Integrand size = 26, antiderivative size = 127 \[ \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {\sqrt {e x}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}+\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \sqrt {b} c^2 \sqrt {e}}+\frac {3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \sqrt {b} c^2 \sqrt {e}} \] Output:
1/2*(e*x)^(1/2)/a^2/c^2/e/(-b^2*x^2+a^2)+3/4*arctan(b^(1/2)*(e*x)^(1/2)/a^ (1/2)/e^(1/2))/a^(7/2)/b^(1/2)/c^2/e^(1/2)+3/4*arctanh(b^(1/2)*(e*x)^(1/2) /a^(1/2)/e^(1/2))/a^(7/2)/b^(1/2)/c^2/e^(1/2)
Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {x}{2 a^2 c^2 \sqrt {e x} \left (a^2-b^2 x^2\right )}+\frac {3 \sqrt {x} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b} c^2 \sqrt {e x}}+\frac {3 \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{7/2} \sqrt {b} c^2 \sqrt {e x}} \] Input:
Integrate[1/(Sqrt[e*x]*(a + b*x)^2*(a*c - b*c*x)^2),x]
Output:
x/(2*a^2*c^2*Sqrt[e*x]*(a^2 - b^2*x^2)) + (3*Sqrt[x]*ArcTan[(Sqrt[b]*Sqrt[ x])/Sqrt[a]])/(4*a^(7/2)*Sqrt[b]*c^2*Sqrt[e*x]) + (3*Sqrt[x]*ArcTanh[(Sqrt [b]*Sqrt[x])/Sqrt[a]])/(4*a^(7/2)*Sqrt[b]*c^2*Sqrt[e*x])
Time = 0.21 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {82, 253, 27, 266, 756, 218, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {1}{\sqrt {e x} \left (a^2 c-b^2 c x^2\right )^2}dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {3 \int \frac {1}{c \sqrt {e x} \left (a^2-b^2 x^2\right )}dx}{4 a^2 c}+\frac {\sqrt {e x}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3 \int \frac {1}{\sqrt {e x} \left (a^2-b^2 x^2\right )}dx}{4 a^2 c^2}+\frac {\sqrt {e x}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {3 \int \frac {1}{a^2-b^2 x^2}d\sqrt {e x}}{2 a^2 c^2 e}+\frac {\sqrt {e x}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {3 \left (\frac {e \int \frac {1}{a e-b e x}d\sqrt {e x}}{2 a}+\frac {e \int \frac {1}{a e+b x e}d\sqrt {e x}}{2 a}\right )}{2 a^2 c^2 e}+\frac {\sqrt {e x}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {3 \left (\frac {e \int \frac {1}{a e-b e x}d\sqrt {e x}}{2 a}+\frac {\sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {b}}\right )}{2 a^2 c^2 e}+\frac {\sqrt {e x}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {e x}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}+\frac {3 \left (\frac {\sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {b}}\right )}{2 a^2 c^2 e}\) |
Input:
Int[1/(Sqrt[e*x]*(a + b*x)^2*(a*c - b*c*x)^2),x]
Output:
Sqrt[e*x]/(2*a^2*c^2*e*(a^2 - b^2*x^2)) + (3*((Sqrt[e]*ArcTan[(Sqrt[b]*Sqr t[e*x])/(Sqrt[a]*Sqrt[e])])/(2*a^(3/2)*Sqrt[b]) + (Sqrt[e]*ArcTanh[(Sqrt[b ]*Sqrt[e*x])/(Sqrt[a]*Sqrt[e])])/(2*a^(3/2)*Sqrt[b])))/(2*a^2*c^2*e)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {\frac {\sqrt {e x}}{e \left (b x +a \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}}+\frac {\sqrt {e x}}{e \left (-b x +a \right )}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}}}{4 c^{2} a^{3}}\) | \(87\) |
derivativedivides | \(\frac {2 e^{3} \left (\frac {\frac {\sqrt {e x}}{-2 b e x +2 a e}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}}{4 a^{3} e^{3}}+\frac {\frac {\sqrt {e x}}{2 b e x +2 a e}+\frac {3 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}}{4 a^{3} e^{3}}\right )}{c^{2}}\) | \(107\) |
default | \(\frac {2 e^{3} \left (\frac {\frac {\sqrt {e x}}{-2 b e x +2 a e}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}}{4 a^{3} e^{3}}+\frac {\frac {\sqrt {e x}}{2 b e x +2 a e}+\frac {3 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}}{4 a^{3} e^{3}}\right )}{c^{2}}\) | \(107\) |
Input:
int(1/(e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x,method=_RETURNVERBOSE)
Output:
1/4/c^2/a^3*((e*x)^(1/2)/e/(b*x+a)+3/(a*e*b)^(1/2)*arctan(b*(e*x)^(1/2)/(a *e*b)^(1/2))+(e*x)^(1/2)/e/(-b*x+a)+3/(a*e*b)^(1/2)*arctanh(b*(e*x)^(1/2)/ (a*e*b)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.19 \[ \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx=\left [-\frac {4 \, \sqrt {e x} a^{2} b + 6 \, {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {a b e} \arctan \left (\frac {\sqrt {a b e} \sqrt {e x}}{b e x}\right ) - 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {a b e} \log \left (\frac {b e x + a e + 2 \, \sqrt {a b e} \sqrt {e x}}{b x - a}\right )}{8 \, {\left (a^{4} b^{3} c^{2} e x^{2} - a^{6} b c^{2} e\right )}}, -\frac {4 \, \sqrt {e x} a^{2} b + 6 \, {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {-a b e} \arctan \left (\frac {\sqrt {-a b e} \sqrt {e x}}{b e x}\right ) + 3 \, {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {-a b e} \log \left (\frac {b e x - a e - 2 \, \sqrt {-a b e} \sqrt {e x}}{b x + a}\right )}{8 \, {\left (a^{4} b^{3} c^{2} e x^{2} - a^{6} b c^{2} e\right )}}\right ] \] Input:
integrate(1/(e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="fricas")
Output:
[-1/8*(4*sqrt(e*x)*a^2*b + 6*(b^2*x^2 - a^2)*sqrt(a*b*e)*arctan(sqrt(a*b*e )*sqrt(e*x)/(b*e*x)) - 3*(b^2*x^2 - a^2)*sqrt(a*b*e)*log((b*e*x + a*e + 2* sqrt(a*b*e)*sqrt(e*x))/(b*x - a)))/(a^4*b^3*c^2*e*x^2 - a^6*b*c^2*e), -1/8 *(4*sqrt(e*x)*a^2*b + 6*(b^2*x^2 - a^2)*sqrt(-a*b*e)*arctan(sqrt(-a*b*e)*s qrt(e*x)/(b*e*x)) + 3*(b^2*x^2 - a^2)*sqrt(-a*b*e)*log((b*e*x - a*e - 2*sq rt(-a*b*e)*sqrt(e*x))/(b*x + a)))/(a^4*b^3*c^2*e*x^2 - a^6*b*c^2*e)]
Result contains complex when optimal does not.
Time = 120.67 (sec) , antiderivative size = 2395, normalized size of antiderivative = 18.86 \[ \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x)**(1/2)/(b*x+a)**2/(-b*c*x+a*c)**2,x)
Output:
Piecewise((-4*a**(17/2)*b**(5/2)*x**(3/2)/(-8*a**(25/2)*b**(5/2)*c**2*sqrt (e)*x + 8*a**(21/2)*b**(9/2)*c**2*sqrt(e)*x**3) - 8*a**9*b**2*x*acoth(sqrt (b)*sqrt(x)/sqrt(a))/(-8*a**(25/2)*b**(5/2)*c**2*sqrt(e)*x + 8*a**(21/2)*b **(9/2)*c**2*sqrt(e)*x**3) + 2*a**9*b**2*x*acoth(sqrt(a)/(sqrt(b)*sqrt(x)) )/(-8*a**(25/2)*b**(5/2)*c**2*sqrt(e)*x + 8*a**(21/2)*b**(9/2)*c**2*sqrt(e )*x**3) + 6*a**9*b**2*x*atan(sqrt(a)/(sqrt(b)*sqrt(x)))/(-8*a**(25/2)*b**( 5/2)*c**2*sqrt(e)*x + 8*a**(21/2)*b**(9/2)*c**2*sqrt(e)*x**3) + I*pi*a**9* b**2*x/(-8*a**(25/2)*b**(5/2)*c**2*sqrt(e)*x + 8*a**(21/2)*b**(9/2)*c**2*s qrt(e)*x**3) + 8*a**7*b**4*x**3*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(-8*a**(25/ 2)*b**(5/2)*c**2*sqrt(e)*x + 8*a**(21/2)*b**(9/2)*c**2*sqrt(e)*x**3) - 2*a **7*b**4*x**3*acoth(sqrt(a)/(sqrt(b)*sqrt(x)))/(-8*a**(25/2)*b**(5/2)*c**2 *sqrt(e)*x + 8*a**(21/2)*b**(9/2)*c**2*sqrt(e)*x**3) - 6*a**7*b**4*x**3*at an(sqrt(a)/(sqrt(b)*sqrt(x)))/(-8*a**(25/2)*b**(5/2)*c**2*sqrt(e)*x + 8*a* *(21/2)*b**(9/2)*c**2*sqrt(e)*x**3) - I*pi*a**7*b**4*x**3/(-8*a**(25/2)*b* *(5/2)*c**2*sqrt(e)*x + 8*a**(21/2)*b**(9/2)*c**2*sqrt(e)*x**3), (Abs(b*x/ a) > 1) & (Abs(a/(b*x)) > 1)), (-2*a**(17/2)*b**(5/2)*x**(3/2)/(-4*a**(25/ 2)*b**(5/2)*c**2*sqrt(e)*x + 4*a**(21/2)*b**(9/2)*c**2*sqrt(e)*x**3) - 4*a **9*b**2*x*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(-4*a**(25/2)*b**(5/2)*c**2*sqrt (e)*x + 4*a**(21/2)*b**(9/2)*c**2*sqrt(e)*x**3) + 3*a**9*b**2*x*atan(sqrt( a)/(sqrt(b)*sqrt(x)))/(-4*a**(25/2)*b**(5/2)*c**2*sqrt(e)*x + 4*a**(21/...
Exception generated. \[ \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx=-\frac {\sqrt {e x} e}{2 \, {\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )} a^{2} c^{2}} + \frac {3 \, \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{4 \, \sqrt {a b e} a^{3} c^{2}} - \frac {3 \, \arctan \left (\frac {\sqrt {e x} b}{\sqrt {-a b e}}\right )}{4 \, \sqrt {-a b e} a^{3} c^{2}} \] Input:
integrate(1/(e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="giac")
Output:
-1/2*sqrt(e*x)*e/((b^2*e^2*x^2 - a^2*e^2)*a^2*c^2) + 3/4*arctan(sqrt(e*x)* b/sqrt(a*b*e))/(sqrt(a*b*e)*a^3*c^2) - 3/4*arctan(sqrt(e*x)*b/sqrt(-a*b*e) )/(sqrt(-a*b*e)*a^3*c^2)
Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {e\,\sqrt {e\,x}}{2\,a^2\,\left (a^2\,c^2\,e^2-b^2\,c^2\,e^2\,x^2\right )}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,a^{7/2}\,\sqrt {b}\,c^2\,\sqrt {e}}+\frac {3\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,a^{7/2}\,\sqrt {b}\,c^2\,\sqrt {e}} \] Input:
int(1/((a*c - b*c*x)^2*(e*x)^(1/2)*(a + b*x)^2),x)
Output:
(e*(e*x)^(1/2))/(2*a^2*(a^2*c^2*e^2 - b^2*c^2*e^2*x^2)) + (3*atan((b^(1/2) *(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(4*a^(7/2)*b^(1/2)*c^2*e^(1/2)) + (3*ata nh((b^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(4*a^(7/2)*b^(1/2)*c^2*e^(1/2 ))
Time = 0.16 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {e x} (a+b x)^2 (a c-b c x)^2} \, dx=\frac {\sqrt {e}\, \left (6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2}-6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} x^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{2} x^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{2} x^{2}+4 \sqrt {x}\, a^{2} b \right )}{8 a^{4} b \,c^{2} e \left (-b^{2} x^{2}+a^{2}\right )} \] Input:
int(1/(e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x)
Output:
(sqrt(e)*(6*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**2 - 6*s qrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*b**2*x**2 - 3*sqrt(b)*s qrt(a)*log( - sqrt(a) + sqrt(x)*sqrt(b))*a**2 + 3*sqrt(b)*sqrt(a)*log( - s qrt(a) + sqrt(x)*sqrt(b))*b**2*x**2 + 3*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt (x)*sqrt(b))*a**2 - 3*sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b))*b**2* x**2 + 4*sqrt(x)*a**2*b))/(8*a**4*b*c**2*e*(a**2 - b**2*x**2))