Integrand size = 26, antiderivative size = 106 \[ \int \frac {1}{(e x)^{3/2} (a+b x) (a c-b c x)} \, dx=-\frac {2}{a^2 c e \sqrt {e x}}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} c e^{3/2}}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{a^{5/2} c e^{3/2}} \]
-arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2))*b^(1/2)/a^(5/2)/c/e^(3/2)+arc tanh(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^(1/2))*b^(1/2)/a^(5/2)/c/e^(3/2)-2/a^2/ c/e/(e*x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(e x)^{3/2} (a+b x) (a c-b c x)} \, dx=\frac {x \left (-2 \sqrt {a}-\sqrt {b} \sqrt {x} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{a^{5/2} c (e x)^{3/2}} \]
(x*(-2*Sqrt[a] - Sqrt[b]*Sqrt[x]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]] + Sqrt[ b]*Sqrt[x]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(a^(5/2)*c*(e*x)^(3/2))
Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {82, 264, 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.
\(\displaystyle \int \frac {1}{(e x)^{3/2} (a+b x) (a c-b c x)} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a^2 c-b^2 c x^2\right )}dx\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {b^2 \int \frac {\sqrt {e x}}{c \left (a^2-b^2 x^2\right )}dx}{a^2 e^2}-\frac {2}{a^2 c e \sqrt {e x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b^2 \int \frac {\sqrt {e x}}{a^2-b^2 x^2}dx}{a^2 c e^2}-\frac {2}{a^2 c e \sqrt {e x}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 b^2 \int \frac {e^3 x}{a^2 e^2-b^2 e^2 x^2}d\sqrt {e x}}{a^2 c e^3}-\frac {2}{a^2 c e \sqrt {e x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 b^2 \int \frac {e x}{a^2 e^2-b^2 e^2 x^2}d\sqrt {e x}}{a^2 c e}-\frac {2}{a^2 c e \sqrt {e x}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {2 b^2 \left (\frac {\int \frac {1}{a e-b e x}d\sqrt {e x}}{2 b}-\frac {\int \frac {1}{a e+b x e}d\sqrt {e x}}{2 b}\right )}{a^2 c e}-\frac {2}{a^2 c e \sqrt {e x}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 b^2 \left (\frac {\int \frac {1}{a e-b e x}d\sqrt {e x}}{2 b}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}\right )}{a^2 c e}-\frac {2}{a^2 c e \sqrt {e x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 b^2 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}\right )}{a^2 c e}-\frac {2}{a^2 c e \sqrt {e x}}\) |
-2/(a^2*c*e*Sqrt[e*x]) + (2*b^2*(-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])))/(a^2*c*e)
3.1.54.3.1 Defintions of rubi rules used
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)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -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 = 0.93 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.70
method | result | size |
pseudoelliptic | \(\frac {b \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right ) \sqrt {e x}-b \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right ) \sqrt {e x}-2 \sqrt {a e b}}{e c \,a^{2} \sqrt {a e b}\, \sqrt {e x}}\) | \(74\) |
risch | \(-\frac {2}{a^{2} c e \sqrt {e x}}+\frac {\frac {b \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{a^{2} \sqrt {a e b}}-\frac {b \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{a^{2} \sqrt {a e b}}}{e c}\) | \(77\) |
derivativedivides | \(-\frac {2 e \left (-\frac {b \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 a^{2} e^{2} \sqrt {a e b}}+\frac {b \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 a^{2} e^{2} \sqrt {a e b}}+\frac {1}{a^{2} e^{2} \sqrt {e x}}\right )}{c}\) | \(78\) |
default | \(\frac {2 e \left (\frac {b \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 a^{2} e^{2} \sqrt {a e b}}-\frac {b \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 a^{2} e^{2} \sqrt {a e b}}-\frac {1}{a^{2} e^{2} \sqrt {e x}}\right )}{c}\) | \(79\) |
(b*arctanh(b*(e*x)^(1/2)/(a*e*b)^(1/2))*(e*x)^(1/2)-b*arctan(b*(e*x)^(1/2) /(a*e*b)^(1/2))*(e*x)^(1/2)-2*(a*e*b)^(1/2))/e/c/a^2/(a*e*b)^(1/2)/(e*x)^( 1/2)
Time = 0.24 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(e x)^{3/2} (a+b x) (a c-b c x)} \, dx=\left [\frac {2 \, e x \sqrt {\frac {b}{a e}} \arctan \left (\frac {\sqrt {e x} a \sqrt {\frac {b}{a e}}}{b x}\right ) + e x \sqrt {\frac {b}{a e}} \log \left (\frac {b x + 2 \, \sqrt {e x} a \sqrt {\frac {b}{a e}} + a}{b x - a}\right ) - 4 \, \sqrt {e x}}{2 \, a^{2} c e^{2} x}, -\frac {2 \, e x \sqrt {-\frac {b}{a e}} \arctan \left (\frac {\sqrt {e x} a \sqrt {-\frac {b}{a e}}}{b x}\right ) - e x \sqrt {-\frac {b}{a e}} \log \left (\frac {b x - 2 \, \sqrt {e x} a \sqrt {-\frac {b}{a e}} - a}{b x + a}\right ) + 4 \, \sqrt {e x}}{2 \, a^{2} c e^{2} x}\right ] \]
[1/2*(2*e*x*sqrt(b/(a*e))*arctan(sqrt(e*x)*a*sqrt(b/(a*e))/(b*x)) + e*x*sq rt(b/(a*e))*log((b*x + 2*sqrt(e*x)*a*sqrt(b/(a*e)) + a)/(b*x - a)) - 4*sqr t(e*x))/(a^2*c*e^2*x), -1/2*(2*e*x*sqrt(-b/(a*e))*arctan(sqrt(e*x)*a*sqrt( -b/(a*e))/(b*x)) - e*x*sqrt(-b/(a*e))*log((b*x - 2*sqrt(e*x)*a*sqrt(-b/(a* e)) - a)/(b*x + a)) + 4*sqrt(e*x))/(a^2*c*e^2*x)]
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 369, normalized size of antiderivative = 3.48 \[ \int \frac {1}{(e x)^{3/2} (a+b x) (a c-b c x)} \, dx=\begin {cases} \frac {1}{3 a b c e^{\frac {3}{2}} x^{\frac {3}{2}}} - \frac {2}{a^{2} c e^{\frac {3}{2}} \sqrt {x}} + \frac {\sqrt {b} \operatorname {acoth}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{a^{\frac {5}{2}} c e^{\frac {3}{2}}} - \frac {\sqrt {b} \operatorname {atan}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{a^{\frac {5}{2}} c e^{\frac {3}{2}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {i \left (1 + i\right )}{6 a b c e^{\frac {3}{2}} x^{\frac {3}{2}}} + \frac {1 + i}{6 a b c e^{\frac {3}{2}} x^{\frac {3}{2}}} + \frac {-6 - 6 i}{6 a^{2} c e^{\frac {3}{2}} \sqrt {x}} - \frac {i \left (-6 - 6 i\right )}{6 a^{2} c e^{\frac {3}{2}} \sqrt {x}} + \frac {\sqrt {b} \left (-3 - 3 i\right ) \operatorname {atan}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{6 a^{\frac {5}{2}} c e^{\frac {3}{2}}} - \frac {i \sqrt {b} \left (-3 - 3 i\right ) \operatorname {atan}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{6 a^{\frac {5}{2}} c e^{\frac {3}{2}}} - \frac {i \sqrt {b} \left (3 + 3 i\right ) \operatorname {atanh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{6 a^{\frac {5}{2}} c e^{\frac {3}{2}}} + \frac {\sqrt {b} \left (3 + 3 i\right ) \operatorname {atanh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{6 a^{\frac {5}{2}} c e^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((1/(3*a*b*c*e**(3/2)*x**(3/2)) - 2/(a**2*c*e**(3/2)*sqrt(x)) + s qrt(b)*acoth(sqrt(b)*sqrt(x)/sqrt(a))/(a**(5/2)*c*e**(3/2)) - sqrt(b)*atan (sqrt(b)*sqrt(x)/sqrt(a))/(a**(5/2)*c*e**(3/2)), Abs(b*x/a) > 1), (-I*(1 + I)/(6*a*b*c*e**(3/2)*x**(3/2)) + (1 + I)/(6*a*b*c*e**(3/2)*x**(3/2)) + (- 6 - 6*I)/(6*a**2*c*e**(3/2)*sqrt(x)) - I*(-6 - 6*I)/(6*a**2*c*e**(3/2)*sqr t(x)) + sqrt(b)*(-3 - 3*I)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**(5/2)*c*e** (3/2)) - I*sqrt(b)*(-3 - 3*I)*atan(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**(5/2)*c* e**(3/2)) - I*sqrt(b)*(3 + 3*I)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**(5/2) *c*e**(3/2)) + sqrt(b)*(3 + 3*I)*atanh(sqrt(b)*sqrt(x)/sqrt(a))/(6*a**(5/2 )*c*e**(3/2)), True))
Exception generated. \[ \int \frac {1}{(e x)^{3/2} (a+b x) (a c-b c x)} \, dx=\text {Exception raised: ValueError} \]
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 = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(e x)^{3/2} (a+b x) (a c-b c x)} \, dx=-\frac {\frac {b \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{\sqrt {a b e} a^{2} c} + \frac {b \arctan \left (\frac {\sqrt {e x} b}{\sqrt {-a b e}}\right )}{\sqrt {-a b e} a^{2} c} + \frac {2}{\sqrt {e x} a^{2} c}}{e} \]
-(b*arctan(sqrt(e*x)*b/sqrt(a*b*e))/(sqrt(a*b*e)*a^2*c) + b*arctan(sqrt(e* x)*b/sqrt(-a*b*e))/(sqrt(-a*b*e)*a^2*c) + 2/(sqrt(e*x)*a^2*c))/e
Time = 0.47 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(e x)^{3/2} (a+b x) (a c-b c x)} \, dx=\frac {\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{a^{5/2}\,c\,e^{3/2}}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{a^{5/2}\,c\,e^{3/2}}-\frac {2}{a^2\,c\,e\,\sqrt {e\,x}} \]