\(\int \frac {e+f x^2}{(a+b x^2)^{9/4} \sqrt [4]{c+d x^2}} \, dx\) [566]

Optimal result

Integrand size = 30, antiderivative size = 168 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx=\frac {2 (b e-a f) x \left (c+d x^2\right )^{3/4}}{5 a (b c-a d) \left (a+b x^2\right )^{5/4}}+\frac {(3 b c e-5 a d e+2 a c f) x \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{5/4} \left (c+d x^2\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (c+d x^2\right )}\right )}{5 a c (b c-a d) \left (a+b x^2\right )^{5/4}} \] Output:

2/5*(-a*f+b*e)*x*(d*x^2+c)^(3/4)/a/(-a*d+b*c)/(b*x^2+a)^(5/4)+1/5*(2*a*c*f 
-5*a*d*e+3*b*c*e)*x*(c*(b*x^2+a)/a/(d*x^2+c))^(5/4)*(d*x^2+c)^(3/4)*hyperg 
eom([1/2, 5/4],[3/2],-(-a*d+b*c)*x^2/a/(d*x^2+c))/a/c/(-a*d+b*c)/(b*x^2+a) 
^(5/4)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 10.36 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.40 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx=\frac {x \left (\frac {16 f x^2 \left (a+b x^2\right )^3 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {9}{4},\frac {5}{2},\frac {(-b c+a d) x^2}{a \left (c+d x^2\right )}\right )}{a^2 \left (1+\frac {d x^2}{c}\right )^{5/4}}+\frac {e \left (c+d x^2\right ) \operatorname {Gamma}\left (\frac {1}{4}\right ) \left (5 c \left (a+b x^2\right ) \left (3 c+2 d x^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {9}{4},\frac {5}{2},\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )+9 (b c-a d) x^2 \left (c+d x^2\right ) \operatorname {Hypergeometric2F1}\left (2,\frac {13}{4},\frac {7}{2},\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )\right )}{c^3 \operatorname {Gamma}\left (\frac {9}{4}\right )}\right )}{48 \left (a+b x^2\right )^{13/4} \sqrt [4]{c+d x^2}} \] Input:

Integrate[(e + f*x^2)/((a + b*x^2)^(9/4)*(c + d*x^2)^(1/4)),x]
 

Output:

(x*((16*f*x^2*(a + b*x^2)^3*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[3/2, 9 
/4, 5/2, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/(a^2*(1 + (d*x^2)/c)^(5/4) 
) + (e*(c + d*x^2)*Gamma[1/4]*(5*c*(a + b*x^2)*(3*c + 2*d*x^2)*Hypergeomet 
ric2F1[1, 9/4, 5/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 9*(b*c - a*d)*x^2 
*(c + d*x^2)*Hypergeometric2F1[2, 13/4, 7/2, ((b*c - a*d)*x^2)/(c*(a + b*x 
^2))]))/(c^3*Gamma[9/4])))/(48*(a + b*x^2)^(13/4)*(c + d*x^2)^(1/4))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {402, 27, 294}

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.

\(\displaystyle \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {2 x \left (c+d x^2\right )^{3/4} (b e-a f)}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}-\frac {2 \int -\frac {3 b c e-5 a d e+2 a c f}{2 \left (b x^2+a\right )^{5/4} \sqrt [4]{d x^2+c}}dx}{5 a (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(2 a c f-5 a d e+3 b c e) \int \frac {1}{\left (b x^2+a\right )^{5/4} \sqrt [4]{d x^2+c}}dx}{5 a (b c-a d)}+\frac {2 x \left (c+d x^2\right )^{3/4} (b e-a f)}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\)

\(\Big \downarrow \) 294

\(\displaystyle \frac {x \left (c+d x^2\right )^{3/4} \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{5/4} (2 a c f-5 a d e+3 b c e) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{5 a c \left (a+b x^2\right )^{5/4} (b c-a d)}+\frac {2 x \left (c+d x^2\right )^{3/4} (b e-a f)}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}\)

Input:

Int[(e + f*x^2)/((a + b*x^2)^(9/4)*(c + d*x^2)^(1/4)),x]
 

Output:

(2*(b*e - a*f)*x*(c + d*x^2)^(3/4))/(5*a*(b*c - a*d)*(a + b*x^2)^(5/4)) + 
((3*b*c*e - 5*a*d*e + 2*a*c*f)*x*((c*(a + b*x^2))/(a*(c + d*x^2)))^(5/4)*( 
c + d*x^2)^(3/4)*Hypergeometric2F1[1/2, 5/4, 3/2, -(((b*c - a*d)*x^2)/(a*( 
c + d*x^2)))])/(5*a*c*(b*c - a*d)*(a + b*x^2)^(5/4))
 

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 294

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[x*((a + b*x^2)^p/(c*(c*((a + b*x^2)/(a*(c + d*x^2))))^p*(c + d*x^2)^(1/2 
+ p)))*Hypergeometric2F1[1/2, -p, 3/2, (-(b*c - a*d))*(x^2/(a*(c + d*x^2))) 
], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q 
+ 1) + 1, 0]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

Maple [F]

\[\int \frac {f \,x^{2}+e}{\left (b \,x^{2}+a \right )^{\frac {9}{4}} \left (x^{2} d +c \right )^{\frac {1}{4}}}d x\]

Input:

int((f*x^2+e)/(b*x^2+a)^(9/4)/(d*x^2+c)^(1/4),x)
 

Output:

int((f*x^2+e)/(b*x^2+a)^(9/4)/(d*x^2+c)^(1/4),x)
 

Fricas [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(9/4)/(d*x^2+c)^(1/4),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^(3/4)*(d*x^2 + c)^(3/4)*(f*x^2 + e)/(b^3*d*x^8 + (b^3 
*c + 3*a*b^2*d)*x^6 + 3*(a*b^2*c + a^2*b*d)*x^4 + a^3*c + (3*a^2*b*c + a^3 
*d)*x^2), x)
 

Sympy [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx=\int \frac {e + f x^{2}}{\left (a + b x^{2}\right )^{\frac {9}{4}} \sqrt [4]{c + d x^{2}}}\, dx \] Input:

integrate((f*x**2+e)/(b*x**2+a)**(9/4)/(d*x**2+c)**(1/4),x)
 

Output:

Integral((e + f*x**2)/((a + b*x**2)**(9/4)*(c + d*x**2)**(1/4)), x)
 

Maxima [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(9/4)/(d*x^2+c)^(1/4),x, algorithm="maxima")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(9/4)*(d*x^2 + c)^(1/4)), x)
 

Giac [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} {\left (d x^{2} + c\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(9/4)/(d*x^2+c)^(1/4),x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(9/4)*(d*x^2 + c)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx=\int \frac {f\,x^2+e}{{\left (b\,x^2+a\right )}^{9/4}\,{\left (d\,x^2+c\right )}^{1/4}} \,d x \] Input:

int((e + f*x^2)/((a + b*x^2)^(9/4)*(c + d*x^2)^(1/4)),x)
 

Output:

int((e + f*x^2)/((a + b*x^2)^(9/4)*(c + d*x^2)^(1/4)), x)
 

Reduce [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{9/4} \sqrt [4]{c+d x^2}} \, dx=\left (\int \frac {x^{2}}{\left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2}+2 \left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} a b \,x^{2}+\left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{2} x^{4}}d x \right ) f +\left (\int \frac {1}{\left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2}+2 \left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} a b \,x^{2}+\left (d \,x^{2}+c \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{2} x^{4}}d x \right ) e \] Input:

int((f*x^2+e)/(b*x^2+a)^(9/4)/(d*x^2+c)^(1/4),x)
 

Output:

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