Integrand size = 26, antiderivative size = 145 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx=-\frac {2 c \left (a+b x^2\right )^{7/4}}{19 a e (e x)^{19/2}}+\frac {2 (12 b c-19 a d) \left (a+b x^2\right )^{7/4}}{285 a^2 e^3 (e x)^{15/2}}-\frac {16 b (12 b c-19 a d) \left (a+b x^2\right )^{7/4}}{3135 a^3 e^5 (e x)^{11/2}}+\frac {64 b^2 (12 b c-19 a d) \left (a+b x^2\right )^{7/4}}{21945 a^4 e^7 (e x)^{7/2}} \] Output:
-2/19*c*(b*x^2+a)^(7/4)/a/e/(e*x)^(19/2)+2/285*(-19*a*d+12*b*c)*(b*x^2+a)^ (7/4)/a^2/e^3/(e*x)^(15/2)-16/3135*b*(-19*a*d+12*b*c)*(b*x^2+a)^(7/4)/a^3/ e^5/(e*x)^(11/2)+64/21945*b^2*(-19*a*d+12*b*c)*(b*x^2+a)^(7/4)/a^4/e^7/(e* x)^(7/2)
Time = 2.45 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.63 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx=-\frac {2 x \left (a+b x^2\right )^{7/4} \left (1155 a^3 c-924 a^2 b c x^2+1463 a^3 d x^2+672 a b^2 c x^4-1064 a^2 b d x^4-384 b^3 c x^6+608 a b^2 d x^6\right )}{21945 a^4 (e x)^{21/2}} \] Input:
Integrate[((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(21/2),x]
Output:
(-2*x*(a + b*x^2)^(7/4)*(1155*a^3*c - 924*a^2*b*c*x^2 + 1463*a^3*d*x^2 + 6 72*a*b^2*c*x^4 - 1064*a^2*b*d*x^4 - 384*b^3*c*x^6 + 608*a*b^2*d*x^6))/(219 45*a^4*(e*x)^(21/2))
Time = 0.24 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {359, 246, 246, 242}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx\) |
\(\Big \downarrow \) 359 |
\(\displaystyle -\frac {(12 b c-19 a d) \int \frac {\left (b x^2+a\right )^{3/4}}{(e x)^{17/2}}dx}{19 a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{19 a e (e x)^{19/2}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {(12 b c-19 a d) \left (-\frac {8 \int \frac {\left (b x^2+a\right )^{7/4}}{(e x)^{17/2}}dx}{7 a}-\frac {2 \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{15/2}}\right )}{19 a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{19 a e (e x)^{19/2}}\) |
\(\Big \downarrow \) 246 |
\(\displaystyle -\frac {(12 b c-19 a d) \left (-\frac {8 \left (-\frac {4 \int \frac {\left (b x^2+a\right )^{11/4}}{(e x)^{17/2}}dx}{11 a}-\frac {2 \left (a+b x^2\right )^{11/4}}{11 a e (e x)^{15/2}}\right )}{7 a}-\frac {2 \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{15/2}}\right )}{19 a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{19 a e (e x)^{19/2}}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {(12 b c-19 a d) \left (-\frac {8 \left (\frac {8 \left (a+b x^2\right )^{15/4}}{165 a^2 e (e x)^{15/2}}-\frac {2 \left (a+b x^2\right )^{11/4}}{11 a e (e x)^{15/2}}\right )}{7 a}-\frac {2 \left (a+b x^2\right )^{7/4}}{7 a e (e x)^{15/2}}\right )}{19 a e^2}-\frac {2 c \left (a+b x^2\right )^{7/4}}{19 a e (e x)^{19/2}}\) |
Input:
Int[((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(21/2),x]
Output:
(-2*c*(a + b*x^2)^(7/4))/(19*a*e*(e*x)^(19/2)) - ((12*b*c - 19*a*d)*((-2*( a + b*x^2)^(7/4))/(7*a*e*(e*x)^(15/2)) - (8*((-2*(a + b*x^2)^(11/4))/(11*a *e*(e*x)^(15/2)) + (8*(a + b*x^2)^(15/4))/(165*a^2*e*(e*x)^(15/2))))/(7*a) ))/(19*a*e^2)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(a*c*2*(p + 1))), x] + Simp[(m + 2*p + 3)/( a*2*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m , p}, x] && ILtQ[Simplify[(m + 1)/2 + p + 1], 0] && NeQ[p, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Time = 0.42 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {7}{4}} \left (608 a \,b^{2} d \,x^{6}-384 b^{3} c \,x^{6}-1064 a^{2} b d \,x^{4}+672 a \,b^{2} c \,x^{4}+1463 a^{3} d \,x^{2}-924 a^{2} b c \,x^{2}+1155 a^{3} c \right )}{21945 a^{4} \left (e x \right )^{\frac {21}{2}}}\) | \(86\) |
orering | \(-\frac {2 x \left (b \,x^{2}+a \right )^{\frac {7}{4}} \left (608 a \,b^{2} d \,x^{6}-384 b^{3} c \,x^{6}-1064 a^{2} b d \,x^{4}+672 a \,b^{2} c \,x^{4}+1463 a^{3} d \,x^{2}-924 a^{2} b c \,x^{2}+1155 a^{3} c \right )}{21945 a^{4} \left (e x \right )^{\frac {21}{2}}}\) | \(86\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (608 a \,b^{3} d \,x^{8}-384 b^{4} c \,x^{8}-456 a^{2} b^{2} d \,x^{6}+288 a \,b^{3} c \,x^{6}+399 a^{3} b d \,x^{4}-252 a^{2} b^{2} c \,x^{4}+1463 a^{4} d \,x^{2}+231 a^{3} b c \,x^{2}+1155 a^{4} c \right )}{21945 e^{10} \sqrt {e x}\, x^{9} a^{4}}\) | \(115\) |
Input:
int((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(21/2),x,method=_RETURNVERBOSE)
Output:
-2/21945*x*(b*x^2+a)^(7/4)*(608*a*b^2*d*x^6-384*b^3*c*x^6-1064*a^2*b*d*x^4 +672*a*b^2*c*x^4+1463*a^3*d*x^2-924*a^2*b*c*x^2+1155*a^3*c)/a^4/(e*x)^(21/ 2)
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx=\frac {2 \, {\left (32 \, {\left (12 \, b^{4} c - 19 \, a b^{3} d\right )} x^{8} - 24 \, {\left (12 \, a b^{3} c - 19 \, a^{2} b^{2} d\right )} x^{6} - 1155 \, a^{4} c + 21 \, {\left (12 \, a^{2} b^{2} c - 19 \, a^{3} b d\right )} x^{4} - 77 \, {\left (3 \, a^{3} b c + 19 \, a^{4} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{21945 \, a^{4} e^{11} x^{10}} \] Input:
integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(21/2),x, algorithm="fricas")
Output:
2/21945*(32*(12*b^4*c - 19*a*b^3*d)*x^8 - 24*(12*a*b^3*c - 19*a^2*b^2*d)*x ^6 - 1155*a^4*c + 21*(12*a^2*b^2*c - 19*a^3*b*d)*x^4 - 77*(3*a^3*b*c + 19* a^4*d)*x^2)*(b*x^2 + a)^(3/4)*sqrt(e*x)/(a^4*e^11*x^10)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(3/4)*(d*x**2+c)/(e*x)**(21/2),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}}{\left (e x\right )^{\frac {21}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(21/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(3/4)*(d*x^2 + c)/(e*x)^(21/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(21/2),x, algorithm="giac")
Output:
Timed out
Time = 0.78 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx=-\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2\,c}{19\,e^{10}}+\frac {x^2\,\left (2926\,d\,a^4+462\,b\,c\,a^3\right )}{21945\,a^4\,e^{10}}-\frac {x^8\,\left (768\,b^4\,c-1216\,a\,b^3\,d\right )}{21945\,a^4\,e^{10}}-\frac {16\,b^2\,x^6\,\left (19\,a\,d-12\,b\,c\right )}{7315\,a^3\,e^{10}}+\frac {2\,b\,x^4\,\left (19\,a\,d-12\,b\,c\right )}{1045\,a^2\,e^{10}}\right )}{x^9\,\sqrt {e\,x}} \] Input:
int(((a + b*x^2)^(3/4)*(c + d*x^2))/(e*x)^(21/2),x)
Output:
-((a + b*x^2)^(3/4)*((2*c)/(19*e^10) + (x^2*(2926*a^4*d + 462*a^3*b*c))/(2 1945*a^4*e^10) - (x^8*(768*b^4*c - 1216*a*b^3*d))/(21945*a^4*e^10) - (16*b ^2*x^6*(19*a*d - 12*b*c))/(7315*a^3*e^10) + (2*b*x^4*(19*a*d - 12*b*c))/(1 045*a^2*e^10)))/(x^9*(e*x)^(1/2))
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )}{(e x)^{21/2}} \, dx=\frac {2 \sqrt {e}\, \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (-608 a \,b^{3} d \,x^{8}+384 b^{4} c \,x^{8}+456 a^{2} b^{2} d \,x^{6}-288 a \,b^{3} c \,x^{6}-399 a^{3} b d \,x^{4}+252 a^{2} b^{2} c \,x^{4}-1463 a^{4} d \,x^{2}-231 a^{3} b c \,x^{2}-1155 a^{4} c \right )}{21945 \sqrt {x}\, a^{4} e^{11} x^{9}} \] Input:
int((b*x^2+a)^(3/4)*(d*x^2+c)/(e*x)^(21/2),x)
Output:
(2*sqrt(e)*(a + b*x**2)**(3/4)*( - 1155*a**4*c - 1463*a**4*d*x**2 - 231*a* *3*b*c*x**2 - 399*a**3*b*d*x**4 + 252*a**2*b**2*c*x**4 + 456*a**2*b**2*d*x **6 - 288*a*b**3*c*x**6 - 608*a*b**3*d*x**8 + 384*b**4*c*x**8))/(21945*sqr t(x)*a**4*e**11*x**9)