Integrand size = 22, antiderivative size = 103 \[ \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx=\frac {2 a^2 (b c-a d) \left (a+b x^2\right )^{7/4}}{7 b^4}-\frac {2 a (2 b c-3 a d) \left (a+b x^2\right )^{11/4}}{11 b^4}+\frac {2 (b c-3 a d) \left (a+b x^2\right )^{15/4}}{15 b^4}+\frac {2 d \left (a+b x^2\right )^{19/4}}{19 b^4} \] Output:
2/7*a^2*(-a*d+b*c)*(b*x^2+a)^(7/4)/b^4-2/11*a*(-3*a*d+2*b*c)*(b*x^2+a)^(11 /4)/b^4+2/15*(-3*a*d+b*c)*(b*x^2+a)^(15/4)/b^4+2/19*d*(b*x^2+a)^(19/4)/b^4
Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.78 \[ \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx=\frac {2 \left (a+b x^2\right )^{7/4} \left (608 a^2 b c-384 a^3 d-1064 a b^2 c x^2+672 a^2 b d x^2+1463 b^3 c x^4-924 a b^2 d x^4+1155 b^3 d x^6\right )}{21945 b^4} \] Input:
Integrate[x^5*(a + b*x^2)^(3/4)*(c + d*x^2),x]
Output:
(2*(a + b*x^2)^(7/4)*(608*a^2*b*c - 384*a^3*d - 1064*a*b^2*c*x^2 + 672*a^2 *b*d*x^2 + 1463*b^3*c*x^4 - 924*a*b^2*d*x^4 + 1155*b^3*d*x^6))/(21945*b^4)
Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {354, 86, 2009}
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.
\(\displaystyle \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int x^4 \left (b x^2+a\right )^{3/4} \left (d x^2+c\right )dx^2\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {1}{2} \int \left (\frac {d \left (b x^2+a\right )^{15/4}}{b^3}+\frac {(b c-3 a d) \left (b x^2+a\right )^{11/4}}{b^3}+\frac {a (3 a d-2 b c) \left (b x^2+a\right )^{7/4}}{b^3}-\frac {a^2 (a d-b c) \left (b x^2+a\right )^{3/4}}{b^3}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {4 a^2 \left (a+b x^2\right )^{7/4} (b c-a d)}{7 b^4}+\frac {4 \left (a+b x^2\right )^{15/4} (b c-3 a d)}{15 b^4}-\frac {4 a \left (a+b x^2\right )^{11/4} (2 b c-3 a d)}{11 b^4}+\frac {4 d \left (a+b x^2\right )^{19/4}}{19 b^4}\right )\) |
Input:
Int[x^5*(a + b*x^2)^(3/4)*(c + d*x^2),x]
Output:
((4*a^2*(b*c - a*d)*(a + b*x^2)^(7/4))/(7*b^4) - (4*a*(2*b*c - 3*a*d)*(a + b*x^2)^(11/4))/(11*b^4) + (4*(b*c - 3*a*d)*(a + b*x^2)^(15/4))/(15*b^4) + (4*d*(a + b*x^2)^(19/4))/(19*b^4))/2
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.41 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(-\frac {256 \left (-\frac {1463 \left (\frac {15 x^{2} d}{19}+c \right ) x^{4} b^{3}}{384}+\frac {133 a \left (\frac {33 x^{2} d}{38}+c \right ) x^{2} b^{2}}{48}-\frac {19 a^{2} \left (\frac {21 x^{2} d}{19}+c \right ) b}{12}+a^{3} d \right ) \left (b \,x^{2}+a \right )^{\frac {7}{4}}}{7315 b^{4}}\) | \(68\) |
gosper | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {7}{4}} \left (-1155 d \,x^{6} b^{3}+924 a \,b^{2} d \,x^{4}-1463 b^{3} c \,x^{4}-672 a^{2} b d \,x^{2}+1064 a \,b^{2} c \,x^{2}+384 a^{3} d -608 a^{2} b c \right )}{21945 b^{4}}\) | \(77\) |
orering | \(-\frac {2 \left (b \,x^{2}+a \right )^{\frac {7}{4}} \left (-1155 d \,x^{6} b^{3}+924 a \,b^{2} d \,x^{4}-1463 b^{3} c \,x^{4}-672 a^{2} b d \,x^{2}+1064 a \,b^{2} c \,x^{2}+384 a^{3} d -608 a^{2} b c \right )}{21945 b^{4}}\) | \(77\) |
trager | \(-\frac {2 \left (-1155 d \,b^{4} x^{8}-231 a \,b^{3} d \,x^{6}-1463 b^{4} c \,x^{6}+252 a^{2} b^{2} d \,x^{4}-399 a \,b^{3} c \,x^{4}-288 a^{3} b d \,x^{2}+456 a^{2} b^{2} c \,x^{2}+384 a^{4} d -608 a^{3} b c \right ) \left (b \,x^{2}+a \right )^{\frac {3}{4}}}{21945 b^{4}}\) | \(101\) |
risch | \(-\frac {2 \left (-1155 d \,b^{4} x^{8}-231 a \,b^{3} d \,x^{6}-1463 b^{4} c \,x^{6}+252 a^{2} b^{2} d \,x^{4}-399 a \,b^{3} c \,x^{4}-288 a^{3} b d \,x^{2}+456 a^{2} b^{2} c \,x^{2}+384 a^{4} d -608 a^{3} b c \right ) \left (b \,x^{2}+a \right )^{\frac {3}{4}}}{21945 b^{4}}\) | \(101\) |
Input:
int(x^5*(b*x^2+a)^(3/4)*(d*x^2+c),x,method=_RETURNVERBOSE)
Output:
-256/7315*(-1463/384*(15/19*x^2*d+c)*x^4*b^3+133/48*a*(33/38*x^2*d+c)*x^2* b^2-19/12*a^2*(21/19*x^2*d+c)*b+a^3*d)*(b*x^2+a)^(7/4)/b^4
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.97 \[ \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx=\frac {2 \, {\left (1155 \, b^{4} d x^{8} + 77 \, {\left (19 \, b^{4} c + 3 \, a b^{3} d\right )} x^{6} + 608 \, a^{3} b c - 384 \, a^{4} d + 21 \, {\left (19 \, a b^{3} c - 12 \, a^{2} b^{2} d\right )} x^{4} - 24 \, {\left (19 \, a^{2} b^{2} c - 12 \, a^{3} b d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}}}{21945 \, b^{4}} \] Input:
integrate(x^5*(b*x^2+a)^(3/4)*(d*x^2+c),x, algorithm="fricas")
Output:
2/21945*(1155*b^4*d*x^8 + 77*(19*b^4*c + 3*a*b^3*d)*x^6 + 608*a^3*b*c - 38 4*a^4*d + 21*(19*a*b^3*c - 12*a^2*b^2*d)*x^4 - 24*(19*a^2*b^2*c - 12*a^3*b *d)*x^2)*(b*x^2 + a)^(3/4)/b^4
Time = 0.88 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.37 \[ \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx=\frac {c \left (\begin {cases} \frac {4 \left (\frac {a^{2} \left (a + b x^{2}\right )^{\frac {7}{4}}}{7} - \frac {2 a \left (a + b x^{2}\right )^{\frac {11}{4}}}{11} + \frac {\left (a + b x^{2}\right )^{\frac {15}{4}}}{15}\right )}{b^{3}} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{4}} x^{6}}{3} & \text {otherwise} \end {cases}\right )}{2} + \frac {d \left (\begin {cases} \frac {4 \left (- \frac {a^{3} \left (a + b x^{2}\right )^{\frac {7}{4}}}{7} + \frac {3 a^{2} \left (a + b x^{2}\right )^{\frac {11}{4}}}{11} - \frac {a \left (a + b x^{2}\right )^{\frac {15}{4}}}{5} + \frac {\left (a + b x^{2}\right )^{\frac {19}{4}}}{19}\right )}{b^{4}} & \text {for}\: b \neq 0 \\\frac {a^{\frac {3}{4}} x^{8}}{4} & \text {otherwise} \end {cases}\right )}{2} \] Input:
integrate(x**5*(b*x**2+a)**(3/4)*(d*x**2+c),x)
Output:
c*Piecewise((4*(a**2*(a + b*x**2)**(7/4)/7 - 2*a*(a + b*x**2)**(11/4)/11 + (a + b*x**2)**(15/4)/15)/b**3, Ne(b, 0)), (a**(3/4)*x**6/3, True))/2 + d* Piecewise((4*(-a**3*(a + b*x**2)**(7/4)/7 + 3*a**2*(a + b*x**2)**(11/4)/11 - a*(a + b*x**2)**(15/4)/5 + (a + b*x**2)**(19/4)/19)/b**4, Ne(b, 0)), (a **(3/4)*x**8/4, True))/2
Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.15 \[ \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx=\frac {2}{7315} \, d {\left (\frac {385 \, {\left (b x^{2} + a\right )}^{\frac {19}{4}}}{b^{4}} - \frac {1463 \, {\left (b x^{2} + a\right )}^{\frac {15}{4}} a}{b^{4}} + \frac {1995 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} a^{2}}{b^{4}} - \frac {1045 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a^{3}}{b^{4}}\right )} + \frac {2}{1155} \, c {\left (\frac {77 \, {\left (b x^{2} + a\right )}^{\frac {15}{4}}}{b^{3}} - \frac {210 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} a}{b^{3}} + \frac {165 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a^{2}}{b^{3}}\right )} \] Input:
integrate(x^5*(b*x^2+a)^(3/4)*(d*x^2+c),x, algorithm="maxima")
Output:
2/7315*d*(385*(b*x^2 + a)^(19/4)/b^4 - 1463*(b*x^2 + a)^(15/4)*a/b^4 + 199 5*(b*x^2 + a)^(11/4)*a^2/b^4 - 1045*(b*x^2 + a)^(7/4)*a^3/b^4) + 2/1155*c* (77*(b*x^2 + a)^(15/4)/b^3 - 210*(b*x^2 + a)^(11/4)*a/b^3 + 165*(b*x^2 + a )^(7/4)*a^2/b^3)
Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (87) = 174\).
Time = 0.12 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.32 \[ \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx=\frac {2 \, {\left (\frac {95 \, {\left (21 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} - 66 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a + 77 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{2}\right )} a c}{b^{2}} + \frac {19 \, {\left (77 \, {\left (b x^{2} + a\right )}^{\frac {15}{4}} - 315 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} a + 495 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a^{2} - 385 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{3}\right )} c}{b^{2}} + \frac {19 \, {\left (77 \, {\left (b x^{2} + a\right )}^{\frac {15}{4}} - 315 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} a + 495 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a^{2} - 385 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{3}\right )} a d}{b^{3}} + \frac {{\left (1155 \, {\left (b x^{2} + a\right )}^{\frac {19}{4}} - 5852 \, {\left (b x^{2} + a\right )}^{\frac {15}{4}} a + 11970 \, {\left (b x^{2} + a\right )}^{\frac {11}{4}} a^{2} - 12540 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} a^{3} + 7315 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{4}\right )} d}{b^{3}}\right )}}{21945 \, b} \] Input:
integrate(x^5*(b*x^2+a)^(3/4)*(d*x^2+c),x, algorithm="giac")
Output:
2/21945*(95*(21*(b*x^2 + a)^(11/4) - 66*(b*x^2 + a)^(7/4)*a + 77*(b*x^2 + a)^(3/4)*a^2)*a*c/b^2 + 19*(77*(b*x^2 + a)^(15/4) - 315*(b*x^2 + a)^(11/4) *a + 495*(b*x^2 + a)^(7/4)*a^2 - 385*(b*x^2 + a)^(3/4)*a^3)*c/b^2 + 19*(77 *(b*x^2 + a)^(15/4) - 315*(b*x^2 + a)^(11/4)*a + 495*(b*x^2 + a)^(7/4)*a^2 - 385*(b*x^2 + a)^(3/4)*a^3)*a*d/b^3 + (1155*(b*x^2 + a)^(19/4) - 5852*(b *x^2 + a)^(15/4)*a + 11970*(b*x^2 + a)^(11/4)*a^2 - 12540*(b*x^2 + a)^(7/4 )*a^3 + 7315*(b*x^2 + a)^(3/4)*a^4)*d/b^3)/b
Time = 0.52 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.93 \[ \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx={\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2\,d\,x^8}{19}-\frac {768\,a^4\,d-1216\,a^3\,b\,c}{21945\,b^4}+\frac {x^6\,\left (2926\,c\,b^4+462\,a\,d\,b^3\right )}{21945\,b^4}-\frac {2\,a\,x^4\,\left (12\,a\,d-19\,b\,c\right )}{1045\,b^2}+\frac {16\,a^2\,x^2\,\left (12\,a\,d-19\,b\,c\right )}{7315\,b^3}\right ) \] Input:
int(x^5*(a + b*x^2)^(3/4)*(c + d*x^2),x)
Output:
(a + b*x^2)^(3/4)*((2*d*x^8)/19 - (768*a^4*d - 1216*a^3*b*c)/(21945*b^4) + (x^6*(2926*b^4*c + 462*a*b^3*d))/(21945*b^4) - (2*a*x^4*(12*a*d - 19*b*c) )/(1045*b^2) + (16*a^2*x^2*(12*a*d - 19*b*c))/(7315*b^3))
Time = 0.23 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.19 \[ \int x^5 \left (a+b x^2\right )^{3/4} \left (c+d x^2\right ) \, dx=\frac {2 \sqrt {\sqrt {b}\, \sqrt {b \,x^{2}+a}\, x +a +b \,x^{2}}\, \sqrt {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}\, \left (384 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a^{4} d x -608 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a^{3} b c x -288 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a^{3} b d \,x^{3}+456 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{3}+252 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a^{2} b^{2} d \,x^{5}-399 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{5}-231 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, a \,b^{3} d \,x^{7}-1463 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, b^{4} c \,x^{7}-1155 \sqrt {b}\, \sqrt {b \,x^{2}+a}\, b^{4} d \,x^{9}-384 a^{5} d +608 a^{4} b c -96 a^{4} b d \,x^{2}+152 a^{3} b^{2} c \,x^{2}+36 a^{3} b^{2} d \,x^{4}-57 a^{2} b^{3} c \,x^{4}-21 a^{2} b^{3} d \,x^{6}+1862 a \,b^{4} c \,x^{6}+1386 a \,b^{4} d \,x^{8}+1463 b^{5} c \,x^{8}+1155 b^{5} d \,x^{10}\right )}{21945 a \,b^{4}} \] Input:
int(x^5*(b*x^2+a)^(3/4)*(d*x^2+c),x)
Output:
(2*sqrt(sqrt(b)*sqrt(a + b*x**2)*x + a + b*x**2)*sqrt(sqrt(a + b*x**2) + s qrt(b)*x)*(384*sqrt(b)*sqrt(a + b*x**2)*a**4*d*x - 608*sqrt(b)*sqrt(a + b* x**2)*a**3*b*c*x - 288*sqrt(b)*sqrt(a + b*x**2)*a**3*b*d*x**3 + 456*sqrt(b )*sqrt(a + b*x**2)*a**2*b**2*c*x**3 + 252*sqrt(b)*sqrt(a + b*x**2)*a**2*b* *2*d*x**5 - 399*sqrt(b)*sqrt(a + b*x**2)*a*b**3*c*x**5 - 231*sqrt(b)*sqrt( a + b*x**2)*a*b**3*d*x**7 - 1463*sqrt(b)*sqrt(a + b*x**2)*b**4*c*x**7 - 11 55*sqrt(b)*sqrt(a + b*x**2)*b**4*d*x**9 - 384*a**5*d + 608*a**4*b*c - 96*a **4*b*d*x**2 + 152*a**3*b**2*c*x**2 + 36*a**3*b**2*d*x**4 - 57*a**2*b**3*c *x**4 - 21*a**2*b**3*d*x**6 + 1862*a*b**4*c*x**6 + 1386*a*b**4*d*x**8 + 14 63*b**5*c*x**8 + 1155*b**5*d*x**10))/(21945*a*b**4)