Integrand size = 28, antiderivative size = 184 \[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x} \sqrt {e+f x}}{(b c-a d) \sqrt {a+b x}}+\frac {2 \sqrt {f} \sqrt {-b e+a f} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {a+b x}}{\sqrt {-b e+a f}}\right )|\frac {d (b e-a f)}{(b c-a d) f}\right )}{b (b c-a d) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}} \]
-2*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(-a*d+b*c)/(b*x+a)^(1/2)+2*EllipticE(f^(1/2 )*(b*x+a)^(1/2)/(a*f-b*e)^(1/2),(d*(-a*f+b*e)/(-a*d+b*c)/f)^(1/2))*f^(1/2) *(a*f-b*e)^(1/2)*(d*x+c)^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b/(-a*d+b*c)/( b*(d*x+c)/(-a*d+b*c))^(1/2)/(f*x+e)^(1/2)
Time = 13.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {a-\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )}{b \sqrt {a-\frac {b c}{d}} \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{f (a+b x)}}} \]
(-2*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[e + f*x]*EllipticE[ArcSin[Sqrt[ a - (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/(b*Sqrt[a - (b*c)/d]*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(f*(a + b*x))])
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {110, 27, 124, 123}
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.
| \(\displaystyle \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2 \int \frac {f \sqrt {c+d x}}{2 \sqrt {a+b x} \sqrt {e+f x}}dx}{b c-a d}-\frac {2 \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f \int \frac {\sqrt {c+d x}}{\sqrt {a+b x} \sqrt {e+f x}}dx}{b c-a d}-\frac {2 \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}} \int \frac {\sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}{\sqrt {a+b x} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{\sqrt {e+f x} (b c-a d) \sqrt {\frac {b (c+d x)}{b c-a d}}}-\frac {2 \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d)}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2 \sqrt {f} \sqrt {c+d x} \sqrt {a f-b e} \sqrt {\frac {b (e+f x)}{b e-a f}} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {a+b x}}{\sqrt {a f-b e}}\right )|\frac {d (b e-a f)}{(b c-a d) f}\right )}{b \sqrt {e+f x} (b c-a d) \sqrt {\frac {b (c+d x)}{b c-a d}}}-\frac {2 \sqrt {c+d x} \sqrt {e+f x}}{\sqrt {a+b x} (b c-a d)}\) |
(-2*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*Sqrt[a + b*x]) + (2*Sqrt[f]* Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticE [ArcSin[(Sqrt[f]*Sqrt[a + b*x])/Sqrt[-(b*e) + a*f]], (d*(b*e - a*f))/((b*c - a*d)*f)])/(b*(b*c - a*d)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x])
3.27.73.3.1 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_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(636\) vs. \(2(161)=322\).
Time = 2.16 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.46
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right )}\, \left (\frac {2 b d f \,x^{2}+2 b c f x +2 b d e x +2 b c e}{\left (a d -b c \right ) b \sqrt {\left (x +\frac {a}{b}\right ) \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}}+\frac {2 \left (\frac {f}{b}-\frac {a d f -b c f -b d e}{b \left (a d -b c \right )}-\frac {b c f +b d e}{\left (a d -b c \right ) b}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, F\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}-\frac {2 f d \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) E\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a F\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{\left (a d -b c \right ) \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}\) | \(637\) |
| default | \(\text {Expression too large to display}\) | \(1023\) |
((b*x+a)*(d*x+c)*(f*x+e))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)* (2*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e)/(a*d-b*c)/b/((x+a/b)*(b*d*f*x^2+b*c*f *x+b*d*e*x+b*c*e))^(1/2)+2*(f/b-1/b*(a*d*f-b*c*f-b*d*e)/(a*d-b*c)-(b*c*f+b *d*e)/(a*d-b*c)/b)*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*((x+a/b)/(-e/f+a/b) )^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^ 2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*EllipticF(((x+e/f)/(e/f-c/d))^(1/2) ,((-e/f+c/d)/(-e/f+a/b))^(1/2))-2*f*d/(a*d-b*c)*(e/f-c/d)*((x+e/f)/(e/f-c/ d))^(1/2)*((x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(b*d*f*x^3 +a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*((-e/f +a/b)*EllipticE(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))-a /b*EllipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 764, normalized size of antiderivative = 4.15 \[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e} b^{2} d f + {\left (a b d e - {\left (2 \, a b c - a^{2} d\right )} f + {\left (b^{2} d e - {\left (2 \, b^{2} c - a b d\right )} f\right )} x\right )} \sqrt {b d f} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} - {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2}\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right ) + 3 \, {\left (b^{2} d f x + a b d f\right )} \sqrt {b d f} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} - {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2}\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3}\right )}}{27 \, b^{3} d^{3} f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} d^{2} e^{2} - {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} - a b c d + a^{2} d^{2}\right )} f^{2}\right )}}{3 \, b^{2} d^{2} f^{2}}, -\frac {4 \, {\left (2 \, b^{3} d^{3} e^{3} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} e^{2} f - 3 \, {\left (b^{3} c^{2} d - 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e f^{2} + {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} f^{3}\right )}}{27 \, b^{3} d^{3} f^{3}}, \frac {3 \, b d f x + b d e + {\left (b c + a d\right )} f}{3 \, b d f}\right )\right )\right )}}{3 \, {\left ({\left (b^{4} c d - a b^{3} d^{2}\right )} f x + {\left (a b^{3} c d - a^{2} b^{2} d^{2}\right )} f\right )}} \]
-2/3*(3*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*b^2*d*f + (a*b*d*e - (2* a*b*c - a^2*d)*f + (b^2*d*e - (2*b^2*c - a*b*d)*f)*x)*sqrt(b*d*f)*weierstr assPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c* d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b ^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c ^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3 *b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 3*(b^2*d*f*x + a*b*d*f)*sqrt( b*d*f)*weierstrassZeta(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c ^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3* c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f ^3), weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2 *c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^ 3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f ^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3 *f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f))))/((b^4*c*d - a*b^ 3*d^2)*f*x + (a*b^3*c*d - a^2*b^2*d^2)*f)
\[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int \frac {\sqrt {e + f x}}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x}}\, dx \]
\[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int { \frac {\sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \]
\[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int { \frac {\sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e+f x}}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx=\int \frac {\sqrt {e+f\,x}}{{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \]