Integrand size = 26, antiderivative size = 266 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {b e-a f} \sqrt {c+d x}}-\frac {2 \sqrt [4]{d e-c f} \sqrt {-\frac {f (c+d x)}{d e-c f}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {b e-a f} \sqrt {c+d x}} \]
2*(-c*f+d*e)^(1/4)*EllipticPi(d^(1/4)*(f*x+e)^(1/4)/(-c*f+d*e)^(1/4),-b^(1 /2)*(-c*f+d*e)^(1/2)/d^(1/2)/(-a*f+b*e)^(1/2),I)*(-f*(d*x+c)/(-c*f+d*e))^( 1/2)/d^(1/4)/b^(1/2)/(-a*f+b*e)^(1/2)/(d*x+c)^(1/2)-2*(-c*f+d*e)^(1/4)*Ell ipticPi(d^(1/4)*(f*x+e)^(1/4)/(-c*f+d*e)^(1/4),b^(1/2)*(-c*f+d*e)^(1/2)/d^ (1/2)/(-a*f+b*e)^(1/2),I)*(-f*(d*x+c)/(-c*f+d*e))^(1/2)/d^(1/4)/b^(1/2)/(- a*f+b*e)^(1/2)/(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=-\frac {4 \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt [4]{\frac {b (e+f x)}{f (a+b x)}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},\frac {1}{4},\frac {7}{4},\frac {-b c+a d}{d (a+b x)},\frac {-b e+a f}{f (a+b x)}\right )}{3 b \sqrt {c+d x} \sqrt [4]{e+f x}} \]
(-4*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*((b*(e + f*x))/(f*(a + b*x)))^(1/4)* AppellF1[3/4, 1/2, 1/4, 7/4, (-(b*c) + a*d)/(d*(a + b*x)), (-(b*e) + a*f)/ (f*(a + b*x))])/(3*b*Sqrt[c + d*x]*(e + f*x)^(1/4))
Time = 0.44 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {117, 116, 993, 1542}
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.
\(\displaystyle \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx\) |
\(\Big \downarrow \) 117 |
\(\displaystyle \frac {\sqrt {-\frac {f (c+d x)}{d e-c f}} \int \frac {1}{(a+b x) \sqrt [4]{e+f x} \sqrt {-\frac {d x f}{d e-c f}-\frac {c f}{d e-c f}}}dx}{\sqrt {c+d x}}\) |
\(\Big \downarrow \) 116 |
\(\displaystyle -\frac {4 \sqrt {-\frac {f (c+d x)}{d e-c f}} \int \frac {\sqrt {e+f x}}{(b e-a f-b (e+f x)) \sqrt {1-\frac {d (e+f x)}{d e-c f}}}d\sqrt [4]{e+f x}}{\sqrt {c+d x}}\) |
\(\Big \downarrow \) 993 |
\(\displaystyle -\frac {4 \sqrt {-\frac {f (c+d x)}{d e-c f}} \left (\frac {\int \frac {1}{\left (\sqrt {b e-a f}-\sqrt {b} \sqrt {e+f x}\right ) \sqrt {1-\frac {d (e+f x)}{d e-c f}}}d\sqrt [4]{e+f x}}{2 \sqrt {b}}-\frac {\int \frac {1}{\left (\sqrt {b e-a f}+\sqrt {b} \sqrt {e+f x}\right ) \sqrt {1-\frac {d (e+f x)}{d e-c f}}}d\sqrt [4]{e+f x}}{2 \sqrt {b}}\right )}{\sqrt {c+d x}}\) |
\(\Big \downarrow \) 1542 |
\(\displaystyle -\frac {4 \sqrt {-\frac {f (c+d x)}{d e-c f}} \left (\frac {\sqrt [4]{d e-c f} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{2 \sqrt {b} \sqrt [4]{d} \sqrt {b e-a f}}-\frac {\sqrt [4]{d e-c f} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {d e-c f}}{\sqrt {d} \sqrt {b e-a f}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt [4]{e+f x}}{\sqrt [4]{d e-c f}}\right ),-1\right )}{2 \sqrt {b} \sqrt [4]{d} \sqrt {b e-a f}}\right )}{\sqrt {c+d x}}\) |
(-4*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*(-1/2*((d*e - c*f)^(1/4)*EllipticPi [-((Sqrt[b]*Sqrt[d*e - c*f])/(Sqrt[d]*Sqrt[b*e - a*f])), ArcSin[(d^(1/4)*( e + f*x)^(1/4))/(d*e - c*f)^(1/4)], -1])/(Sqrt[b]*d^(1/4)*Sqrt[b*e - a*f]) + ((d*e - c*f)^(1/4)*EllipticPi[(Sqrt[b]*Sqrt[d*e - c*f])/(Sqrt[d]*Sqrt[b *e - a*f]), ArcSin[(d^(1/4)*(e + f*x)^(1/4))/(d*e - c*f)^(1/4)], -1])/(2*S qrt[b]*d^(1/4)*Sqrt[b*e - a*f])))/Sqrt[c + d*x]
3.31.44.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 1/4)), x_] :> Simp[-4 Subst[Int[x^2/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[-f/(d*e - c*f), 0]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 1/4)), x_] :> Simp[Sqrt[(-f)*((c + d*x)/(d*e - c*f))]/Sqrt[c + d*x] Int[1 /((a + b*x)*Sqrt[(-c)*(f/(d*e - c*f)) - d*f*(x/(d*e - c*f))]*(e + f*x)^(1/4 )), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[-f/(d*e - c*f), 0]
Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(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)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b) Int[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
\[\int \frac {1}{\left (b x +a \right ) \left (f x +e \right )^{\frac {1}{4}} \sqrt {d x +c}}d x\]
Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\int \frac {1}{\left (a + b x\right ) \sqrt {c + d x} \sqrt [4]{e + f x}}\, dx \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} \sqrt {d x + c} {\left (f x + e\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} \sqrt {d x + c} {\left (f x + e\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt [4]{e+f x}} \, dx=\int \frac {1}{{\left (e+f\,x\right )}^{1/4}\,\left (a+b\,x\right )\,\sqrt {c+d\,x}} \,d x \]