\(\int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} (a+b x^4)} \, dx\) [278]

Optimal result

Integrand size = 28, antiderivative size = 69 \[ \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx=-\frac {2 \sqrt {c+d x^4} \operatorname {AppellF1}\left (-\frac {1}{8},1,-\frac {1}{2},\frac {7}{8},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {e x} \sqrt {1+\frac {d x^4}{c}}} \] Output:

-2*(d*x^4+c)^(1/2)*AppellF1(-1/8,1,-1/2,7/8,-b*x^4/a,-d*x^4/c)/a/e/(e*x)^( 
1/2)/(1+d*x^4/c)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(69)=138\).

Time = 11.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx=\frac {x \left (-70 a \left (c+d x^4\right )-10 (b c-4 a d) x^4 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{2},1,\frac {15}{8},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+14 b d x^8 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {15}{8},\frac {1}{2},1,\frac {23}{8},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )}{35 a^2 (e x)^{3/2} \sqrt {c+d x^4}} \] Input:

Integrate[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]
 

Output:

(x*(-70*a*(c + d*x^4) - 10*(b*c - 4*a*d)*x^4*Sqrt[1 + (d*x^4)/c]*AppellF1[ 
7/8, 1/2, 1, 15/8, -((d*x^4)/c), -((b*x^4)/a)] + 14*b*d*x^8*Sqrt[1 + (d*x^ 
4)/c]*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^4)/c), -((b*x^4)/a)]))/(35*a^2*( 
e*x)^(3/2)*Sqrt[c + d*x^4])
 

Rubi [A] (verified)

Time = 0.44 (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, 1013, 1012}

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.

\(\displaystyle \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx\)

\(\Big \downarrow \) 966

\(\displaystyle \frac {2 \int \frac {e^3 \sqrt {d x^4+c}}{x \left (b x^4 e^4+a e^4\right )}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \int \frac {\sqrt {d x^4+c}}{e x \left (b x^4 e^4+a e^4\right )}d\sqrt {e x}\)

\(\Big \downarrow \) 1013

\(\displaystyle \frac {2 e^3 \sqrt {c+d x^4} \int \frac {\sqrt {\frac {d x^4}{c}+1}}{e x \left (b x^4 e^4+a e^4\right )}d\sqrt {e x}}{\sqrt {\frac {d x^4}{c}+1}}\)

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {2 \sqrt {c+d x^4} \operatorname {AppellF1}\left (-\frac {1}{8},1,-\frac {1}{2},\frac {7}{8},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{a e \sqrt {e x} \sqrt {\frac {d x^4}{c}+1}}\)

Input:

Int[Sqrt[c + d*x^4]/((e*x)^(3/2)*(a + b*x^4)),x]
 

Output:

(-2*Sqrt[c + d*x^4]*AppellF1[-1/8, 1, -1/2, 7/8, -((b*x^4)/a), -((d*x^4)/c 
)])/(a*e*Sqrt[e*x]*Sqrt[1 + (d*x^4)/c])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 966

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]
 

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; 
 FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & 
& NeQ[m, n - 1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [F]

\[\int \frac {\sqrt {d \,x^{4}+c}}{\left (e x \right )^{\frac {3}{2}} \left (b \,x^{4}+a \right )}d x\]

Input:

int((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x)
 

Output:

int((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x)
 

Fricas [F]

\[ \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x, algorithm="fricas")
 

Output:

integral(sqrt(d*x^4 + c)*sqrt(e*x)/(b*e^2*x^6 + a*e^2*x^2), x)
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {c + d x^{4}}}{\left (e x\right )^{\frac {3}{2}} \left (a + b x^{4}\right )}\, dx \] Input:

integrate((d*x**4+c)**(1/2)/(e*x)**(3/2)/(b*x**4+a),x)
 

Output:

Integral(sqrt(c + d*x**4)/((e*x)**(3/2)*(a + b*x**4)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)), x)
 

Giac [F]

\[ \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx=\int { \frac {\sqrt {d x^{4} + c}}{{\left (b x^{4} + a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x, algorithm="giac")
 

Output:

integrate(sqrt(d*x^4 + c)/((b*x^4 + a)*(e*x)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx=\int \frac {\sqrt {d\,x^4+c}}{{\left (e\,x\right )}^{3/2}\,\left (b\,x^4+a\right )} \,d x \] Input:

int((c + d*x^4)^(1/2)/((e*x)^(3/2)*(a + b*x^4)),x)
 

Output:

int((c + d*x^4)^(1/2)/((e*x)^(3/2)*(a + b*x^4)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {c+d x^4}}{(e x)^{3/2} \left (a+b x^4\right )} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {d \,x^{4}+c}}{\sqrt {x}\, a x +\sqrt {x}\, b \,x^{5}}d x \right )}{e^{2}} \] Input:

int((d*x^4+c)^(1/2)/(e*x)^(3/2)/(b*x^4+a),x)
 

Output:

(sqrt(e)*int(sqrt(c + d*x**4)/(sqrt(x)*a*x + sqrt(x)*b*x**5),x))/e**2