Integrand size = 24, antiderivative size = 427 \[ \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=-\frac {\left (45 b^2 c^2-32 a b c d-8 a^2 d^2\right ) x}{10 b d^3 (b c-a d) \sqrt [4]{a+b x^2}}+\frac {(9 b c-4 a d) x \left (a+b x^2\right )^{3/4}}{10 b d^2 (b c-a d)}-\frac {c x^3 \left (a+b x^2\right )^{3/4}}{2 d (b c-a d) \left (c+d x^2\right )}+\frac {\sqrt {a} \left (45 b^2 c^2-32 a b c d-8 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{10 b^{3/2} d^3 (b c-a d) \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} c^2 (9 b c-10 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 d^{7/2} (-b c+a d)^{3/2} x}+\frac {\sqrt [4]{a} c^2 (9 b c-10 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 d^{7/2} (-b c+a d)^{3/2} x} \] Output:
-1/10*(-8*a^2*d^2-32*a*b*c*d+45*b^2*c^2)*x/b/d^3/(-a*d+b*c)/(b*x^2+a)^(1/4 )+1/10*(-4*a*d+9*b*c)*x*(b*x^2+a)^(3/4)/b/d^2/(-a*d+b*c)-1/2*c*x^3*(b*x^2+ a)^(3/4)/d/(-a*d+b*c)/(d*x^2+c)+1/10*a^(1/2)*(-8*a^2*d^2-32*a*b*c*d+45*b^2 *c^2)*(1+b*x^2/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x/a^(1/2))),2^(1/ 2))/b^(3/2)/d^3/(-a*d+b*c)/(b*x^2+a)^(1/4)-1/4*a^(1/4)*c^2*(-10*a*d+9*b*c) *(-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d -b*c)^(1/2),I)/d^(7/2)/(a*d-b*c)^(3/2)/x+1/4*a^(1/4)*c^2*(-10*a*d+9*b*c)*( -b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d-b* c)^(1/2),I)/d^(7/2)/(a*d-b*c)^(3/2)/x
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 9.58 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.92 \[ \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\frac {x^3 \left (\frac {\left (-45 b^2 c^2+32 a b c d+8 a^2 d^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}-\frac {6 \left (-6 a c \left (4 a^2 d^2+4 a b d^2 x^2-b^2 c \left (9 c+4 d x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\left (a+b x^2\right ) \left (4 a d \left (c+d x^2\right )-b c \left (9 c+4 d x^2\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{60 b d^2 (b c-a d) \sqrt [4]{a+b x^2}} \] Input:
Integrate[x^6/((a + b*x^2)^(1/4)*(c + d*x^2)^2),x]
Output:
(x^3*(((-45*b^2*c^2 + 32*a*b*c*d + 8*a^2*d^2)*(1 + (b*x^2)/a)^(1/4)*Appell F1[3/2, 1/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])/c - (6*(-6*a*c*(4*a^2*d^ 2 + 4*a*b*d^2*x^2 - b^2*c*(9*c + 4*d*x^2))*AppellF1[1/2, 1/4, 1, 3/2, -((b *x^2)/a), -((d*x^2)/c)] + (a + b*x^2)*(4*a*d*(c + d*x^2) - b*c*(9*c + 4*d* x^2))*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c* AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))/((c + d*x^2)*(-6 *a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + x^2*(4*a*d*A ppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(60*b*d^2*(b*c - a*d)*(a + b *x^2)^(1/4))
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {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.
| \(\displaystyle \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 395 |
| \(\displaystyle \frac {\sqrt [4]{\frac {b x^2}{a}+1} \int \frac {x^6}{\sqrt [4]{\frac {b x^2}{a}+1} \left (d x^2+c\right )^2}dx}{\sqrt [4]{a+b x^2}}\) |
| \(\Big \downarrow \) 394 |
| \(\displaystyle \frac {x^7 \sqrt [4]{\frac {b x^2}{a}+1} \operatorname {AppellF1}\left (\frac {7}{2},\frac {1}{4},2,\frac {9}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{7 c^2 \sqrt [4]{a+b x^2}}\) |
Input:
Int[x^6/((a + b*x^2)^(1/4)*(c + d*x^2)^2),x]
Output:
(x^7*(1 + (b*x^2)/a)^(1/4)*AppellF1[7/2, 1/4, 2, 9/2, -((b*x^2)/a), -((d*x ^2)/c)])/(7*c^2*(a + b*x^2)^(1/4))
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])
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])
\[\int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (x^{2} d +c \right )^{2}}d x\]
Input:
int(x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x)
Output:
int(x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x)
Timed out. \[ \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int \frac {x^{6}}{\sqrt [4]{a + b x^{2}} \left (c + d x^{2}\right )^{2}}\, dx \] Input:
integrate(x**6/(b*x**2+a)**(1/4)/(d*x**2+c)**2,x)
Output:
Integral(x**6/((a + b*x**2)**(1/4)*(c + d*x**2)**2), x)
\[ \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x, algorithm="maxima")
Output:
integrate(x^6/((b*x^2 + a)^(1/4)*(d*x^2 + c)^2), x)
\[ \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int { \frac {x^{6}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \] Input:
integrate(x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x, algorithm="giac")
Output:
integrate(x^6/((b*x^2 + a)^(1/4)*(d*x^2 + c)^2), x)
Timed out. \[ \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int \frac {x^6}{{\left (b\,x^2+a\right )}^{1/4}\,{\left (d\,x^2+c\right )}^2} \,d x \] Input:
int(x^6/((a + b*x^2)^(1/4)*(c + d*x^2)^2),x)
Output:
int(x^6/((a + b*x^2)^(1/4)*(c + d*x^2)^2), x)
\[ \int \frac {x^6}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int \frac {x^{6}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} c^{2}+2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} c d \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} d^{2} x^{4}}d x \] Input:
int(x^6/(b*x^2+a)^(1/4)/(d*x^2+c)^2,x)
Output:
int(x**6/((a + b*x**2)**(1/4)*c**2 + 2*(a + b*x**2)**(1/4)*c*d*x**2 + (a + b*x**2)**(1/4)*d**2*x**4),x)