3.5.76 \(\int \frac {1}{d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}} \, dx\) [476]
Optimal. Leaf size=215 \[ -\frac {f^2 \left (4 a e^2-b^2 f^2\right )}{2 e \left (2 d e-b f^2\right ) \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {2 \left (d^2 e-b d f^2+a e f^2\right ) \log \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}{\left (2 d e-b f^2\right )^2}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}{2 e \left (2 d e-b f^2\right )^2} \]
[Out]
2*(a*e*f^2-b*d*f^2+d^2*e)*ln(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2))/(-b*f^2+2*d*e)^2-1/2*f^2*(-b^2*f^2+4*a*e^2)*ln
(b*f^2+2*e*(e*x+f*(a+x*(b*f^2+e^2*x)/f^2)^(1/2)))/e/(-b*f^2+2*d*e)^2-1/2*f^2*(-b^2*f^2+4*a*e^2)/e/(-b*f^2+2*d*
e)/(b*f^2+2*e*(e*x+f*(a+x*(b*f^2+e^2*x)/f^2)^(1/2)))
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2141, 907}
\begin {gather*} -\frac {f^2 \left (4 a e^2-b^2 f^2\right )}{2 e \left (2 d e-b f^2\right ) \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}{2 e \left (2 d e-b f^2\right )^2}+\frac {2 \left (a e f^2-b d f^2+d^2 e\right ) \log \left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )}{\left (2 d e-b f^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-1),x]
[Out]
-1/2*(f^2*(4*a*e^2 - b^2*f^2))/(e*(2*d*e - b*f^2)*(b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2]))) +
(2*(d^2*e - b*d*f^2 + a*e*f^2)*Log[d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2]])/(2*d*e - b*f^2)^2 - (f^2*(4*a*
e^2 - b^2*f^2)*Log[b*f^2 + 2*e*(e*x + f*Sqrt[a + (x*(b*f^2 + e^2*x))/f^2])])/(2*e*(2*d*e - b*f^2)^2)
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 2141
Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol]
:> Dist[2, Subst[Int[(g + h*x^n)^p*((d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2)/(-2*d*e + b*f^2 + 2
*e*x)^2), x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e, f, g, h, n}, x] && EqQ[e^2 -
c*f^2, 0] && IntegerQ[p]
Rubi steps
\begin {align*} \int \frac {1}{d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}} \, dx &=2 \text {Subst}\left (\int \frac {d^2 e-(b d-a e) f^2-\left (2 d e-b f^2\right ) x+e x^2}{x \left (-2 d e+b f^2+2 e x\right )^2} \, dx,x,d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {d^2 e-b d f^2+a e f^2}{\left (2 d e-b f^2\right )^2 x}+\frac {4 a e^2 f^2-b^2 f^4}{2 \left (2 d e-b f^2\right ) \left (2 d e-b f^2-2 e x\right )^2}+\frac {4 a e^2 f^2-b^2 f^4}{2 \left (2 d e-b f^2\right )^2 \left (2 d e-b f^2-2 e x\right )}\right ) \, dx,x,d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )\\ &=-\frac {f^2 \left (4 a e^2-b^2 f^2\right )}{2 e \left (2 d e-b f^2\right ) \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {2 \left (d^2 e-b d f^2+a e f^2\right ) \log \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )}{\left (2 d e-b f^2\right )^2}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \log \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}{2 e \left (2 d e-b f^2\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(570\) vs. \(2(215)=430\).
time = 1.80, size = 570, normalized size = 2.65 \begin {gather*} \frac {e x}{2 d e-b f^2}+\frac {f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}{-2 d e+b f^2}-\frac {\log \left (b f+2 e \sqrt {\frac {e^2}{f^2}} x-2 e \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )}{4 e}-\frac {b d e f \log \left (b d f-2 a e f+2 d e \sqrt {\frac {e^2}{f^2}} x-b \sqrt {\frac {e^2}{f^2}} f^2 x+\left (-2 d e+b f^2\right ) \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )}{\sqrt {\frac {e^2}{f^2}} \left (-2 d e+b f^2\right )^2}+\frac {\left (-b d f^2+d^2 \left (e+\sqrt {\frac {e^2}{f^2}} f\right )+a f^2 \left (e+\sqrt {\frac {e^2}{f^2}} f\right )\right ) \log \left (b d f-2 a e f+\sqrt {\frac {e^2}{f^2}} \left (2 d e-b f^2\right ) x+\left (-2 d e+b f^2\right ) \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )}{\left (-2 d e+b f^2\right )^2}-\frac {\sqrt {\frac {e^2}{f^2}} f \log \left (e \left (-b f+2 e \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )\right )}{4 e^2}-\frac {f^2 \left (-e+\sqrt {\frac {e^2}{f^2}} f\right ) \left (-4 a e^2+b^2 f^2\right ) \log \left (b f+2 e \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{4 e^2 \left (-2 d e+b f^2\right )^2}+\frac {\left (e-\sqrt {\frac {e^2}{f^2}} f\right ) \left (d^2 e-b d f^2+a e f^2\right ) \log \left (d+f \left (-\sqrt {\frac {e^2}{f^2}} x+\sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}{e \left (-2 d e+b f^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x + f*Sqrt[a + b*x + (e^2*x^2)/f^2])^(-1),x]
[Out]
(e*x)/(2*d*e - b*f^2) + (f*Sqrt[a + x*(b + (e^2*x)/f^2)])/(-2*d*e + b*f^2) - Log[b*f + 2*e*Sqrt[e^2/f^2]*x - 2
*e*Sqrt[a + x*(b + (e^2*x)/f^2)]]/(4*e) - (b*d*e*f*Log[b*d*f - 2*a*e*f + 2*d*e*Sqrt[e^2/f^2]*x - b*Sqrt[e^2/f^
2]*f^2*x + (-2*d*e + b*f^2)*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(Sqrt[e^2/f^2]*(-2*d*e + b*f^2)^2) + ((-(b*d*f^2)
+ d^2*(e + Sqrt[e^2/f^2]*f) + a*f^2*(e + Sqrt[e^2/f^2]*f))*Log[b*d*f - 2*a*e*f + Sqrt[e^2/f^2]*(2*d*e - b*f^2)
*x + (-2*d*e + b*f^2)*Sqrt[a + x*(b + (e^2*x)/f^2)]])/(-2*d*e + b*f^2)^2 - (Sqrt[e^2/f^2]*f*Log[e*(-(b*f) + 2*
e*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + x*(b + (e^2*x)/f^2)]))])/(4*e^2) - (f^2*(-e + Sqrt[e^2/f^2]*f)*(-4*a*e^2 + b^
2*f^2)*Log[b*f + 2*e*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + x*(b + (e^2*x)/f^2)])])/(4*e^2*(-2*d*e + b*f^2)^2) + ((e -
Sqrt[e^2/f^2]*f)*(d^2*e - b*d*f^2 + a*e*f^2)*Log[d + f*(-(Sqrt[e^2/f^2]*x) + Sqrt[a + x*(b + (e^2*x)/f^2)])])
/(e*(-2*d*e + b*f^2)^2)
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4917\) vs.
\(2(205)=410\).
time = 0.04, size = 4918, normalized size = 22.87
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(4918\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d+e*x+f*(a+b*x+1/f^2*e^2*x^2)^(1/2)),x,method=_RETURNVERBOSE)
[Out]
f/(b*f^2-2*d*e)*(1/f^2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b
*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2
+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)+1/2*f^3/(b*f^2-2*d*e)^2*ln((-1/2*(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*
e^2)/f^2/(b*f^2-2*d*e)+1/f^2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e)))/(1/f^2*e^2)^(1/2)+(1/f^2*e^2*(x+(a*f^2-d^2)/(b
*f^2-2*d*e))^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a
^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(1/f^2
*e^2)^(1/2)*b^2-f/(b*f^2-2*d*e)^2*ln((-1/2*(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)+1/f^
2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e)))/(1/f^2*e^2)^(1/2)+(1/f^2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2-(-b^2*f^4+2*
a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^
2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(1/f^2*e^2)^(1/2)*a*e^2-f/(b*f^2-
2*d*e)^2*ln((-1/2*(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)+1/f^2*e^2*(x+(a*f^2-d^2)/(b*f
^2-2*d*e)))/(1/f^2*e^2)^(1/2)+(1/f^2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d
^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2
-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(1/f^2*e^2)^(1/2)*b*d*e+1/f/(b*f^2-2*d*e)^2*ln((-1/2*(-b^2
*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)+1/f^2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e)))/(1/f^2*e^2)
^(1/2)+(1/f^2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*
e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)
/f^2/(b*f^2-2*d*e)^2)^(1/2))/(1/f^2*e^2)^(1/2)*d^2*e^2-f^3/(b*f^2-2*d*e)^3/((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2
*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^
2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/
f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*
d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*(1/f^2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2-(-b^2*f^4+2*a*e^2*f^2
+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4
+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(x+(a*f^2-d^2)/(b*f^2-2*d*e)))*a^2*e^2+2*f
^3/(b*f^2-2*d*e)^3/((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2
*d*e)^2)^(1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-
2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*
e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*(1/f^2*e^2
*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)
/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e
)^2)^(1/2))/(x+(a*f^2-d^2)/(b*f^2-2*d*e)))*a*b*d*e-f^3/(b*f^2-2*d*e)^3/((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4
+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^
4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/
(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*
e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*(1/f^2*e^2*(x+(a*f^2-d^2)/(b*f^2-2*d*e))^2-(-b^2*f^4+2*a*e^2*f^2+2*b
*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a
*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2))/(x+(a*f^2-d^2)/(b*f^2-2*d*e)))*b^2*d^2-2*f/(b*
f^2-2*d*e)^3/((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^
2)^(1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)
^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^2-2*d*e))+2*((a^2*e^2*f^
4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*(1/f^2*e^2*(x+(a
*f^2-d^2)/(b*f^2-2*d*e))^2-(-b^2*f^4+2*a*e^2*f^2+2*b*d*e*f^2-2*d^2*e^2)/f^2/(b*f^2-2*d*e)*(x+(a*f^2-d^2)/(b*f^
2-2*d*e))+(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(
1/2))/(x+(a*f^2-d^2)/(b*f^2-2*d*e)))*a*d^2*e^2+2*f/(b*f^2-2*d*e)^3/((a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*a
*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2-2*d*e)^2)^(1/2)*ln((2*(a^2*e^2*f^4-2*a*b*d*e*f^4+b^2*d^2*f^4+2*
a*d^2*e^2*f^2-2*b*d^3*e*f^2+d^4*e^2)/f^2/(b*f^2...
________________________________________________________________________________________
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+b*x+e^2*x^2/f^2)^(1/2)),x, algorithm="maxima")
[Out]
integrate(1/(x*e + sqrt(b*x + a + x^2*e^2/f^2)*f + d), x)
________________________________________________________________________________________
Fricas [A]
time = 5.67, size = 362, normalized size = 1.68 \begin {gather*} -\frac {2 \, b f^{2} x e^{2} - 4 \, d x e^{3} + 2 \, {\left (b d f^{2} e - {\left (a f^{2} + d^{2}\right )} e^{2}\right )} \log \left (-b d f^{2} - 2 \, d x e^{2} + {\left (b f^{2} x + 2 \, a f^{2}\right )} e - {\left (b f^{3} - 2 \, d f e\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right ) + 2 \, {\left (b d f^{2} e - {\left (a f^{2} + d^{2}\right )} e^{2}\right )} \log \left (-b f^{2} x - a f^{2} + 2 \, d x e + d^{2}\right ) + {\left (b^{2} f^{4} - 2 \, b d f^{2} e - 2 \, {\left (a f^{2} - d^{2}\right )} e^{2}\right )} \log \left (-b f^{2} + 2 \, f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} e - 2 \, x e^{2}\right ) - 2 \, {\left (b d f^{2} e - {\left (a f^{2} + d^{2}\right )} e^{2}\right )} \log \left (-x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} - d\right ) - 2 \, {\left (b f^{3} e - 2 \, d f e^{2}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}}{2 \, {\left (b^{2} f^{4} e - 4 \, b d f^{2} e^{2} + 4 \, d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2)),x, algorithm="fricas")
[Out]
-1/2*(2*b*f^2*x*e^2 - 4*d*x*e^3 + 2*(b*d*f^2*e - (a*f^2 + d^2)*e^2)*log(-b*d*f^2 - 2*d*x*e^2 + (b*f^2*x + 2*a*
f^2)*e - (b*f^3 - 2*d*f*e)*sqrt((b*f^2*x + a*f^2 + x^2*e^2)/f^2)) + 2*(b*d*f^2*e - (a*f^2 + d^2)*e^2)*log(-b*f
^2*x - a*f^2 + 2*d*x*e + d^2) + (b^2*f^4 - 2*b*d*f^2*e - 2*(a*f^2 - d^2)*e^2)*log(-b*f^2 + 2*f*sqrt((b*f^2*x +
a*f^2 + x^2*e^2)/f^2)*e - 2*x*e^2) - 2*(b*d*f^2*e - (a*f^2 + d^2)*e^2)*log(-x*e + f*sqrt((b*f^2*x + a*f^2 + x
^2*e^2)/f^2) - d) - 2*(b*f^3*e - 2*d*f*e^2)*sqrt((b*f^2*x + a*f^2 + x^2*e^2)/f^2))/(b^2*f^4*e - 4*b*d*f^2*e^2
+ 4*d^2*e^3)
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d+e*x+f*(a+b*x+e**2*x**2/f**2)**(1/2)),x)
[Out]
Integral(1/(d + e*x + f*sqrt(a + b*x + e**2*x**2/f**2)), x)
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 523 vs.
\(2 (204) = 408\).
time = 3.88, size = 523, normalized size = 2.43 \begin {gather*} -\frac {x e}{b f^{2} - 2 \, d e} - \frac {{\left (b d f^{2} {\left | f \right |} - a f^{2} {\left | f \right |} e - d^{2} {\left | f \right |} e\right )} \log \left ({\left | -{\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} b f^{2} + b d f^{2} - 2 \, a f^{2} e + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} d e \right |}\right )}{b^{2} f^{5} - 4 \, b d f^{3} e + 4 \, d^{2} f e^{2}} - \frac {{\left (b d f^{2} - a f^{2} e - d^{2} e\right )} \log \left ({\left | -b f^{2} x - a f^{2} + 2 \, d x e + d^{2} \right |}\right )}{b^{2} f^{4} - 4 \, b d f^{2} e + 4 \, d^{2} e^{2}} - \frac {{\left (b^{2} f^{4} {\left | f \right |} - 2 \, b d f^{2} {\left | f \right |} e - 2 \, a f^{2} {\left | f \right |} e^{2} + 2 \, d^{2} {\left | f \right |} e^{2}\right )} \log \left ({\left | b f^{2} + 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right )}{2 \, {\left (b^{2} f^{5} e - 4 \, b d f^{3} e^{2} + 4 \, d^{2} f e^{3}\right )}} + \frac {{\left (b d f^{2} {\left | f \right |} - a f^{2} {\left | f \right |} e - d^{2} {\left | f \right |} e\right )} \log \left ({\left | x e + d - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}} \right |}\right )}{b^{2} f^{5} - 4 \, b d f^{3} e + 4 \, d^{2} f e^{2}} + \frac {{\left (b^{2} f^{5} {\left | f \right |} - 4 \, b d f^{3} {\left | f \right |} e + 4 \, d^{2} f {\left | f \right |} e^{2}\right )} \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}}{b^{3} f^{8} - 6 \, b^{2} d f^{6} e + 12 \, b d^{2} f^{4} e^{2} - 8 \, d^{3} f^{2} e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(d+e*x+f*(a+b*x+e^2*x^2/f^2)^(1/2)),x, algorithm="giac")
[Out]
-x*e/(b*f^2 - 2*d*e) - (b*d*f^2*abs(f) - a*f^2*abs(f)*e - d^2*abs(f)*e)*log(abs(-(x*e - sqrt(b*f^2*x + a*f^2 +
x^2*e^2))*b*f^2 + b*d*f^2 - 2*a*f^2*e + 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*d*e))/(b^2*f^5 - 4*b*d*f^3*
e + 4*d^2*f*e^2) - (b*d*f^2 - a*f^2*e - d^2*e)*log(abs(-b*f^2*x - a*f^2 + 2*d*x*e + d^2))/(b^2*f^4 - 4*b*d*f^2
*e + 4*d^2*e^2) - 1/2*(b^2*f^4*abs(f) - 2*b*d*f^2*abs(f)*e - 2*a*f^2*abs(f)*e^2 + 2*d^2*abs(f)*e^2)*log(abs(b*
f^2 + 2*(x*e - sqrt(b*f^2*x + a*f^2 + x^2*e^2))*e))/(b^2*f^5*e - 4*b*d*f^3*e^2 + 4*d^2*f*e^3) + (b*d*f^2*abs(f
) - a*f^2*abs(f)*e - d^2*abs(f)*e)*log(abs(x*e + d - sqrt(b*f^2*x + a*f^2 + x^2*e^2)))/(b^2*f^5 - 4*b*d*f^3*e
+ 4*d^2*f*e^2) + (b^2*f^5*abs(f) - 4*b*d*f^3*abs(f)*e + 4*d^2*f*abs(f)*e^2)*sqrt(b*f^2*x + a*f^2 + x^2*e^2)/(b
^3*f^8 - 6*b^2*d*f^6*e + 12*b*d^2*f^4*e^2 - 8*d^3*f^2*e^3)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(d + e*x + f*(a + b*x + (e^2*x^2)/f^2)^(1/2)),x)
[Out]
int(1/(d + e*x + f*(a + b*x + (e^2*x^2)/f^2)^(1/2)), x)
________________________________________________________________________________________