Integrand size = 26, antiderivative size = 127 \[ \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {(e x)^{3/2}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}-\frac {\sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} b^{3/2} c^2}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{5/2} b^{3/2} c^2} \] Output:
1/2*(e*x)^(3/2)/a^2/c^2/e/(-b^2*x^2+a^2)-1/4*e^(1/2)*arctan(b^(1/2)*(e*x)^ (1/2)/a^(1/2)/e^(1/2))/a^(5/2)/b^(3/2)/c^2+1/4*e^(1/2)*arctanh(b^(1/2)*(e* x)^(1/2)/a^(1/2)/e^(1/2))/a^(5/2)/b^(3/2)/c^2
Time = 0.10 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {x \sqrt {e x}}{2 a^2 c^2 \left (a^2-b^2 x^2\right )}-\frac {\sqrt {e x} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2} c^2 \sqrt {x}}+\frac {\sqrt {e x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} b^{3/2} c^2 \sqrt {x}} \] Input:
Integrate[Sqrt[e*x]/((a + b*x)^2*(a*c - b*c*x)^2),x]
Output:
(x*Sqrt[e*x])/(2*a^2*c^2*(a^2 - b^2*x^2)) - (Sqrt[e*x]*ArcTan[(Sqrt[b]*Sqr t[x])/Sqrt[a]])/(4*a^(5/2)*b^(3/2)*c^2*Sqrt[x]) + (Sqrt[e*x]*ArcTanh[(Sqrt [b]*Sqrt[x])/Sqrt[a]])/(4*a^(5/2)*b^(3/2)*c^2*Sqrt[x])
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {82, 253, 27, 266, 27, 827, 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.
\(\displaystyle \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {\sqrt {e x}}{\left (a^2 c-b^2 c x^2\right )^2}dx\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {\int \frac {\sqrt {e x}}{c \left (a^2-b^2 x^2\right )}dx}{4 a^2 c}+\frac {(e x)^{3/2}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {e x}}{a^2-b^2 x^2}dx}{4 a^2 c^2}+\frac {(e x)^{3/2}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\int \frac {e^3 x}{a^2 e^2-b^2 e^2 x^2}d\sqrt {e x}}{2 a^2 c^2 e}+\frac {(e x)^{3/2}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e \int \frac {e x}{a^2 e^2-b^2 e^2 x^2}d\sqrt {e x}}{2 a^2 c^2}+\frac {(e x)^{3/2}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {e \left (\frac {\int \frac {1}{a e-b e x}d\sqrt {e x}}{2 b}-\frac {\int \frac {1}{a e+b x e}d\sqrt {e x}}{2 b}\right )}{2 a^2 c^2}+\frac {(e x)^{3/2}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {e \left (\frac {\int \frac {1}{a e-b e x}d\sqrt {e x}}{2 b}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}\right )}{2 a^2 c^2}+\frac {(e x)^{3/2}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {e \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}\right )}{2 a^2 c^2}+\frac {(e x)^{3/2}}{2 a^2 c^2 e \left (a^2-b^2 x^2\right )}\) |
Input:
Int[Sqrt[e*x]/((a + b*x)^2*(a*c - b*c*x)^2),x]
Output:
(e*x)^(3/2)/(2*a^2*c^2*e*(a^2 - b^2*x^2)) + (e*(-1/2*ArcTan[(Sqrt[b]*Sqrt[ e*x])/(Sqrt[a]*Sqrt[e])]/(Sqrt[a]*b^(3/2)*Sqrt[e]) + ArcTanh[(Sqrt[b]*Sqrt [e*x])/(Sqrt[a]*Sqrt[e])]/(2*Sqrt[a]*b^(3/2)*Sqrt[e])))/(2*a^2*c^2)
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {e x}}{b x +a}-\frac {e \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}}+\frac {\sqrt {e x}}{-b x +a}+\frac {e \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}}}{4 c^{2} a^{2} b}\) | \(86\) |
derivativedivides | \(\frac {2 e^{3} \left (\frac {\frac {\sqrt {e x}}{2 a e \left (-b e x +a e \right )}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 a e \sqrt {a e b}}}{4 a e b}-\frac {\frac {\sqrt {e x}}{2 a e \left (b e x +a e \right )}+\frac {\arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 a e \sqrt {a e b}}}{4 a e b}\right )}{c^{2}}\) | \(137\) |
default | \(\frac {2 e^{3} \left (\frac {\frac {\sqrt {e x}}{2 a e \left (-b e x +a e \right )}+\frac {\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 a e \sqrt {a e b}}}{4 a e b}-\frac {\frac {\sqrt {e x}}{2 a e \left (b e x +a e \right )}+\frac {\arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 a e \sqrt {a e b}}}{4 a e b}\right )}{c^{2}}\) | \(137\) |
Input:
int((e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x,method=_RETURNVERBOSE)
Output:
1/4/c^2/a^2/b*(-(e*x)^(1/2)/(b*x+a)-e/(a*e*b)^(1/2)*arctan(b*(e*x)^(1/2)/( a*e*b)^(1/2))+(e*x)^(1/2)/(-b*x+a)+e/(a*e*b)^(1/2)*arctanh(b*(e*x)^(1/2)/( a*e*b)^(1/2)))
Time = 0.11 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.33 \[ \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx=\left [-\frac {4 \, \sqrt {e x} b x + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {\frac {e}{a b}} \arctan \left (\frac {\sqrt {e x} b \sqrt {\frac {e}{a b}}}{e}\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {\frac {e}{a b}} \log \left (\frac {b e x + 2 \, \sqrt {e x} a b \sqrt {\frac {e}{a b}} + a e}{b x - a}\right )}{8 \, {\left (a^{2} b^{3} c^{2} x^{2} - a^{4} b c^{2}\right )}}, -\frac {4 \, \sqrt {e x} b x + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {-\frac {e}{a b}} \arctan \left (\frac {\sqrt {e x} b \sqrt {-\frac {e}{a b}}}{e}\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \sqrt {-\frac {e}{a b}} \log \left (\frac {b e x - 2 \, \sqrt {e x} a b \sqrt {-\frac {e}{a b}} - a e}{b x + a}\right )}{8 \, {\left (a^{2} b^{3} c^{2} x^{2} - a^{4} b c^{2}\right )}}\right ] \] Input:
integrate((e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="fricas")
Output:
[-1/8*(4*sqrt(e*x)*b*x + 2*(b^2*x^2 - a^2)*sqrt(e/(a*b))*arctan(sqrt(e*x)* b*sqrt(e/(a*b))/e) - (b^2*x^2 - a^2)*sqrt(e/(a*b))*log((b*e*x + 2*sqrt(e*x )*a*b*sqrt(e/(a*b)) + a*e)/(b*x - a)))/(a^2*b^3*c^2*x^2 - a^4*b*c^2), -1/8 *(4*sqrt(e*x)*b*x + 2*(b^2*x^2 - a^2)*sqrt(-e/(a*b))*arctan(sqrt(e*x)*b*sq rt(-e/(a*b))/e) - (b^2*x^2 - a^2)*sqrt(-e/(a*b))*log((b*e*x - 2*sqrt(e*x)* a*b*sqrt(-e/(a*b)) - a*e)/(b*x + a)))/(a^2*b^3*c^2*x^2 - a^4*b*c^2)]
Result contains complex when optimal does not.
Time = 3.72 (sec) , antiderivative size = 2740, normalized size of antiderivative = 21.57 \[ \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x)**(1/2)/(b*x+a)**2/(-b*c*x+a*c)**2,x)
Output:
Piecewise((2*a**(13/2)*b**8*sqrt(e)*x**(17/2)*acoth(sqrt(a)/(sqrt(b)*sqrt( x)))/(8*a**9*b**(19/2)*c**2*x**(17/2) - 24*a**7*b**(23/2)*c**2*x**(21/2) + 24*a**5*b**(27/2)*c**2*x**(25/2) - 8*a**3*b**(31/2)*c**2*x**(29/2)) + 2*a **(13/2)*b**8*sqrt(e)*x**(17/2)*atan(sqrt(a)/(sqrt(b)*sqrt(x)))/(8*a**9*b* *(19/2)*c**2*x**(17/2) - 24*a**7*b**(23/2)*c**2*x**(21/2) + 24*a**5*b**(27 /2)*c**2*x**(25/2) - 8*a**3*b**(31/2)*c**2*x**(29/2)) + I*pi*a**(13/2)*b** 8*sqrt(e)*x**(17/2)/(8*a**9*b**(19/2)*c**2*x**(17/2) - 24*a**7*b**(23/2)*c **2*x**(21/2) + 24*a**5*b**(27/2)*c**2*x**(25/2) - 8*a**3*b**(31/2)*c**2*x **(29/2)) - 6*a**(9/2)*b**10*sqrt(e)*x**(21/2)*acoth(sqrt(a)/(sqrt(b)*sqrt (x)))/(8*a**9*b**(19/2)*c**2*x**(17/2) - 24*a**7*b**(23/2)*c**2*x**(21/2) + 24*a**5*b**(27/2)*c**2*x**(25/2) - 8*a**3*b**(31/2)*c**2*x**(29/2)) - 6* a**(9/2)*b**10*sqrt(e)*x**(21/2)*atan(sqrt(a)/(sqrt(b)*sqrt(x)))/(8*a**9*b **(19/2)*c**2*x**(17/2) - 24*a**7*b**(23/2)*c**2*x**(21/2) + 24*a**5*b**(2 7/2)*c**2*x**(25/2) - 8*a**3*b**(31/2)*c**2*x**(29/2)) - 3*I*pi*a**(9/2)*b **10*sqrt(e)*x**(21/2)/(8*a**9*b**(19/2)*c**2*x**(17/2) - 24*a**7*b**(23/2 )*c**2*x**(21/2) + 24*a**5*b**(27/2)*c**2*x**(25/2) - 8*a**3*b**(31/2)*c** 2*x**(29/2)) + 6*a**(5/2)*b**12*sqrt(e)*x**(25/2)*acoth(sqrt(a)/(sqrt(b)*s qrt(x)))/(8*a**9*b**(19/2)*c**2*x**(17/2) - 24*a**7*b**(23/2)*c**2*x**(21/ 2) + 24*a**5*b**(27/2)*c**2*x**(25/2) - 8*a**3*b**(31/2)*c**2*x**(29/2)) + 6*a**(5/2)*b**12*sqrt(e)*x**(25/2)*atan(sqrt(a)/(sqrt(b)*sqrt(x)))/(8*...
Exception generated. \[ \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((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 = 112, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx=-\frac {\frac {2 \, \sqrt {e x} e^{3} x}{{\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )} a^{2} c^{2}} + \frac {e^{2} \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{\sqrt {a b e} a^{2} b c^{2}} + \frac {e^{2} \arctan \left (\frac {\sqrt {e x} b}{\sqrt {-a b e}}\right )}{\sqrt {-a b e} a^{2} b c^{2}}}{4 \, e} \] Input:
integrate((e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="giac")
Output:
-1/4*(2*sqrt(e*x)*e^3*x/((b^2*e^2*x^2 - a^2*e^2)*a^2*c^2) + e^2*arctan(sqr t(e*x)*b/sqrt(a*b*e))/(sqrt(a*b*e)*a^2*b*c^2) + e^2*arctan(sqrt(e*x)*b/sqr t(-a*b*e))/(sqrt(-a*b*e)*a^2*b*c^2))/e
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {e\,{\left (e\,x\right )}^{3/2}}{2\,a^2\,\left (a^2\,c^2\,e^2-b^2\,c^2\,e^2\,x^2\right )}-\frac {\sqrt {e}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,a^{5/2}\,b^{3/2}\,c^2}+\frac {\sqrt {e}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,a^{5/2}\,b^{3/2}\,c^2} \] Input:
int((e*x)^(1/2)/((a*c - b*c*x)^2*(a + b*x)^2),x)
Output:
(e*(e*x)^(3/2))/(2*a^2*(a^2*c^2*e^2 - b^2*c^2*e^2*x^2)) - (e^(1/2)*atan((b ^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(4*a^(5/2)*b^(3/2)*c^2) + (e^(1/2) *atanh((b^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(4*a^(5/2)*b^(3/2)*c^2)
Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {e x}}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2}+2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} x^{2}-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2}+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{2} x^{2}+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2}-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{2} x^{2}+4 \sqrt {x}\, a \,b^{2} x \right )}{8 a^{3} b^{2} c^{2} \left (-b^{2} x^{2}+a^{2}\right )} \] Input:
int((e*x)^(1/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x)
Output:
(sqrt(e)*( - 2*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**2 + 2*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*b**2*x**2 - sqrt(b)* sqrt(a)*log( - sqrt(a) + sqrt(x)*sqrt(b))*a**2 + sqrt(b)*sqrt(a)*log( - sq rt(a) + sqrt(x)*sqrt(b))*b**2*x**2 + sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x) *sqrt(b))*a**2 - sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b))*b**2*x**2 + 4*sqrt(x)*a*b**2*x))/(8*a**3*b**2*c**2*(a**2 - b**2*x**2))