Integrand size = 36, antiderivative size = 133 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\sqrt {e x} \sqrt {b c^2-b d^2 x^2}}{2 b c d (c+d x)^{3/2}}+\frac {\sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {d} \sqrt {e x} \sqrt {c+d x}}{\sqrt {e} \sqrt {b c^2-b d^2 x^2}}\right )}{2 \sqrt {2} \sqrt {b} c d^{3/2}} \] Output:
-1/2*(e*x)^(1/2)*(-b*d^2*x^2+b*c^2)^(1/2)/b/c/d/(d*x+c)^(3/2)+1/4*e^(1/2)* arctan(1/e^(1/2)/(-b*d^2*x^2+b*c^2)^(1/2)*2^(1/2)*b^(1/2)*d^(1/2)*(e*x)^(1 /2)*(d*x+c)^(1/2))*2^(1/2)/b^(1/2)/c/d^(3/2)
Time = 4.75 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {e x} \sqrt {c-d x} \left (-2 \sqrt {d} \sqrt {x} \sqrt {c-d x}+\sqrt {2} (c+d x) \arctan \left (\frac {\sqrt {2} \sqrt {d} \sqrt {x}}{\sqrt {c-d x}}\right )\right )}{4 c d^{3/2} \sqrt {x} \sqrt {c+d x} \sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:
Integrate[Sqrt[e*x]/((c + d*x)^(3/2)*Sqrt[b*c^2 - b*d^2*x^2]),x]
Output:
(Sqrt[e*x]*Sqrt[c - d*x]*(-2*Sqrt[d]*Sqrt[x]*Sqrt[c - d*x] + Sqrt[2]*(c + d*x)*ArcTan[(Sqrt[2]*Sqrt[d]*Sqrt[x])/Sqrt[c - d*x]]))/(4*c*d^(3/2)*Sqrt[x ]*Sqrt[c + d*x]*Sqrt[b*(c^2 - d^2*x^2)])
Time = 0.44 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {586, 105, 104, 218}
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.
\(\displaystyle \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx\) |
\(\Big \downarrow \) 586 |
\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \int \frac {\sqrt {e x}}{(c+d x)^2 \sqrt {b c-b d x}}dx}{\sqrt {b c^2-b d^2 x^2}}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e \int \frac {1}{\sqrt {e x} (c+d x) \sqrt {b c-b d x}}dx}{4 d}-\frac {\sqrt {e x} \sqrt {b c-b d x}}{2 b c d (c+d x)}\right )}{\sqrt {b c^2-b d^2 x^2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {e \int \frac {1}{c e+\frac {2 b c d x e}{b c-b d x}}d\frac {\sqrt {e x}}{\sqrt {b c-b d x}}}{2 d}-\frac {\sqrt {e x} \sqrt {b c-b d x}}{2 b c d (c+d x)}\right )}{\sqrt {b c^2-b d^2 x^2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (\frac {\sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {b c-b d x}}\right )}{2 \sqrt {2} \sqrt {b} c d^{3/2}}-\frac {\sqrt {e x} \sqrt {b c-b d x}}{2 b c d (c+d x)}\right )}{\sqrt {b c^2-b d^2 x^2}}\) |
Input:
Int[Sqrt[e*x]/((c + d*x)^(3/2)*Sqrt[b*c^2 - b*d^2*x^2]),x]
Output:
(Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*(-1/2*(Sqrt[e*x]*Sqrt[b*c - b*d*x])/(b*c* d*(c + d*x)) + (Sqrt[e]*ArcTan[(Sqrt[2]*Sqrt[b]*Sqrt[d]*Sqrt[e*x])/(Sqrt[e ]*Sqrt[b*c - b*d*x])])/(2*Sqrt[2]*Sqrt[b]*c*d^(3/2))))/Sqrt[b*c^2 - b*d^2* x^2]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
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[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_) , x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]/((c + d*x)^FracPart[p]*(a/c + ( b*x)/d)^FracPart[p]) Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(217\) vs. \(2(101)=202\).
Time = 0.30 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.64
method | result | size |
default | \(-\frac {\sqrt {e x}\, \sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (\ln \left (\frac {3 b c d e x -b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d}{d x +c}\right ) b c d e x +\ln \left (\frac {3 b c d e x -b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d}{d x +c}\right ) b \,c^{2} e +2 \sqrt {2}\, \sqrt {-\frac {b \,c^{2} e}{d}}\, \sqrt {\left (-d x +c \right ) b e x}\, d \right ) \sqrt {2}}{8 b c \left (d x +c \right )^{\frac {3}{2}} d^{2} \sqrt {\left (-d x +c \right ) b e x}\, \sqrt {-\frac {b \,c^{2} e}{d}}}\) | \(218\) |
Input:
int((e*x)^(1/2)/(d*x+c)^(3/2)/(-b*d^2*x^2+b*c^2)^(1/2),x,method=_RETURNVER BOSE)
Output:
-1/8*(e*x)^(1/2)*(b*(-d^2*x^2+c^2))^(1/2)/b/c*(ln((3*b*c*d*e*x-b*c^2*e+2*2 ^(1/2)*(-1/d*b*c^2*e)^(1/2)*((-d*x+c)*b*e*x)^(1/2)*d)/(d*x+c))*b*c*d*e*x+l n((3*b*c*d*e*x-b*c^2*e+2*2^(1/2)*(-1/d*b*c^2*e)^(1/2)*((-d*x+c)*b*e*x)^(1/ 2)*d)/(d*x+c))*b*c^2*e+2*2^(1/2)*(-1/d*b*c^2*e)^(1/2)*((-d*x+c)*b*e*x)^(1/ 2)*d)*2^(1/2)/(d*x+c)^(3/2)/d^2/((-d*x+c)*b*e*x)^(1/2)/(-1/d*b*c^2*e)^(1/2 )
Time = 0.20 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.89 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\left [\frac {\sqrt {\frac {1}{2}} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {-\frac {e}{b d}} \log \left (-\frac {17 \, d^{3} e x^{3} + 3 \, c d^{2} e x^{2} - 13 \, c^{2} d e x + c^{3} e + 8 \, \sqrt {\frac {1}{2}} \sqrt {-b d^{2} x^{2} + b c^{2}} {\left (3 \, d^{2} x - c d\right )} \sqrt {d x + c} \sqrt {e x} \sqrt {-\frac {e}{b d}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 4 \, \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x}}{8 \, {\left (b c d^{3} x^{2} + 2 \, b c^{2} d^{2} x + b c^{3} d\right )}}, -\frac {\sqrt {\frac {1}{2}} {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {\frac {e}{b d}} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x} d \sqrt {\frac {e}{b d}}}{3 \, d^{2} e x^{2} + 2 \, c d e x - c^{2} e}\right ) + 2 \, \sqrt {-b d^{2} x^{2} + b c^{2}} \sqrt {d x + c} \sqrt {e x}}{4 \, {\left (b c d^{3} x^{2} + 2 \, b c^{2} d^{2} x + b c^{3} d\right )}}\right ] \] Input:
integrate((e*x)^(1/2)/(d*x+c)^(3/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm= "fricas")
Output:
[1/8*(sqrt(1/2)*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(-e/(b*d))*log(-(17*d^ 3*e*x^3 + 3*c*d^2*e*x^2 - 13*c^2*d*e*x + c^3*e + 8*sqrt(1/2)*sqrt(-b*d^2*x ^2 + b*c^2)*(3*d^2*x - c*d)*sqrt(d*x + c)*sqrt(e*x)*sqrt(-e/(b*d)))/(d^3*x ^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - 4*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*sqrt(e*x))/(b*c*d^3*x^2 + 2*b*c^2*d^2*x + b*c^3*d), -1/4*(sqrt(1/2)* (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(e/(b*d))*arctan(4*sqrt(1/2)*sqrt(-b*d ^2*x^2 + b*c^2)*sqrt(d*x + c)*sqrt(e*x)*d*sqrt(e/(b*d))/(3*d^2*e*x^2 + 2*c *d*e*x - c^2*e)) + 2*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*sqrt(e*x))/(b* c*d^3*x^2 + 2*b*c^2*d^2*x + b*c^3*d)]
\[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\int \frac {\sqrt {e x}}{\sqrt {- b \left (- c + d x\right ) \left (c + d x\right )} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x)**(1/2)/(d*x+c)**(3/2)/(-b*d**2*x**2+b*c**2)**(1/2),x)
Output:
Integral(sqrt(e*x)/(sqrt(-b*(-c + d*x)*(c + d*x))*(c + d*x)**(3/2)), x)
\[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\int { \frac {\sqrt {e x}}{\sqrt {-b d^{2} x^{2} + b c^{2}} {\left (d x + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x)^(1/2)/(d*x+c)^(3/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm= "maxima")
Output:
integrate(sqrt(e*x)/(sqrt(-b*d^2*x^2 + b*c^2)*(d*x + c)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (101) = 202\).
Time = 0.23 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {\frac {\sqrt {2} \sqrt {-b d e} e^{2} \log \left (\frac {{\left | -4 \, \sqrt {2} e^{2} {\left | b \right |} {\left | c \right |} - 6 \, b c e^{2} + 2 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} \right |}}{{\left | 4 \, \sqrt {2} e^{2} {\left | b \right |} {\left | c \right |} - 6 \, b c e^{2} + 2 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} \right |}}\right )}{d^{2} {\left | b \right |} {\left | c \right |} {\left | e \right |}} + \frac {8 \, {\left (\sqrt {-b d e} b c e^{6} - 3 \, \sqrt {-b d e} {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} e^{4}\right )}}{{\left (b^{2} c^{2} e^{4} - 6 \, {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{2} b c e^{2} + {\left (\sqrt {-b d e} \sqrt {e x} - \sqrt {-b d e^{2} x + b c e^{2}}\right )}^{4}\right )} d^{2} {\left | e \right |}}}{8 \, e} \] Input:
integrate((e*x)^(1/2)/(d*x+c)^(3/2)/(-b*d^2*x^2+b*c^2)^(1/2),x, algorithm= "giac")
Output:
-1/8*(sqrt(2)*sqrt(-b*d*e)*e^2*log(abs(-4*sqrt(2)*e^2*abs(b)*abs(c) - 6*b* c*e^2 + 2*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^2)/abs(4*s qrt(2)*e^2*abs(b)*abs(c) - 6*b*c*e^2 + 2*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b *d*e^2*x + b*c*e^2))^2))/(d^2*abs(b)*abs(c)*abs(e)) + 8*(sqrt(-b*d*e)*b*c* e^6 - 3*sqrt(-b*d*e)*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2)) ^2*e^4)/((b^2*c^2*e^4 - 6*(sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c* e^2))^2*b*c*e^2 + (sqrt(-b*d*e)*sqrt(e*x) - sqrt(-b*d*e^2*x + b*c*e^2))^4) *d^2*abs(e)))/e
Timed out. \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\int \frac {\sqrt {e\,x}}{\sqrt {b\,c^2-b\,d^2\,x^2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:
int((e*x)^(1/2)/((b*c^2 - b*d^2*x^2)^(1/2)*(c + d*x)^(3/2)),x)
Output:
int((e*x)^(1/2)/((b*c^2 - b*d^2*x^2)^(1/2)*(c + d*x)^(3/2)), x)
Time = 0.21 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.46 \[ \int \frac {\sqrt {e x}}{(c+d x)^{3/2} \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {e}\, \sqrt {b}\, \left (-4 \sqrt {x}\, \sqrt {-d x +c}\, d +\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i +\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x -\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i -\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x -\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i -\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}-\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x +\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) c i +\sqrt {d}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}+\sqrt {c}+\sqrt {x}\, \sqrt {d}\, i}{\sqrt {c}}\right ) d i x \right )}{8 b c \,d^{2} \left (d x +c \right )} \] Input:
int((e*x)^(1/2)/(d*x+c)^(3/2)/(-b*d^2*x^2+b*c^2)^(1/2),x)
Output:
(sqrt(e)*sqrt(b)*( - 4*sqrt(x)*sqrt(c - d*x)*d + sqrt(d)*sqrt(2)*log((sqrt (c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*i + sqrt(d)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sqrt(2) - sqrt(c) + sqrt(x)*s qrt(d)*i)/sqrt(c))*d*i*x - sqrt(d)*sqrt(2)*log((sqrt(c - d*x) - sqrt(c)*sq rt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*c*i - sqrt(d)*sqrt(2)*log((s qrt(c - d*x) - sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d*i *x - sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) - sqrt(c) + sqrt (x)*sqrt(d)*i)/sqrt(c))*c*i - sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c) *sqrt(2) - sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c))*d*i*x + sqrt(d)*sqrt(2)*l og((sqrt(c - d*x) + sqrt(c)*sqrt(2) + sqrt(c) + sqrt(x)*sqrt(d)*i)/sqrt(c) )*c*i + sqrt(d)*sqrt(2)*log((sqrt(c - d*x) + sqrt(c)*sqrt(2) + sqrt(c) + s qrt(x)*sqrt(d)*i)/sqrt(c))*d*i*x))/(8*b*c*d**2*(c + d*x))