Integrand size = 26, antiderivative size = 125 \[ \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {e (e x)^{3/2}}{2 b^2 c^2 \left (a^2-b^2 x^2\right )}+\frac {3 e^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 \sqrt {a} b^{7/2} c^2}-\frac {3 e^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{4 \sqrt {a} b^{7/2} c^2} \] Output:
1/2*e*(e*x)^(3/2)/b^2/c^2/(-b^2*x^2+a^2)+3/4*e^(5/2)*arctan(b^(1/2)*(e*x)^ (1/2)/a^(1/2)/e^(1/2))/a^(1/2)/b^(7/2)/c^2-3/4*e^(5/2)*arctanh(b^(1/2)*(e* x)^(1/2)/a^(1/2)/e^(1/2))/a^(1/2)/b^(7/2)/c^2
Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.99 \[ \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {(e x)^{5/2} \left (-2 \sqrt {a} b^{3/2} x^{3/2}-3 \left (a^2-b^2 x^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+3 \left (a^2-b^2 x^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{4 \sqrt {a} b^{7/2} c^2 x^{5/2} \left (-a^2+b^2 x^2\right )} \] Input:
Integrate[(e*x)^(5/2)/((a + b*x)^2*(a*c - b*c*x)^2),x]
Output:
((e*x)^(5/2)*(-2*Sqrt[a]*b^(3/2)*x^(3/2) - 3*(a^2 - b^2*x^2)*ArcTan[(Sqrt[ b]*Sqrt[x])/Sqrt[a]] + 3*(a^2 - b^2*x^2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a] ]))/(4*Sqrt[a]*b^(7/2)*c^2*x^(5/2)*(-a^2 + b^2*x^2))
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {82, 252, 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.
\(\displaystyle \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \int \frac {(e x)^{5/2}}{\left (a^2 c-b^2 c x^2\right )^2}dx\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {e (e x)^{3/2}}{2 b^2 c^2 \left (a^2-b^2 x^2\right )}-\frac {3 e^2 \int \frac {\sqrt {e x}}{c \left (a^2-b^2 x^2\right )}dx}{4 b^2 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e (e x)^{3/2}}{2 b^2 c^2 \left (a^2-b^2 x^2\right )}-\frac {3 e^2 \int \frac {\sqrt {e x}}{a^2-b^2 x^2}dx}{4 b^2 c^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {e (e x)^{3/2}}{2 b^2 c^2 \left (a^2-b^2 x^2\right )}-\frac {3 e \int \frac {e^3 x}{a^2 e^2-b^2 e^2 x^2}d\sqrt {e x}}{2 b^2 c^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e (e x)^{3/2}}{2 b^2 c^2 \left (a^2-b^2 x^2\right )}-\frac {3 e^3 \int \frac {e x}{a^2 e^2-b^2 e^2 x^2}d\sqrt {e x}}{2 b^2 c^2}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {e (e x)^{3/2}}{2 b^2 c^2 \left (a^2-b^2 x^2\right )}-\frac {3 e^3 \left (\frac {\int \frac {1}{a e-b e x}d\sqrt {e x}}{2 b}-\frac {\int \frac {1}{a e+b x e}d\sqrt {e x}}{2 b}\right )}{2 b^2 c^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {e (e x)^{3/2}}{2 b^2 c^2 \left (a^2-b^2 x^2\right )}-\frac {3 e^3 \left (\frac {\int \frac {1}{a e-b e x}d\sqrt {e x}}{2 b}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}\right )}{2 b^2 c^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {e (e x)^{3/2}}{2 b^2 c^2 \left (a^2-b^2 x^2\right )}-\frac {3 e^3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}-\frac {\arctan \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{2 \sqrt {a} b^{3/2} \sqrt {e}}\right )}{2 b^2 c^2}\) |
Input:
Int[(e*x)^(5/2)/((a + b*x)^2*(a*c - b*c*x)^2),x]
Output:
(e*(e*x)^(3/2))/(2*b^2*c^2*(a^2 - b^2*x^2)) - (3*e^3*(-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*b^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*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[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 = 0.53 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(\frac {e^{3} \left (-\frac {\sqrt {e x}}{e \left (b x +a \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}}+\frac {\sqrt {e x}}{e \left (-b x +a \right )}-\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{\sqrt {a e b}}\right )}{4 b^{3} c^{2}}\) | \(91\) |
derivativedivides | \(\frac {2 e^{3} \left (-\frac {-\frac {\sqrt {e x}}{2 \left (-b e x +a e \right )}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}}{4 b^{3}}+\frac {-\frac {\sqrt {e x}}{2 \left (b e x +a e \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}}{4 b^{3}}\right )}{c^{2}}\) | \(101\) |
default | \(\frac {2 e^{3} \left (-\frac {-\frac {\sqrt {e x}}{2 \left (-b e x +a e \right )}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}}{4 b^{3}}+\frac {-\frac {\sqrt {e x}}{2 \left (b e x +a e \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 \sqrt {a e b}}}{4 b^{3}}\right )}{c^{2}}\) | \(101\) |
Input:
int((e*x)^(5/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x,method=_RETURNVERBOSE)
Output:
1/4*e^3/b^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)))/c^2
Time = 0.10 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.59 \[ \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx=\left [-\frac {4 \, \sqrt {e x} b e^{2} x - 6 \, {\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )} \sqrt {\frac {e}{a b}} \arctan \left (\frac {\sqrt {e x} b \sqrt {\frac {e}{a b}}}{e}\right ) - 3 \, {\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )} \sqrt {\frac {e}{a b}} \log \left (\frac {b e x - 2 \, \sqrt {e x} a b \sqrt {\frac {e}{a b}} + a e}{b x - a}\right )}{8 \, {\left (b^{5} c^{2} x^{2} - a^{2} b^{3} c^{2}\right )}}, -\frac {4 \, \sqrt {e x} b e^{2} x - 6 \, {\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )} \sqrt {-\frac {e}{a b}} \arctan \left (\frac {\sqrt {e x} b \sqrt {-\frac {e}{a b}}}{e}\right ) - 3 \, {\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )} \sqrt {-\frac {e}{a b}} \log \left (\frac {b e x + 2 \, \sqrt {e x} a b \sqrt {-\frac {e}{a b}} - a e}{b x + a}\right )}{8 \, {\left (b^{5} c^{2} x^{2} - a^{2} b^{3} c^{2}\right )}}\right ] \] Input:
integrate((e*x)^(5/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="fricas")
Output:
[-1/8*(4*sqrt(e*x)*b*e^2*x - 6*(b^2*e^2*x^2 - a^2*e^2)*sqrt(e/(a*b))*arcta n(sqrt(e*x)*b*sqrt(e/(a*b))/e) - 3*(b^2*e^2*x^2 - a^2*e^2)*sqrt(e/(a*b))*l og((b*e*x - 2*sqrt(e*x)*a*b*sqrt(e/(a*b)) + a*e)/(b*x - a)))/(b^5*c^2*x^2 - a^2*b^3*c^2), -1/8*(4*sqrt(e*x)*b*e^2*x - 6*(b^2*e^2*x^2 - a^2*e^2)*sqrt (-e/(a*b))*arctan(sqrt(e*x)*b*sqrt(-e/(a*b))/e) - 3*(b^2*e^2*x^2 - a^2*e^2 )*sqrt(-e/(a*b))*log((b*e*x + 2*sqrt(e*x)*a*b*sqrt(-e/(a*b)) - a*e)/(b*x + a)))/(b^5*c^2*x^2 - a^2*b^3*c^2)]
Timed out. \[ \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx=\text {Timed out} \] Input:
integrate((e*x)**(5/2)/(b*x+a)**2/(-b*c*x+a*c)**2,x)
Output:
Timed out
Exception generated. \[ \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x)^(5/2)/(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 = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx=-\frac {1}{4} \, e^{2} {\left (\frac {2 \, \sqrt {e x} e^{2} x}{{\left (b^{2} e^{2} x^{2} - a^{2} e^{2}\right )} b^{2} c^{2}} - \frac {3 \, e \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{\sqrt {a b e} b^{3} c^{2}} - \frac {3 \, e \arctan \left (\frac {\sqrt {e x} b}{\sqrt {-a b e}}\right )}{\sqrt {-a b e} b^{3} c^{2}}\right )} \] Input:
integrate((e*x)^(5/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x, algorithm="giac")
Output:
-1/4*e^2*(2*sqrt(e*x)*e^2*x/((b^2*e^2*x^2 - a^2*e^2)*b^2*c^2) - 3*e*arctan (sqrt(e*x)*b/sqrt(a*b*e))/(sqrt(a*b*e)*b^3*c^2) - 3*e*arctan(sqrt(e*x)*b/s qrt(-a*b*e))/(sqrt(-a*b*e)*b^3*c^2))
Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.81 \[ \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {e^3\,{\left (e\,x\right )}^{3/2}}{2\,b^2\,\left (a^2\,c^2\,e^2-b^2\,c^2\,e^2\,x^2\right )}+\frac {3\,e^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,\sqrt {a}\,b^{7/2}\,c^2}-\frac {3\,e^{5/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{4\,\sqrt {a}\,b^{7/2}\,c^2} \] Input:
int((e*x)^(5/2)/((a*c - b*c*x)^2*(a + b*x)^2),x)
Output:
(e^3*(e*x)^(3/2))/(2*b^2*(a^2*c^2*e^2 - b^2*c^2*e^2*x^2)) + (3*e^(5/2)*ata n((b^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(4*a^(1/2)*b^(7/2)*c^2) - (3*e ^(5/2)*atanh((b^(1/2)*(e*x)^(1/2))/(a^(1/2)*e^(1/2))))/(4*a^(1/2)*b^(7/2)* c^2)
Time = 0.15 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.35 \[ \int \frac {(e x)^{5/2}}{(a+b x)^2 (a c-b c x)^2} \, dx=\frac {\sqrt {e}\, e^{2} \left (6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2}-6 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{2} x^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{2} x^{2}-3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) a^{2}+3 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {x}\, \sqrt {b}\right ) b^{2} x^{2}+4 \sqrt {x}\, a \,b^{2} x \right )}{8 a \,b^{4} c^{2} \left (-b^{2} x^{2}+a^{2}\right )} \] Input:
int((e*x)^(5/2)/(b*x+a)^2/(-b*c*x+a*c)^2,x)
Output:
(sqrt(e)*e**2*(6*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**2 - 6*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*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 - 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*b**2*x))/(8*a*b**4*c**2*(a**2 - b**2*x**2))