Integrand size = 26, antiderivative size = 103 \[ \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx=-\frac {2 e \sqrt {e x}}{b^2 c}+\frac {\sqrt {a} e^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} c}+\frac {\sqrt {a} e^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} c} \] Output:
-2*e*(e*x)^(1/2)/b^2/c+a^(1/2)*e^(3/2)*arctan(b^(1/2)*(e*x)^(1/2)/a^(1/2)/ e^(1/2))/b^(5/2)/c+a^(1/2)*e^(3/2)*arctanh(b^(1/2)*(e*x)^(1/2)/a^(1/2)/e^( 1/2))/b^(5/2)/c
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx=\frac {(e x)^{3/2} \left (-2 \sqrt {b} \sqrt {x}+\sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{b^{5/2} c x^{3/2}} \] Input:
Integrate[(e*x)^(3/2)/((a + b*x)*(a*c - b*c*x)),x]
Output:
((e*x)^(3/2)*(-2*Sqrt[b]*Sqrt[x] + Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a ]] + Sqrt[a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(b^(5/2)*c*x^(3/2))
Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {82, 262, 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.
\(\displaystyle \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {(e x)^{3/2}}{a^2 c-b^2 c x^2}dx\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {a^2 e^2 \int \frac {1}{c \sqrt {e x} \left (a^2-b^2 x^2\right )}dx}{b^2}-\frac {2 e \sqrt {e x}}{b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^2 e^2 \int \frac {1}{\sqrt {e x} \left (a^2-b^2 x^2\right )}dx}{b^2 c}-\frac {2 e \sqrt {e x}}{b^2 c}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {2 a^2 e \int \frac {1}{a^2-b^2 x^2}d\sqrt {e x}}{b^2 c}-\frac {2 e \sqrt {e x}}{b^2 c}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {2 a^2 e \left (\frac {e \int \frac {1}{a e-b e x}d\sqrt {e x}}{2 a}+\frac {e \int \frac {1}{a e+b x e}d\sqrt {e x}}{2 a}\right )}{b^2 c}-\frac {2 e \sqrt {e x}}{b^2 c}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 a^2 e \left (\frac {e \int \frac {1}{a e-b e x}d\sqrt {e x}}{2 a}+\frac {\sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {b}}\right )}{b^2 c}-\frac {2 e \sqrt {e x}}{b^2 c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 a^2 e \left (\frac {\sqrt {e} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{3/2} \sqrt {b}}\right )}{b^2 c}-\frac {2 e \sqrt {e x}}{b^2 c}\) |
Input:
Int[(e*x)^(3/2)/((a + b*x)*(a*c - b*c*x)),x]
Output:
(-2*e*Sqrt[e*x])/(b^2*c) + (2*a^2*e*((Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[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])))/(b^2*c)
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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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 = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.61
method | result | size |
pseudoelliptic | \(\frac {e \left (\arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right ) a e +\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right ) a e -2 \sqrt {e x}\, \sqrt {a e b}\right )}{c \,b^{2} \sqrt {a e b}}\) | \(63\) |
derivativedivides | \(-\frac {2 e \left (\frac {\sqrt {e x}}{b^{2}}-\frac {a e \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 b^{2} \sqrt {a e b}}-\frac {a e \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 b^{2} \sqrt {a e b}}\right )}{c}\) | \(71\) |
default | \(\frac {2 e \left (-\frac {\sqrt {e x}}{b^{2}}+\frac {a e \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 b^{2} \sqrt {a e b}}+\frac {a e \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 b^{2} \sqrt {a e b}}\right )}{c}\) | \(72\) |
risch | \(-\frac {2 x \,e^{2}}{b^{2} \sqrt {e x}\, c}+\frac {\left (\frac {a \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{b^{2} \sqrt {a e b}}+\frac {a \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{b^{2} \sqrt {a e b}}\right ) e^{2}}{c}\) | \(77\) |
Input:
int((e*x)^(3/2)/(b*x+a)/(-b*c*x+a*c),x,method=_RETURNVERBOSE)
Output:
e*(arctan(b*(e*x)^(1/2)/(a*e*b)^(1/2))*a*e+arctanh(b*(e*x)^(1/2)/(a*e*b)^( 1/2))*a*e-2*(e*x)^(1/2)*(a*e*b)^(1/2))/c/b^2/(a*e*b)^(1/2)
Time = 0.12 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.89 \[ \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx=\left [\frac {2 \, \sqrt {\frac {a e}{b}} e \arctan \left (\frac {\sqrt {e x} b \sqrt {\frac {a e}{b}}}{a e}\right ) + \sqrt {\frac {a e}{b}} e \log \left (\frac {b e x + 2 \, \sqrt {e x} b \sqrt {\frac {a e}{b}} + a e}{b x - a}\right ) - 4 \, \sqrt {e x} e}{2 \, b^{2} c}, -\frac {2 \, \sqrt {-\frac {a e}{b}} e \arctan \left (\frac {\sqrt {e x} b \sqrt {-\frac {a e}{b}}}{a e}\right ) - \sqrt {-\frac {a e}{b}} e \log \left (\frac {b e x + 2 \, \sqrt {e x} b \sqrt {-\frac {a e}{b}} - a e}{b x + a}\right ) + 4 \, \sqrt {e x} e}{2 \, b^{2} c}\right ] \] Input:
integrate((e*x)^(3/2)/(b*x+a)/(-b*c*x+a*c),x, algorithm="fricas")
Output:
[1/2*(2*sqrt(a*e/b)*e*arctan(sqrt(e*x)*b*sqrt(a*e/b)/(a*e)) + sqrt(a*e/b)* e*log((b*e*x + 2*sqrt(e*x)*b*sqrt(a*e/b) + a*e)/(b*x - a)) - 4*sqrt(e*x)*e )/(b^2*c), -1/2*(2*sqrt(-a*e/b)*e*arctan(sqrt(e*x)*b*sqrt(-a*e/b)/(a*e)) - sqrt(-a*e/b)*e*log((b*e*x + 2*sqrt(e*x)*b*sqrt(-a*e/b) - a*e)/(b*x + a)) + 4*sqrt(e*x)*e)/(b^2*c)]
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (95) = 190\).
Time = 1.65 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.01 \[ \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx=\begin {cases} \frac {\sqrt {a} e^{\frac {3}{2}} \operatorname {acoth}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{b^{\frac {5}{2}} c} - \frac {\sqrt {a} e^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{b^{\frac {5}{2}} c} - \frac {2 e^{\frac {3}{2}} \sqrt {x}}{b^{2} c} - \frac {e^{\frac {3}{2}} x^{\frac {3}{2}}}{3 a b c} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {\sqrt {a} e^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{b^{\frac {5}{2}} c} + \frac {\sqrt {a} e^{\frac {3}{2}} \operatorname {atanh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{b^{\frac {5}{2}} c} - \frac {2 e^{\frac {3}{2}} \sqrt {x}}{b^{2} c} - \frac {e^{\frac {3}{2}} x^{\frac {3}{2}}}{3 a b c} & \text {otherwise} \end {cases} \] Input:
integrate((e*x)**(3/2)/(b*x+a)/(-b*c*x+a*c),x)
Output:
Piecewise((sqrt(a)*e**(3/2)*acoth(sqrt(a)/(sqrt(b)*sqrt(x)))/(b**(5/2)*c) - sqrt(a)*e**(3/2)*atan(sqrt(a)/(sqrt(b)*sqrt(x)))/(b**(5/2)*c) - 2*e**(3/ 2)*sqrt(x)/(b**2*c) - e**(3/2)*x**(3/2)/(3*a*b*c), Abs(a/(b*x)) > 1), (-sq rt(a)*e**(3/2)*atan(sqrt(a)/(sqrt(b)*sqrt(x)))/(b**(5/2)*c) + sqrt(a)*e**( 3/2)*atanh(sqrt(a)/(sqrt(b)*sqrt(x)))/(b**(5/2)*c) - 2*e**(3/2)*sqrt(x)/(b **2*c) - e**(3/2)*x**(3/2)/(3*a*b*c), True))
Exception generated. \[ \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(3/2)/(b*x+a)/(-b*c*x+a*c),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 = 86, normalized size of antiderivative = 0.83 \[ \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx=\frac {\frac {a e^{3} \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{\sqrt {a b e} b^{2} c} - \frac {a e^{3} \arctan \left (\frac {\sqrt {e x} b}{\sqrt {-a b e}}\right )}{\sqrt {-a b e} b^{2} c} - \frac {2 \, \sqrt {e x} e^{2}}{b^{2} c}}{e} \] Input:
integrate((e*x)^(3/2)/(b*x+a)/(-b*c*x+a*c),x, algorithm="giac")
Output:
(a*e^3*arctan(sqrt(e*x)*b/sqrt(a*b*e))/(sqrt(a*b*e)*b^2*c) - a*e^3*arctan( sqrt(e*x)*b/sqrt(-a*b*e))/(sqrt(-a*b*e)*b^2*c) - 2*sqrt(e*x)*e^2/(b^2*c))/ e
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71 \[ \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx=\frac {\sqrt {a}\,e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{b^{5/2}\,c}-\frac {2\,e\,\sqrt {e\,x}}{b^2\,c}+\frac {\sqrt {a}\,e^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{b^{5/2}\,c} \] Input:
int((e*x)^(3/2)/((a*c - b*c*x)*(a + b*x)),x)
Output:
(a^(1/2)*e^(3/2)*atan((b^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(b^(5/2)*c ) - (2*e*(e*x)^(1/2))/(b^2*c) + (a^(1/2)*e^(3/2)*atanh((b^(1/2)*(e*x)^(1/2 ))/(a^(1/2)*e^(1/2))))/(b^(5/2)*c)
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.65 \[ \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx=\frac {\sqrt {e}\, e \left (2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right )-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right )+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right )-4 \sqrt {x}\, b \right )}{2 b^{3} c} \] Input:
int((e*x)^(3/2)/(b*x+a)/(-b*c*x+a*c),x)
Output:
(sqrt(e)*e*(2*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a))) - sqrt(b )*sqrt(a)*log( - sqrt(a) + sqrt(x)*sqrt(b)) + sqrt(b)*sqrt(a)*log(sqrt(a) + sqrt(x)*sqrt(b)) - 4*sqrt(x)*b))/(2*b**3*c)