Integrand size = 35, antiderivative size = 449 \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 d \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 (b c-a d) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b c-a d) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]
2*(-a*d+b*c)*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h /f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(d*(h*x +g)/(-c*h+d*g))^(1/2)/b^2/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*(-a*d+b*c) *EllipticPi(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),-b*(-c*f+d*e)/(-a*d+b*c) /f,((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e)/(-c*f+d*e ))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/b^2/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1 /2)+2*d*EllipticE(h^(1/2)*(f*x+e)^(1/2)/(e*h-f*g)^(1/2),(-d*(-e*h+f*g)/(-c *f+d*e)/h)^(1/2))*(e*h-f*g)^(1/2)*(d*x+c)^(1/2)*(f*(h*x+g)/(-e*h+f*g))^(1/ 2)/b/f/h^(1/2)/(-f*(d*x+c)/(-c*f+d*e))^(1/2)/(h*x+g)^(1/2)
Result contains complex when optimal does not.
Time = 21.30 (sec) , antiderivative size = 1176, normalized size of antiderivative = 2.62 \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \left (b^2 d^2 e^2 f \sqrt {-e+\frac {c f}{d}} g-b^2 c d e f^2 \sqrt {-e+\frac {c f}{d}} g-a b d^2 e f^2 \sqrt {-e+\frac {c f}{d}} g+a b c d f^3 \sqrt {-e+\frac {c f}{d}} g-b^2 d^2 e^3 \sqrt {-e+\frac {c f}{d}} h+b^2 c d e^2 f \sqrt {-e+\frac {c f}{d}} h+a b d^2 e^2 f \sqrt {-e+\frac {c f}{d}} h-a b c d e f^2 \sqrt {-e+\frac {c f}{d}} h-b^2 d^2 e f \sqrt {-e+\frac {c f}{d}} g (e+f x)+a b d^2 f^2 \sqrt {-e+\frac {c f}{d}} g (e+f x)+2 b^2 d^2 e^2 \sqrt {-e+\frac {c f}{d}} h (e+f x)-b^2 c d e f \sqrt {-e+\frac {c f}{d}} h (e+f x)-2 a b d^2 e f \sqrt {-e+\frac {c f}{d}} h (e+f x)+a b c d f^2 \sqrt {-e+\frac {c f}{d}} h (e+f x)-b^2 d^2 e \sqrt {-e+\frac {c f}{d}} h (e+f x)^2+a b d^2 f \sqrt {-e+\frac {c f}{d}} h (e+f x)^2+i b d (b e-a f) (d e-c f) h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right )|\frac {d (-f g+e h)}{(d e-c f) h}\right )+i b (-b c+a d) f (d e-c f) h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d (-f g+e h)}{(d e-c f) h}\right )-i b^2 c^2 f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticPi}\left (\frac {b d e-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d (-f g+e h)}{(d e-c f) h}\right )+2 i a b c d f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticPi}\left (\frac {b d e-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d (-f g+e h)}{(d e-c f) h}\right )-i a^2 d^2 f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticPi}\left (\frac {b d e-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d (-f g+e h)}{(d e-c f) h}\right )\right )}{b^2 f^2 (-b e+a f) \sqrt {-e+\frac {c f}{d}} h \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]
(2*(b^2*d^2*e^2*f*Sqrt[-e + (c*f)/d]*g - b^2*c*d*e*f^2*Sqrt[-e + (c*f)/d]* g - a*b*d^2*e*f^2*Sqrt[-e + (c*f)/d]*g + a*b*c*d*f^3*Sqrt[-e + (c*f)/d]*g - b^2*d^2*e^3*Sqrt[-e + (c*f)/d]*h + b^2*c*d*e^2*f*Sqrt[-e + (c*f)/d]*h + a*b*d^2*e^2*f*Sqrt[-e + (c*f)/d]*h - a*b*c*d*e*f^2*Sqrt[-e + (c*f)/d]*h - b^2*d^2*e*f*Sqrt[-e + (c*f)/d]*g*(e + f*x) + a*b*d^2*f^2*Sqrt[-e + (c*f)/d ]*g*(e + f*x) + 2*b^2*d^2*e^2*Sqrt[-e + (c*f)/d]*h*(e + f*x) - b^2*c*d*e*f *Sqrt[-e + (c*f)/d]*h*(e + f*x) - 2*a*b*d^2*e*f*Sqrt[-e + (c*f)/d]*h*(e + f*x) + a*b*c*d*f^2*Sqrt[-e + (c*f)/d]*h*(e + f*x) - b^2*d^2*e*Sqrt[-e + (c *f)/d]*h*(e + f*x)^2 + a*b*d^2*f*Sqrt[-e + (c*f)/d]*h*(e + f*x)^2 + I*b*d* (b*e - a*f)*(d*e - c*f)*h*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2 )*Sqrt[(f*(g + h*x))/(h*(e + f*x))]*EllipticE[I*ArcSinh[Sqrt[-e + (c*f)/d] /Sqrt[e + f*x]], (d*(-(f*g) + e*h))/((d*e - c*f)*h)] + I*b*(-(b*c) + a*d)* f*(d*e - c*f)*h*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Sqrt[(f* (g + h*x))/(h*(e + f*x))]*EllipticF[I*ArcSinh[Sqrt[-e + (c*f)/d]/Sqrt[e + f*x]], (d*(-(f*g) + e*h))/((d*e - c*f)*h)] - I*b^2*c^2*f^2*h*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Sqrt[(f*(g + h*x))/(h*(e + f*x))]*Ell ipticPi[(b*d*e - a*d*f)/(b*d*e - b*c*f), I*ArcSinh[Sqrt[-e + (c*f)/d]/Sqrt [e + f*x]], (d*(-(f*g) + e*h))/((d*e - c*f)*h)] + (2*I)*a*b*c*d*f^2*h*Sqrt [(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Sqrt[(f*(g + h*x))/(h*(e + f *x))]*EllipticPi[(b*d*e - a*d*f)/(b*d*e - b*c*f), I*ArcSinh[Sqrt[-e + (...
Time = 0.83 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {197, 2009}
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.
| \(\displaystyle \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
| \(\Big \downarrow \) 197 |
| \(\displaystyle \int \left (\frac {(b c-a d)^2}{b^2 (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}+\frac {d (b c-a d)}{b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}+\frac {d \sqrt {c+d x}}{b \sqrt {e+f x} \sqrt {g+h x}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 (b c-a d) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b c-a d) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 d \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {g+h x} \sqrt {-\frac {f (c+d x)}{d e-c f}}}\) |
(2*d*Sqrt[-(f*g) + e*h]*Sqrt[c + d*x]*Sqrt[(f*(g + h*x))/(f*g - e*h)]*Elli pticE[ArcSin[(Sqrt[h]*Sqrt[e + f*x])/Sqrt[-(f*g) + e*h]], -((d*(f*g - e*h) )/((d*e - c*f)*h))])/(b*f*Sqrt[h]*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*Sqrt[ g + h*x]) + (2*(b*c - a*d)*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c* f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x ])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b^2*Sqrt[f]*Sqr t[e + f*x]*Sqrt[g + h*x]) - (2*(b*c - a*d)*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticPi[-((b*(d*e - c*f))/((b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f] ], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(b^2*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h *x])
3.1.59.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_))/(Sqrt[(e_.) + (f _.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Int[ExpandIntegrand[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), (a + b*x)^m*(c + d*x)^(n + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[m] && IntegerQ[n + 1/2]
Time = 1.48 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.71
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (-\frac {2 d \left (a d -2 b c \right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{b^{2} \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 d^{2} \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{b \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \Pi \left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {a}{b}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{b^{3} \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}\, \left (-\frac {g}{h}+\frac {a}{b}\right )}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(769\) |
| default | \(\text {Expression too large to display}\) | \(1551\) |
((d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)* (-2*d*(a*d-2*b*c)/b^2*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c /d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g *x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*EllipticF(((x+g/h)/(g/h-e/f))^(1 /2),((-g/h+e/f)/(-g/h+c/d))^(1/2))+2*d^2/b*(g/h-e/f)*((x+g/h)/(g/h-e/f))^( 1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f* h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*((-g/h+c/d) *EllipticE(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))-c/d*El lipticF(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2)))+2*(a^2*d ^2-2*a*b*c*d+b^2*c^2)/b^3*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g /h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d *f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)/(-g/h+a/b)*EllipticPi(((x+g/ h)/(g/h-e/f))^(1/2),(-g/h+e/f)/(-g/h+a/b),((-g/h+e/f)/(-g/h+c/d))^(1/2)))
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
\[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (a + b x\right ) \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
\[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (b x + a\right )} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
\[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (b x + a\right )} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )} \,d x \]