Integrand size = 28, antiderivative size = 69 \[ \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx=\frac {2 \sqrt {e x} \sqrt {c+d x^4} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {1}{2},\frac {9}{8},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {1+\frac {d x^4}{c}}} \] Output:
2*(e*x)^(1/2)*(d*x^4+c)^(1/2)*AppellF1(1/8,1,-1/2,9/8,-b*x^4/a,-d*x^4/c)/a /e/(1+d*x^4/c)^(1/2)
Time = 11.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx=\frac {2 x \sqrt {c+d x^4} \operatorname {AppellF1}\left (\frac {1}{8},-\frac {1}{2},1,\frac {9}{8},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{a \sqrt {e x} \sqrt {\frac {c+d x^4}{c}}} \] Input:
Integrate[Sqrt[c + d*x^4]/(Sqrt[e*x]*(a + b*x^4)),x]
Output:
(2*x*Sqrt[c + d*x^4]*AppellF1[1/8, -1/2, 1, 9/8, -((d*x^4)/c), -((b*x^4)/a )])/(a*Sqrt[e*x]*Sqrt[(c + d*x^4)/c])
Time = 0.38 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {966, 27, 937, 936}
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.
\(\displaystyle \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx\) |
\(\Big \downarrow \) 966 |
\(\displaystyle \frac {2 \int \frac {e^4 \sqrt {d x^4+c}}{b x^4 e^4+a e^4}d\sqrt {e x}}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 e^3 \int \frac {\sqrt {d x^4+c}}{b x^4 e^4+a e^4}d\sqrt {e x}\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {2 e^3 \sqrt {c+d x^4} \int \frac {\sqrt {\frac {d x^4}{c}+1}}{b x^4 e^4+a e^4}d\sqrt {e x}}{\sqrt {\frac {d x^4}{c}+1}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {2 \sqrt {e x} \sqrt {c+d x^4} \operatorname {AppellF1}\left (\frac {1}{8},1,-\frac {1}{2},\frac {9}{8},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {\frac {d x^4}{c}+1}}\) |
Input:
Int[Sqrt[c + d*x^4]/(Sqrt[e*x]*(a + b*x^4)),x]
Output:
(2*Sqrt[e*x]*Sqrt[c + d*x^4]*AppellF1[1/8, 1, -1/2, 9/8, -((b*x^4)/a), -(( d*x^4)/c)])/(a*e*Sqrt[1 + (d*x^4)/c])
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_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*( m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*x)^( 1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && FractionQ[m] && IntegerQ[p]
\[\int \frac {\sqrt {d \,x^{4}+c}}{\sqrt {e x}\, \left (b \,x^{4}+a \right )}d x\]
Input:
int((d*x^4+c)^(1/2)/(e*x)^(1/2)/(b*x^4+a),x)
Output:
int((d*x^4+c)^(1/2)/(e*x)^(1/2)/(b*x^4+a),x)
Timed out. \[ \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx=\text {Timed out} \] Input:
integrate((d*x^4+c)^(1/2)/(e*x)^(1/2)/(b*x^4+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {c + d x^{4}}}{\sqrt {e x} \left (a + b x^{4}\right )}\, dx \] Input:
integrate((d*x**4+c)**(1/2)/(e*x)**(1/2)/(b*x**4+a),x)
Output:
Integral(sqrt(c + d*x**4)/(sqrt(e*x)*(a + b*x**4)), x)
\[ \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} \sqrt {e x}} \,d x } \] Input:
integrate((d*x^4+c)^(1/2)/(e*x)^(1/2)/(b*x^4+a),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*sqrt(e*x)), x)
\[ \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} \sqrt {e x}} \,d x } \] Input:
integrate((d*x^4+c)^(1/2)/(e*x)^(1/2)/(b*x^4+a),x, algorithm="giac")
Output:
integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*sqrt(e*x)), x)
Timed out. \[ \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {d\,x^4+c}}{\sqrt {e\,x}\,\left (b\,x^4+a\right )} \,d x \] Input:
int((c + d*x^4)^(1/2)/((e*x)^(1/2)*(a + b*x^4)),x)
Output:
int((c + d*x^4)^(1/2)/((e*x)^(1/2)*(a + b*x^4)), x)
\[ \int \frac {\sqrt {c+d x^4}}{\sqrt {e x} \left (a+b x^4\right )} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d \,x^{4}+c}}{\sqrt {x}\, a +\sqrt {x}\, b \,x^{4}}d x \right )}{e} \] Input:
int((d*x^4+c)^(1/2)/(e*x)^(1/2)/(b*x^4+a),x)
Output:
(sqrt(e)*int(sqrt(c + d*x**4)/(sqrt(x)*a + sqrt(x)*b*x**4),x))/e