\(\int \frac {1}{(a+b x^2)^{3/4} (c+d x^2)^{19/4}} \, dx\) [525]

Optimal result

Integrand size = 23, antiderivative size = 1 \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx=0 \] Output:

0
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 1 in optimal.

Time = 11.01 (sec) , antiderivative size = 309, normalized size of antiderivative = 309.00 \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx=-\frac {x \left (2 c d \left (a+b x^2\right ) \left (77 c^3 (b c-a d)^3+7 c^2 (27 b c-13 a d) (b c-a d)^2 \left (c+d x^2\right )+c (b c-a d) \left (381 b^2 c^2-358 a b c d+117 a^2 d^2\right ) \left (c+d x^2\right )^2+\left (893 b^3 c^3-1201 a b^2 c^2 d+783 a^2 b c d^2-195 a^3 d^3\right ) \left (c+d x^2\right )^3\right )-5 \left (231 b^4 c^4-308 a b^3 c^3 d+330 a^2 b^2 c^2 d^2-180 a^3 b c d^3+39 a^4 d^4\right ) \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{3/4} \left (c+d x^2\right )^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {(-b c+a d) x^2}{a \left (c+d x^2\right )}\right )\right )}{1155 c^5 (b c-a d)^4 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{15/4}} \] Input:

Integrate[1/((a + b*x^2)^(3/4)*(c + d*x^2)^(19/4)),x]
 

Output:

-1/1155*(x*(2*c*d*(a + b*x^2)*(77*c^3*(b*c - a*d)^3 + 7*c^2*(27*b*c - 13*a 
*d)*(b*c - a*d)^2*(c + d*x^2) + c*(b*c - a*d)*(381*b^2*c^2 - 358*a*b*c*d + 
 117*a^2*d^2)*(c + d*x^2)^2 + (893*b^3*c^3 - 1201*a*b^2*c^2*d + 783*a^2*b* 
c*d^2 - 195*a^3*d^3)*(c + d*x^2)^3) - 5*(231*b^4*c^4 - 308*a*b^3*c^3*d + 3 
30*a^2*b^2*c^2*d^2 - 180*a^3*b*c*d^3 + 39*a^4*d^4)*((c*(a + b*x^2))/(a*(c 
+ d*x^2)))^(3/4)*(c + d*x^2)^4*Hypergeometric2F1[1/2, 3/4, 3/2, ((-(b*c) + 
 a*d)*x^2)/(a*(c + d*x^2))]))/(c^5*(b*c - a*d)^4*(a + b*x^2)^(3/4)*(c + d* 
x^2)^(15/4))
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 1 in optimal.

Time = 7.23 (sec) , antiderivative size = 1547, normalized size of antiderivative = 1547.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {334, 334, 333}

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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {\left (\frac {b x^2}{a}+1\right )^{3/4} \int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{3/4} \left (d x^2+c\right )^{19/4}}dx}{\left (a+b x^2\right )^{3/4}}\)

\(\Big \downarrow \) 334

\(\displaystyle \frac {\left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {d x^2}{c}+1\right )^{3/4} \int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{3/4} \left (\frac {d x^2}{c}+1\right )^{19/4}}dx}{c^4 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}}\)

\(\Big \downarrow \) 333

\(\displaystyle \frac {x \left (1408 b c d^4 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}-400 a d^5 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}+400 b c d^4 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}-8 a d^5 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}+8 b c d^4 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^{10}+1408 a c d^4 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8+6336 b c^2 d^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8-1776 a c d^4 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8+1776 b c^2 d^3 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8-32 a c d^4 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8+32 b c^2 d^3 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^8+6336 a c^2 d^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6+11088 b c^3 d^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6-3036 a c^2 d^3 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6+3036 b c^3 d^2 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6-48 a c^2 d^3 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6+48 b c^3 d^2 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^6+11088 a c^3 d^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4+9240 b c^4 d \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4-2404 a c^3 d^2 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4+2404 b c^4 d \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4-32 a c^3 d^2 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4+32 b c^4 d \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^4+3465 b c^5 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2+9240 a c^4 d \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2+744 b c^5 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2-744 a c^4 d \operatorname {Hypergeometric2F1}\left (\frac {7}{4},2,\frac {13}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2+4 (b c-a d) \left (d x^2+c\right )^2 \left (70 d^2 x^4+164 c d x^2+103 c^2\right ) \, _3F_2\left (\frac {7}{4},2,2;1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2+16 (b c-a d) \left (d x^2+c\right )^3 \left (5 d x^2+6 c\right ) \, _4F_3\left (\frac {7}{4},2,2,2;1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2+8 b c^5 \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2-8 a c^4 d \, _5F_4\left (\frac {7}{4},2,2,2,2;1,1,1,\frac {13}{2};\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right ) x^2+3465 a c^5 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (b x^2+a\right )}\right )\right )}{3465 c^9 \left (b x^2+a\right )^{7/4} \left (d x^2+c\right )^{3/4} \left (\frac {d x^2}{c}+1\right )^3}\)

Input:

Int[1/((a + b*x^2)^(3/4)*(c + d*x^2)^(19/4)),x]
 

Output:

(x*(3465*a*c^5*Hypergeometric2F1[3/4, 1, 11/2, ((b*c - a*d)*x^2)/(c*(a + b 
*x^2))] + 3465*b*c^5*x^2*Hypergeometric2F1[3/4, 1, 11/2, ((b*c - a*d)*x^2) 
/(c*(a + b*x^2))] + 9240*a*c^4*d*x^2*Hypergeometric2F1[3/4, 1, 11/2, ((b*c 
 - a*d)*x^2)/(c*(a + b*x^2))] + 9240*b*c^4*d*x^4*Hypergeometric2F1[3/4, 1, 
 11/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 11088*a*c^3*d^2*x^4*Hypergeome 
tric2F1[3/4, 1, 11/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 11088*b*c^3*d^2 
*x^6*Hypergeometric2F1[3/4, 1, 11/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 
6336*a*c^2*d^3*x^6*Hypergeometric2F1[3/4, 1, 11/2, ((b*c - a*d)*x^2)/(c*(a 
 + b*x^2))] + 6336*b*c^2*d^3*x^8*Hypergeometric2F1[3/4, 1, 11/2, ((b*c - a 
*d)*x^2)/(c*(a + b*x^2))] + 1408*a*c*d^4*x^8*Hypergeometric2F1[3/4, 1, 11/ 
2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 1408*b*c*d^4*x^10*Hypergeometric2F 
1[3/4, 1, 11/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 744*b*c^5*x^2*Hyperge 
ometric2F1[7/4, 2, 13/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 744*a*c^4*d* 
x^2*Hypergeometric2F1[7/4, 2, 13/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 2 
404*b*c^4*d*x^4*Hypergeometric2F1[7/4, 2, 13/2, ((b*c - a*d)*x^2)/(c*(a + 
b*x^2))] - 2404*a*c^3*d^2*x^4*Hypergeometric2F1[7/4, 2, 13/2, ((b*c - a*d) 
*x^2)/(c*(a + b*x^2))] + 3036*b*c^3*d^2*x^6*Hypergeometric2F1[7/4, 2, 13/2 
, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 3036*a*c^2*d^3*x^6*Hypergeometric2F 
1[7/4, 2, 13/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] + 1776*b*c^2*d^3*x^8*Hy 
pergeometric2F1[7/4, 2, 13/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] - 1776...
 

Defintions of rubi rules used

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

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

Maple [F]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )^{\frac {19}{4}}}d x\]

Input:

int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(19/4),x)
 

Output:

int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(19/4),x)
 

Fricas [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{\frac {19}{4}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(19/4),x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^(1/4)*(d*x^2 + c)^(1/4)/(b*d^5*x^12 + (5*b*c*d^4 + a* 
d^5)*x^10 + 5*(2*b*c^2*d^3 + a*c*d^4)*x^8 + 10*(b*c^3*d^2 + a*c^2*d^3)*x^6 
 + a*c^5 + 5*(b*c^4*d + 2*a*c^3*d^2)*x^4 + (b*c^5 + 5*a*c^4*d)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x**2+a)**(3/4)/(d*x**2+c)**(19/4),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{\frac {19}{4}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(19/4),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^(19/4)), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{\frac {19}{4}}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(19/4),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^(19/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/4}\,{\left (d\,x^2+c\right )}^{19/4}} \,d x \] Input:

int(1/((a + b*x^2)^(3/4)*(c + d*x^2)^(19/4)),x)
 

Output:

int(1/((a + b*x^2)^(3/4)*(c + d*x^2)^(19/4)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{19/4}} \, dx=\int \frac {1}{\left (d \,x^{2}+c \right )^{\frac {3}{4}} \left (b \,x^{2}+a \right )^{\frac {3}{4}} c^{4}+4 \left (d \,x^{2}+c \right )^{\frac {3}{4}} \left (b \,x^{2}+a \right )^{\frac {3}{4}} c^{3} d \,x^{2}+6 \left (d \,x^{2}+c \right )^{\frac {3}{4}} \left (b \,x^{2}+a \right )^{\frac {3}{4}} c^{2} d^{2} x^{4}+4 \left (d \,x^{2}+c \right )^{\frac {3}{4}} \left (b \,x^{2}+a \right )^{\frac {3}{4}} c \,d^{3} x^{6}+\left (d \,x^{2}+c \right )^{\frac {3}{4}} \left (b \,x^{2}+a \right )^{\frac {3}{4}} d^{4} x^{8}}d x \] Input:

int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(19/4),x)
 

Output:

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