/f
^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+(a^2*f^4+2*a*d^2*f^2+d^4)/d^2
/f^2)^(1/2))/(x+1/2*(-a*f^2+d^2)/d/e))-1/4/e*f/d^2*(4/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2
*(x+1/2*(-a*f^2+d^2)/d/e)+(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)+1/4/f/d*ln((1/2*e*(a*f^2-d^2)/d/f^2+1/f^2*e
^2*(x+1/2*(-a*f^2+d^2)/d/e))/(1/f^2*e^2)^(1/2)+(1/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+e*(a*f^2-d^2)/d/f^2*(x+1/
2*(-a*f^2+d^2)/d/e)+1/4*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)+1/4/e/f/((a^2*f^4+2*a*d^2*
f^2+d^4)/d^2/f^2)^(1/2)*ln((1/2*(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2+e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)
+1/2*((a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)*(4/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+
1/2*(-a*f^2+d^2)/d/e)+(a^2*f^4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2))/(x+1/2*(-a*f^2+d^2)/d/e))+1/4*d^2*f/e/(a^2*f^4
+2*a*d^2*f^2+d^4)*(4/f^2*e^2*(x+1/2*(-a*f^2+d^2)/d/e)^2+4*e*(a*f^2-d^2)/d/f^2*(x+1/2*(-a*f^2+d^2)/d/e)+(a^2*f^
4+2*a*d^2*f^2+d^4)/d^2/f^2)^(1/2)-1/4*d^3/f/(a^...

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

[Out]

integrate((x*e + sqrt(a + x^2*e^2/f^2)*f + d)^(-2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (135) = 270\).
time = 0.38, size = 290, normalized size = 1.92 \begin {gather*} -\frac {a^{2} f^{4} + a d^{2} f^{2} - 2 \, d^{2} x^{2} e^{2} - 2 \, d^{3} x e + {\left (a^{2} f^{4} - 2 \, a d f^{2} x e - a d^{2} f^{2}\right )} \log \left (-a f^{2} x e + a d f^{2} + 2 \, d x^{2} e^{2} + {\left (a f^{3} - 2 \, d f x e\right )} \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}}\right ) + {\left (a^{2} f^{4} - 2 \, a d f^{2} x e - a d^{2} f^{2}\right )} \log \left (-a f^{2} + 2 \, d x e + d^{2}\right ) - {\left (a^{2} f^{4} - 2 \, a d f^{2} x e - a d^{2} f^{2}\right )} \log \left (-x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}} - d\right ) - 2 \, {\left (a d f^{3} - d^{2} f x e\right )} \sqrt {\frac {a f^{2} + x^{2} e^{2}}{f^{2}}}}{2 \, {\left (2 \, d^{4} x e^{2} - {\left (a d^{3} f^{2} - d^{5}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^2*f^4 + a*d^2*f^2 - 2*d^2*x^2*e^2 - 2*d^3*x*e + (a^2*f^4 - 2*a*d*f^2*x*e - a*d^2*f^2)*log(-a*f^2*x*e +
a*d*f^2 + 2*d*x^2*e^2 + (a*f^3 - 2*d*f*x*e)*sqrt((a*f^2 + x^2*e^2)/f^2)) + (a^2*f^4 - 2*a*d*f^2*x*e - a*d^2*f
^2)*log(-a*f^2 + 2*d*x*e + d^2) - (a^2*f^4 - 2*a*d*f^2*x*e - a*d^2*f^2)*log(-x*e + f*sqrt((a*f^2 + x^2*e^2)/f^
2) - d) - 2*(a*d*f^3 - d^2*f*x*e)*sqrt((a*f^2 + x^2*e^2)/f^2))/(2*d^4*x*e^2 - (a*d^3*f^2 - d^5)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x + f \sqrt {a + \frac {e^{2} x^{2}}{f^{2}}}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**(-2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (135) = 270\).
time = 4.25, size = 398, normalized size = 2.64 \begin {gather*} \frac {a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | a f^{2} - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} d \right |}\right )}{2 \, d^{3}} + \frac {a f^{2} e^{\left (-1\right )} \log \left ({\left | -a f^{2} + 2 \, d x e + d^{2} \right |}\right )}{2 \, d^{3}} - \frac {a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e - d + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right )}{2 \, d^{3}} + \frac {a f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right )}{2 \, d^{3}} - \frac {\sqrt {a f^{2} + x^{2} e^{2}} {\left | f \right |} e^{\left (-1\right )}}{2 \, d^{2} f} + \frac {x}{2 \, d^{2}} + \frac {{\left (a^{2} f^{4} + 2 \, a d^{2} f^{2} + d^{4}\right )} e^{\left (-1\right )}}{4 \, {\left (a f^{2} - 2 \, d x e - d^{2}\right )} d^{3}} + \frac {{\left ({\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} a^{2} f^{4} {\left | f \right |} + 2 \, a^{2} d f^{4} {\left | f \right |} + 2 \, a d^{3} f^{2} {\left | f \right |} - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} d^{4} {\left | f \right |}\right )} e^{\left (-1\right )}}{4 \, {\left ({\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} a f^{2} + a d f^{2} - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )}^{2} d - {\left (x e - \sqrt {a f^{2} + x^{2} e^{2}}\right )} d^{2}\right )} d^{3} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*f*abs(f)*e^(-1)*log(abs(a*f^2 - (x*e - sqrt(a*f^2 + x^2*e^2))*d))/d^3 + 1/2*a*f^2*e^(-1)*log(abs(-a*f^2
+ 2*d*x*e + d^2))/d^3 - 1/2*a*f*abs(f)*e^(-1)*log(abs(-x*e - d + sqrt(a*f^2 + x^2*e^2)))/d^3 + 1/2*a*f*abs(f)*
e^(-1)*log(abs(-x*e + sqrt(a*f^2 + x^2*e^2)))/d^3 - 1/2*sqrt(a*f^2 + x^2*e^2)*abs(f)*e^(-1)/(d^2*f) + 1/2*x/d^
2 + 1/4*(a^2*f^4 + 2*a*d^2*f^2 + d^4)*e^(-1)/((a*f^2 - 2*d*x*e - d^2)*d^3) + 1/4*((x*e - sqrt(a*f^2 + x^2*e^2)
)*a^2*f^4*abs(f) + 2*a^2*d*f^4*abs(f) + 2*a*d^3*f^2*abs(f) - (x*e - sqrt(a*f^2 + x^2*e^2))*d^4*abs(f))*e^(-1)/
(((x*e - sqrt(a*f^2 + x^2*e^2))*a*f^2 + a*d*f^2 - (x*e - sqrt(a*f^2 + x^2*e^2))^2*d - (x*e - sqrt(a*f^2 + x^2*
e^2))*d^2)*d^3*f)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (d+e\,x+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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