\(\int \frac {(c+d x^2)^{7/4} (e+f x^2)}{(a+b x^2)^{17/4}} \, dx\) [568]

Optimal result

Integrand size = 30, antiderivative size = 168 \[ \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx=\frac {2 (b e-a f) x \left (c+d x^2\right )^{11/4}}{13 a (b c-a d) \left (a+b x^2\right )^{13/4}}+\frac {(11 b c e-13 a d e+2 a c f) x \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{13/4} \left (c+d x^2\right )^{11/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {13}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (c+d x^2\right )}\right )}{13 a c (b c-a d) \left (a+b x^2\right )^{13/4}} \] Output:

2/13*(-a*f+b*e)*x*(d*x^2+c)^(11/4)/a/(-a*d+b*c)/(b*x^2+a)^(13/4)+1/13*(2*a 
*c*f-13*a*d*e+11*b*c*e)*x*(c*(b*x^2+a)/a/(d*x^2+c))^(13/4)*(d*x^2+c)^(11/4 
)*hypergeom([1/2, 13/4],[3/2],-(-a*d+b*c)*x^2/a/(d*x^2+c))/a/c/(-a*d+b*c)/ 
(b*x^2+a)^(13/4)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 12.19 (sec) , antiderivative size = 1290, normalized size of antiderivative = 7.68 \[ \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx =\text {Too large to display} \] Input:

Integrate[((c + d*x^2)^(7/4)*(e + f*x^2))/(a + b*x^2)^(17/4),x]
 

Output:

(-2*x*(693*a*b^4*c^5*e*(a + b*x^2)^3*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2 
)/a), -((d*x^2)/c)] - 126*a^2*b^3*c^4*d*e*(a + b*x^2)^3*AppellF1[1/2, 1/4, 
 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] - 819*a^3*b^2*c^3*d^2*e*(a + b*x^2) 
^3*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + 126*a^2*b^3* 
c^5*f*(a + b*x^2)^3*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x^2)/a), -((d*x^2)/c 
)] + 126*a^3*b^2*c^4*d*f*(a + b*x^2)^3*AppellF1[1/2, 1/4, 1/4, 3/2, -((b*x 
^2)/a), -((d*x^2)/c)] - (c + d*x^2)*(45*a^3*(-(b*c) + a*d)^3*(-(b*e) + a*f 
) + 5*a^2*(b*c - a*d)^2*(11*b^2*c*e - 18*a^2*d*f + a*b*(5*d*e + 2*c*f))*(a 
 + b*x^2) + a*(-(b*c) + a*d)*(-77*b^3*c^2*e + 45*a^3*d^2*f + 2*a*b^2*c*(18 
*d*e - 7*c*f) + 10*a^2*b*d*(2*d*e - c*f))*(a + b*x^2)^2 + 21*b^3*c^2*(11*b 
*c*e - 13*a*d*e + 2*a*c*f)*(a + b*x^2)^3)*(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)])) + 154*b^4*c^3*d*e*x^2*(a + b*x^2)^3*(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*Appe 
llF1[3/2, 5/4, 1/4, 5/2, -((b*x^2)/a), -((d*x^2)/c)])) + 28*a*b^3*c^3*d*f* 
x^2*(a + b*x^2)^3*(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, ...
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {401, 27, 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.

\(\displaystyle \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {2 x \left (c+d x^2\right )^{7/4} (b e-a f)}{13 a b \left (a+b x^2\right )^{13/4}}-\frac {2 \int -\frac {\left (d x^2+c\right )^{3/4} \left (d (4 b e+9 a f) x^2+c (11 b e+2 a f)\right )}{2 \left (b x^2+a\right )^{13/4}}dx}{13 a b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (d x^2+c\right )^{3/4} \left (d (4 b e+9 a f) x^2+c (11 b e+2 a f)\right )}{\left (b x^2+a\right )^{13/4}}dx}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b e-a f)}{13 a b \left (a+b x^2\right )^{13/4}}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {2 x \left (c+d x^2\right )^{3/4} (b c (2 a f+11 b e)-a d (9 a f+4 b e))}{9 a b \left (a+b x^2\right )^{9/4}}-\frac {2 \int -\frac {d \left (45 d f a^2+20 b d e a+8 b c f a+44 b^2 c e\right ) x^2+c (7 b c (11 b e+2 a f)+2 a d (4 b e+9 a f))}{2 \left (b x^2+a\right )^{9/4} \sqrt [4]{d x^2+c}}dx}{9 a b}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b e-a f)}{13 a b \left (a+b x^2\right )^{13/4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {d \left (45 d f a^2+20 b d e a+8 b c f a+44 b^2 c e\right ) x^2+c (7 b c (11 b e+2 a f)+2 a d (4 b e+9 a f))}{\left (b x^2+a\right )^{9/4} \sqrt [4]{d x^2+c}}dx}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} (b c (2 a f+11 b e)-a d (9 a f+4 b e))}{9 a b \left (a+b x^2\right )^{9/4}}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b e-a f)}{13 a b \left (a+b x^2\right )^{13/4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {\frac {2 x \left (c+d x^2\right )^{3/4} \left (-45 a^3 d^2 f-10 a^2 b d (2 d e-c f)-2 a b^2 c (18 d e-7 c f)+77 b^3 c^2 e\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}-\frac {2 \int -\frac {21 b^2 c^2 (11 b c e-13 a d e+2 a c f)}{2 \left (b x^2+a\right )^{5/4} \sqrt [4]{d x^2+c}}dx}{5 a (b c-a d)}}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} (b c (2 a f+11 b e)-a d (9 a f+4 b e))}{9 a b \left (a+b x^2\right )^{9/4}}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b e-a f)}{13 a b \left (a+b x^2\right )^{13/4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {21 b^2 c^2 (2 a c f-13 a d e+11 b c e) \int \frac {1}{\left (b x^2+a\right )^{5/4} \sqrt [4]{d x^2+c}}dx}{5 a (b c-a d)}+\frac {2 x \left (c+d x^2\right )^{3/4} \left (-45 a^3 d^2 f-10 a^2 b d (2 d e-c f)-2 a b^2 c (18 d e-7 c f)+77 b^3 c^2 e\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} (b c (2 a f+11 b e)-a d (9 a f+4 b e))}{9 a b \left (a+b x^2\right )^{9/4}}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b e-a f)}{13 a b \left (a+b x^2\right )^{13/4}}\)

\(\Big \downarrow \) 294

\(\displaystyle \frac {\frac {\frac {2 x \left (c+d x^2\right )^{3/4} \left (-45 a^3 d^2 f-10 a^2 b d (2 d e-c f)-2 a b^2 c (18 d e-7 c f)+77 b^3 c^2 e\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}+\frac {21 b^2 c x \left (c+d x^2\right )^{3/4} \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{5/4} (2 a c f-13 a d e+11 b c e) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{5 a \left (a+b x^2\right )^{5/4} (b c-a d)}}{9 a b}+\frac {2 x \left (c+d x^2\right )^{3/4} (b c (2 a f+11 b e)-a d (9 a f+4 b e))}{9 a b \left (a+b x^2\right )^{9/4}}}{13 a b}+\frac {2 x \left (c+d x^2\right )^{7/4} (b e-a f)}{13 a b \left (a+b x^2\right )^{13/4}}\)

Input:

Int[((c + d*x^2)^(7/4)*(e + f*x^2))/(a + b*x^2)^(17/4),x]
 

Output:

(2*(b*e - a*f)*x*(c + d*x^2)^(7/4))/(13*a*b*(a + b*x^2)^(13/4)) + ((2*(b*c 
*(11*b*e + 2*a*f) - a*d*(4*b*e + 9*a*f))*x*(c + d*x^2)^(3/4))/(9*a*b*(a + 
b*x^2)^(9/4)) + ((2*(77*b^3*c^2*e - 45*a^3*d^2*f - 2*a*b^2*c*(18*d*e - 7*c 
*f) - 10*a^2*b*d*(2*d*e - c*f))*x*(c + d*x^2)^(3/4))/(5*a*(b*c - a*d)*(a + 
 b*x^2)^(5/4)) + (21*b^2*c*(11*b*c*e - 13*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))/(13*a*b)
 

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 401

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]
 

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]

\[\int \frac {\left (x^{2} d +c \right )^{\frac {7}{4}} \left (f \,x^{2}+e \right )}{\left (b \,x^{2}+a \right )^{\frac {17}{4}}}d x\]

Input:

int((d*x^2+c)^(7/4)*(f*x^2+e)/(b*x^2+a)^(17/4),x)
 

Output:

int((d*x^2+c)^(7/4)*(f*x^2+e)/(b*x^2+a)^(17/4),x)
 

Fricas [F]

\[ \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {7}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {17}{4}}} \,d x } \] Input:

integrate((d*x^2+c)^(7/4)*(f*x^2+e)/(b*x^2+a)^(17/4),x, algorithm="fricas" 
)
 

Output:

integral((d*f*x^4 + (d*e + c*f)*x^2 + c*e)*(b*x^2 + a)^(3/4)*(d*x^2 + c)^( 
3/4)/(b^5*x^10 + 5*a*b^4*x^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x 
^2 + a^5), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx=\text {Timed out} \] Input:

integrate((d*x**2+c)**(7/4)*(f*x**2+e)/(b*x**2+a)**(17/4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {7}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {17}{4}}} \,d x } \] Input:

integrate((d*x^2+c)^(7/4)*(f*x^2+e)/(b*x^2+a)^(17/4),x, algorithm="maxima" 
)
 

Output:

integrate((d*x^2 + c)^(7/4)*(f*x^2 + e)/(b*x^2 + a)^(17/4), x)
 

Giac [F]

\[ \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {7}{4}} {\left (f x^{2} + e\right )}}{{\left (b x^{2} + a\right )}^{\frac {17}{4}}} \,d x } \] Input:

integrate((d*x^2+c)^(7/4)*(f*x^2+e)/(b*x^2+a)^(17/4),x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)^(7/4)*(f*x^2 + e)/(b*x^2 + a)^(17/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{7/4}\,\left (f\,x^2+e\right )}{{\left (b\,x^2+a\right )}^{17/4}} \,d x \] Input:

int(((c + d*x^2)^(7/4)*(e + f*x^2))/(a + b*x^2)^(17/4),x)
 

Output:

int(((c + d*x^2)^(7/4)*(e + f*x^2))/(a + b*x^2)^(17/4), x)
 

Reduce [F]

\[ \int \frac {\left (c+d x^2\right )^{7/4} \left (e+f x^2\right )}{\left (a+b x^2\right )^{17/4}} \, dx=\left (\int \frac {\left (d \,x^{2}+c \right )^{\frac {3}{4}}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{4}+4 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} b \,x^{2}+6 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{4}+4 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{3} x^{6}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{4} x^{8}}d x \right ) c e +\left (\int \frac {\left (d \,x^{2}+c \right )^{\frac {3}{4}} x^{4}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{4}+4 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} b \,x^{2}+6 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{4}+4 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{3} x^{6}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{4} x^{8}}d x \right ) d f +\left (\int \frac {\left (d \,x^{2}+c \right )^{\frac {3}{4}} x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{4}+4 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} b \,x^{2}+6 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{4}+4 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{3} x^{6}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{4} x^{8}}d x \right ) c f +\left (\int \frac {\left (d \,x^{2}+c \right )^{\frac {3}{4}} x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{4}+4 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{3} b \,x^{2}+6 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a^{2} b^{2} x^{4}+4 \left (b \,x^{2}+a \right )^{\frac {1}{4}} a \,b^{3} x^{6}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b^{4} x^{8}}d x \right ) d e \] Input:

int((d*x^2+c)^(7/4)*(f*x^2+e)/(b*x^2+a)^(17/4),x)
 

Output:

int((c + d*x**2)**(3/4)/((a + b*x**2)**(1/4)*a**4 + 4*(a + b*x**2)**(1/4)* 
a**3*b*x**2 + 6*(a + b*x**2)**(1/4)*a**2*b**2*x**4 + 4*(a + b*x**2)**(1/4) 
*a*b**3*x**6 + (a + b*x**2)**(1/4)*b**4*x**8),x)*c*e + int(((c + d*x**2)** 
(3/4)*x**4)/((a + b*x**2)**(1/4)*a**4 + 4*(a + b*x**2)**(1/4)*a**3*b*x**2 
+ 6*(a + b*x**2)**(1/4)*a**2*b**2*x**4 + 4*(a + b*x**2)**(1/4)*a*b**3*x**6 
 + (a + b*x**2)**(1/4)*b**4*x**8),x)*d*f + int(((c + d*x**2)**(3/4)*x**2)/ 
((a + b*x**2)**(1/4)*a**4 + 4*(a + b*x**2)**(1/4)*a**3*b*x**2 + 6*(a + b*x 
**2)**(1/4)*a**2*b**2*x**4 + 4*(a + b*x**2)**(1/4)*a*b**3*x**6 + (a + b*x* 
*2)**(1/4)*b**4*x**8),x)*c*f + int(((c + d*x**2)**(3/4)*x**2)/((a + b*x**2 
)**(1/4)*a**4 + 4*(a + b*x**2)**(1/4)*a**3*b*x**2 + 6*(a + b*x**2)**(1/4)* 
a**2*b**2*x**4 + 4*(a + b*x**2)**(1/4)*a*b**3*x**6 + (a + b*x**2)**(1/4)*b 
**4*x**8),x)*d*e