\(\int \frac {x^6}{(a+b x^2)^{3/4} (c+d x^2)^{3/4}} \, dx\) [1569]

Optimal result

Integrand size = 26, antiderivative size = 88 \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\frac {x^7 \left (1+\frac {b x^2}{a}\right )^{3/4} \left (1+\frac {d x^2}{c}\right )^{3/4} \operatorname {AppellF1}\left (\frac {7}{2},\frac {3}{4},\frac {3}{4},\frac {9}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{7 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \] Output:

1/7*x^7*(1+b*x^2/a)^(3/4)*(1+d*x^2/c)^(3/4)*AppellF1(7/2,3/4,3/4,9/2,-b*x^ 
2/a,-d*x^2/c)/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(224\) vs. \(2(88)=176\).

Time = 6.25 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.55 \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\frac {\left (7 b^2 c^2+6 a b c d+7 a^2 d^2\right ) x^3 \left (1+\frac {b x^2}{a}\right )^{3/4} \left (1+\frac {d x^2}{c}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},\frac {3}{4},\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+2 x \left (c+d x^2\right ) \left (-\left (\left (a+b x^2\right ) \left (7 b c+7 a d-4 b d x^2\right )\right )+7 a (b c+a d) \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {(-b c+a d) x^2}{a \left (c+d x^2\right )}\right )\right )}{32 b^2 d^2 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \] Input:

Integrate[x^6/((a + b*x^2)^(3/4)*(c + d*x^2)^(3/4)),x]
 

Output:

((7*b^2*c^2 + 6*a*b*c*d + 7*a^2*d^2)*x^3*(1 + (b*x^2)/a)^(3/4)*(1 + (d*x^2 
)/c)^(3/4)*AppellF1[3/2, 3/4, 3/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + 2*x* 
(c + d*x^2)*(-((a + b*x^2)*(7*b*c + 7*a*d - 4*b*d*x^2)) + 7*a*(b*c + a*d)* 
((c*(a + b*x^2))/(a*(c + d*x^2)))^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, ( 
(-(b*c) + a*d)*x^2)/(a*(c + d*x^2))]))/(32*b^2*d^2*(a + b*x^2)^(3/4)*(c + 
d*x^2)^(3/4))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {395, 395, 394}

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.

\(\displaystyle \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {\left (\frac {b x^2}{a}+1\right )^{3/4} \int \frac {x^6}{\left (\frac {b x^2}{a}+1\right )^{3/4} \left (d x^2+c\right )^{3/4}}dx}{\left (a+b x^2\right )^{3/4}}\)

\(\Big \downarrow \) 395

\(\displaystyle \frac {\left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {d x^2}{c}+1\right )^{3/4} \int \frac {x^6}{\left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {d x^2}{c}+1\right )^{3/4}}dx}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}}\)

\(\Big \downarrow \) 394

\(\displaystyle \frac {x^7 \left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {d x^2}{c}+1\right )^{3/4} \operatorname {AppellF1}\left (\frac {7}{2},\frac {3}{4},\frac {3}{4},\frac {9}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{7 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}}\)

Input:

Int[x^6/((a + b*x^2)^(3/4)*(c + d*x^2)^(3/4)),x]
 

Output:

(x^7*(1 + (b*x^2)/a)^(3/4)*(1 + (d*x^2)/c)^(3/4)*AppellF1[7/2, 3/4, 3/4, 9 
/2, -((b*x^2)/a), -((d*x^2)/c)])/(7*(a + b*x^2)^(3/4)*(c + d*x^2)^(3/4))
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ 
FracPart[p])   Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ 
[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 
1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [F]

\[\int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )^{\frac {3}{4}}}d x\]

Input:

int(x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x)
 

Output:

int(x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\text {Timed out} \] Input:

integrate(x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int \frac {x^{6}}{\left (a + b x^{2}\right )^{\frac {3}{4}} \left (c + d x^{2}\right )^{\frac {3}{4}}}\, dx \] Input:

integrate(x**6/(b*x**2+a)**(3/4)/(d*x**2+c)**(3/4),x)
 

Output:

Integral(x**6/((a + b*x**2)**(3/4)*(c + d*x**2)**(3/4)), x)
 

Maxima [F]

\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x, algorithm="maxima")
 

Output:

integrate(x^6/((b*x^2 + a)^(3/4)*(d*x^2 + c)^(3/4)), x)
 

Giac [F]

\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{\frac {3}{4}}} \,d x } \] Input:

integrate(x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x, algorithm="giac")
 

Output:

integrate(x^6/((b*x^2 + a)^(3/4)*(d*x^2 + c)^(3/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int \frac {x^6}{{\left (b\,x^2+a\right )}^{3/4}\,{\left (d\,x^2+c\right )}^{3/4}} \,d x \] Input:

int(x^6/((a + b*x^2)^(3/4)*(c + d*x^2)^(3/4)),x)
 

Output:

int(x^6/((a + b*x^2)^(3/4)*(c + d*x^2)^(3/4)), x)
 

Reduce [F]

\[ \int \frac {x^6}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int \frac {x^{6}}{\left (d \,x^{2}+c \right )^{\frac {3}{4}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x \] Input:

int(x^6/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x)
 

Output:

int(x**6/((c + d*x**2)**(3/4)*(a + b*x**2)**(3/4)),x)