Integrand size = 30, antiderivative size = 168 \[ \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx=\frac {2 (b e-a f) x \left (c+d x^2\right )^{7/4}}{9 a (b c-a d) \left (a+b x^2\right )^{9/4}}+\frac {(7 b c e-9 a d e+2 a c f) x \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{9/4} \left (c+d x^2\right )^{7/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {9}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (c+d x^2\right )}\right )}{9 a c (b c-a d) \left (a+b x^2\right )^{9/4}} \] Output:
2/9*(-a*f+b*e)*x*(d*x^2+c)^(7/4)/a/(-a*d+b*c)/(b*x^2+a)^(9/4)+1/9*(2*a*c*f -9*a*d*e+7*b*c*e)*x*(c*(b*x^2+a)/a/(d*x^2+c))^(9/4)*(d*x^2+c)^(7/4)*hyperg eom([1/2, 9/4],[3/2],-(-a*d+b*c)*x^2/a/(d*x^2+c))/a/c/(-a*d+b*c)/(b*x^2+a) ^(9/4)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 11.91 (sec) , antiderivative size = 1207, normalized size of antiderivative = 7.18 \[ \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx =\text {Too large to display} \] Input:
Integrate[((c + d*x^2)^(3/4)*(e + f*x^2))/(a + b*x^2)^(13/4),x]
Output:
(-2*x*(63*a*b^3*c^4*e*(a + b*x^2)^2*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2) /a), -((d*x^2)/c)] - 18*a^2*b^2*c^3*d*e*(a + b*x^2)^2*AppellF1[1/2, 1/4, 1 /4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 81*a^3*b*c^2*d^2*e*(a + b*x^2)^2*Ap pellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 18*a^2*b^2*c^4*f* (a + b*x^2)^2*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 1 8*a^3*b*c^3*d*f*(a + b*x^2)^2*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), - ((d*x^2)/c)] - (c + d*x^2)*(5*a^2*(b*c - a*d)^2*(b*e - a*f) + a*(-(b*c) + a*d)*(-7*b^2*c*e + 5*a^2*d*f + a*b*(4*d*e - 2*c*f))*(a + b*x^2) + 3*b^2*c* (7*b*c*e - 9*a*d*e + 2*a*c*f)*(a + b*x^2)^2)*(6*a*c*AppellF1[1/2, 1/4, 1/4 , 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(a*d*AppellF1[3/2, 1/4, 5/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1/4, 5/2, -((b*x^2)/ a), -((d*x^2)/c)])) + 14*b^3*c^2*d*e*x^2*(a + b*x^2)^2*(1 + (b*x^2)/a)^(1/ 4)*(1 + (d*x^2)/c)^(1/4)*AppellF1[3/2, 1/4, 1/4, 5/2, -((b*x^2)/a), -((d*x ^2)/c)]*(6*a*c*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(a*d*AppellF1[3/2, 1/4, 5/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)] + b*c*Ap pellF1[3/2, 5/4, 1/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)])) + 4*a*b^2*c^2*d*f *x^2*(a + b*x^2)^2*(1 + (b*x^2)/a)^(1/4)*(1 + (d*x^2)/c)^(1/4)*AppellF1[3/ 2, 1/4, 1/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)]*(6*a*c*AppellF1[1/2, 1/4, 1/ 4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - x^2*(a*d*AppellF1[3/2, 1/4, 5/4, 5/2 , -((b*x^2)/a), -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1/4, 5/2, -((b*x...
Time = 0.34 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.40, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {401, 27, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {2 x \left (c+d x^2\right )^{3/4} (b e-a f)}{9 a b \left (a+b x^2\right )^{9/4}}-\frac {2 \int -\frac {d (4 b e+5 a f) x^2+c (7 b e+2 a f)}{2 \left (b x^2+a\right )^{9/4} \sqrt [4]{d x^2+c}}dx}{9 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {d (4 b e+5 a f) x^2+c (7 b e+2 a f)}{\left (b x^2+a\right )^{9/4} \sqrt [4]{d x^2+c}}dx}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} (b e-a f)}{9 a b \left (a+b x^2\right )^{9/4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {2 x \left (c+d x^2\right )^{3/4} (b c (2 a f+7 b e)-a d (5 a f+4 b e))}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}-\frac {2 \int -\frac {3 b c (7 b c e-9 a d e+2 a c f)}{2 \left (b x^2+a\right )^{5/4} \sqrt [4]{d x^2+c}}dx}{5 a (b c-a d)}}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} (b e-a f)}{9 a b \left (a+b x^2\right )^{9/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 b c (2 a c f-9 a d e+7 b c e) \int \frac {1}{\left (b x^2+a\right )^{5/4} \sqrt [4]{d x^2+c}}dx}{5 a (b c-a d)}+\frac {2 x \left (c+d x^2\right )^{3/4} (b c (2 a f+7 b e)-a d (5 a f+4 b e))}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} (b e-a f)}{9 a b \left (a+b x^2\right )^{9/4}}\) |
| \(\Big \downarrow \) 294 |
| \(\displaystyle \frac {\frac {3 b x \left (c+d x^2\right )^{3/4} \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{5/4} (2 a c f-9 a d e+7 b c e) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}+\frac {2 x \left (c+d x^2\right )^{3/4} (b c (2 a f+7 b e)-a d (5 a f+4 b e))}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} (b e-a f)}{9 a b \left (a+b x^2\right )^{9/4}}\) |
Input:
Int[((c + d*x^2)^(3/4)*(e + f*x^2))/(a + b*x^2)^(13/4),x]
Output:
(2*(b*e - a*f)*x*(c + d*x^2)^(3/4))/(9*a*b*(a + b*x^2)^(9/4)) + ((2*(b*c*( 7*b*e + 2*a*f) - a*d*(4*b*e + 5*a*f))*x*(c + d*x^2)^(3/4))/(5*a*(b*c - a*d )*(a + b*x^2)^(5/4)) + (3*b*(7*b*c*e - 9*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*(b*c - a*d)*(a + b*x^2)^(5/ 4)))/(9*a*b)
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_)^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]
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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
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]
\[\int \frac {\left (x^{2} d +c \right )^{\frac {3}{4}} \left (f \,x^{2}+e \right )}{\left (b \,x^{2}+a \right )^{\frac {13}{4}}}d x\]
Input:
int((d*x^2+c)^(3/4)*(f*x^2+e)/(b*x^2+a)^(13/4),x)
Output:
int((d*x^2+c)^(3/4)*(f*x^2+e)/(b*x^2+a)^(13/4),x)
\[ \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {13}{4}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/4)*(f*x^2+e)/(b*x^2+a)^(13/4),x, algorithm="fricas" )
Output:
integral((b*x^2 + a)^(3/4)*(d*x^2 + c)^(3/4)*(f*x^2 + e)/(b^4*x^8 + 4*a*b^ 3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx=\text {Timed out} \] Input:
integrate((d*x**2+c)**(3/4)*(f*x**2+e)/(b*x**2+a)**(13/4),x)
Output:
Timed out
\[ \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {13}{4}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/4)*(f*x^2+e)/(b*x^2+a)^(13/4),x, algorithm="maxima" )
Output:
integrate((d*x^2 + c)^(3/4)*(f*x^2 + e)/(b*x^2 + a)^(13/4), x)
\[ \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {13}{4}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/4)*(f*x^2+e)/(b*x^2+a)^(13/4),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(3/4)*(f*x^2 + e)/(b*x^2 + a)^(13/4), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/4}\,\left (f\,x^2+e\right )}{{\left (b\,x^2+a\right )}^{13/4}} \,d x \] Input:
int(((c + d*x^2)^(3/4)*(e + f*x^2))/(a + b*x^2)^(13/4),x)
Output:
int(((c + d*x^2)^(3/4)*(e + f*x^2))/(a + b*x^2)^(13/4), x)
\[ \int \frac {\left (c+d x^2\right )^{3/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{13/4}} \, dx=\left (\int \frac {\left (d \,x^{2}+c \right )^{\frac {3}{4}}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{3} x^{6}}d x \right ) e +\left (\int \frac {\left (d \,x^{2}+c \right )^{\frac {3}{4}} x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{3} x^{6}}d x \right ) f \] Input:
int((d*x^2+c)^(3/4)*(f*x^2+e)/(b*x^2+a)^(13/4),x)
Output:
int((c + d*x**2)**(3/4)/((a + b*x**2)**(1/4)*a**3 + 3*(a + b*x**2)**(1/4)* a**2*b*x**2 + 3*(a + b*x**2)**(1/4)*a*b**2*x**4 + (a + b*x**2)**(1/4)*b**3 *x**6),x)*e + int(((c + d*x**2)**(3/4)*x**2)/((a + b*x**2)**(1/4)*a**3 + 3 *(a + b*x**2)**(1/4)*a**2*b*x**2 + 3*(a + b*x**2)**(1/4)*a*b**2*x**4 + (a + b*x**2)**(1/4)*b**3*x**6),x)*f