\(\int x^{5/2} (a+b x^2)^2 (c+d x^2)^2 \, dx\) [400]

Optimal result

Integrand size = 24, antiderivative size = 97 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{7} a^2 c^2 x^{7/2}+\frac {4}{11} a c (b c+a d) x^{11/2}+\frac {2}{15} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{15/2}+\frac {4}{19} b d (b c+a d) x^{19/2}+\frac {2}{23} b^2 d^2 x^{23/2} \]

[Out]

2/7*a^2*c^2*x^(7/2)+4/11*a*c*(a*d+b*c)*x^(11/2)+2/15*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^(15/2)+4/19*b*d*(a*d+b*c)*x
^(19/2)+2/23*b^2*d^2*x^(23/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {459} \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac {2}{7} a^2 c^2 x^{7/2}+\frac {4}{19} b d x^{19/2} (a d+b c)+\frac {4}{11} a c x^{11/2} (a d+b c)+\frac {2}{23} b^2 d^2 x^{23/2} \]

[In]

Int[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

(2*a^2*c^2*x^(7/2))/7 + (4*a*c*(b*c + a*d)*x^(11/2))/11 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (4
*b*d*(b*c + a*d)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2))/23

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c^2 x^{5/2}+2 a c (b c+a d) x^{9/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{13/2}+2 b d (b c+a d) x^{17/2}+b^2 d^2 x^{21/2}\right ) \, dx \\ & = \frac {2}{7} a^2 c^2 x^{7/2}+\frac {4}{11} a c (b c+a d) x^{11/2}+\frac {2}{15} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{15/2}+\frac {4}{19} b d (b c+a d) x^{19/2}+\frac {2}{23} b^2 d^2 x^{23/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2 x^{7/2} \left (437 a^2 \left (165 c^2+210 c d x^2+77 d^2 x^4\right )+322 a b x^2 \left (285 c^2+418 c d x^2+165 d^2 x^4\right )+77 b^2 x^4 \left (437 c^2+690 c d x^2+285 d^2 x^4\right )\right )}{504735} \]

[In]

Integrate[x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

(2*x^(7/2)*(437*a^2*(165*c^2 + 210*c*d*x^2 + 77*d^2*x^4) + 322*a*b*x^2*(285*c^2 + 418*c*d*x^2 + 165*d^2*x^4) +
77*b^2*x^4*(437*c^2 + 690*c*d*x^2 + 285*d^2*x^4)))/504735

Maple [A] (verified)

Time = 2.79 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {2 b^{2} d^{2} x^{\frac {23}{2}}}{23}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} c^{2} x^{\frac {7}{2}}}{7}\)	\(90\)
default	\(\frac {2 b^{2} d^{2} x^{\frac {23}{2}}}{23}+\frac {2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{2} c^{2} x^{\frac {7}{2}}}{7}\)	\(90\)
gosper	\(\frac {2 x^{\frac {7}{2}} \left (21945 b^{2} d^{2} x^{8}+53130 a b \,d^{2} x^{6}+53130 b^{2} c d \,x^{6}+33649 a^{2} d^{2} x^{4}+134596 x^{4} a b c d +33649 b^{2} c^{2} x^{4}+91770 a^{2} c d \,x^{2}+91770 x^{2} b \,c^{2} a +72105 a^{2} c^{2}\right )}{504735}\)	\(97\)
trager	\(\frac {2 x^{\frac {7}{2}} \left (21945 b^{2} d^{2} x^{8}+53130 a b \,d^{2} x^{6}+53130 b^{2} c d \,x^{6}+33649 a^{2} d^{2} x^{4}+134596 x^{4} a b c d +33649 b^{2} c^{2} x^{4}+91770 a^{2} c d \,x^{2}+91770 x^{2} b \,c^{2} a +72105 a^{2} c^{2}\right )}{504735}\)	\(97\)
risch	\(\frac {2 x^{\frac {7}{2}} \left (21945 b^{2} d^{2} x^{8}+53130 a b \,d^{2} x^{6}+53130 b^{2} c d \,x^{6}+33649 a^{2} d^{2} x^{4}+134596 x^{4} a b c d +33649 b^{2} c^{2} x^{4}+91770 a^{2} c d \,x^{2}+91770 x^{2} b \,c^{2} a +72105 a^{2} c^{2}\right )}{504735}\)	\(97\)

[In]

int(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

2/23*b^2*d^2*x^(23/2)+2/19*(2*a*b*d^2+2*b^2*c*d)*x^(19/2)+2/15*(a^2*d^2+4*a*b*c*d+b^2*c^2)*x^(15/2)+2/11*(2*a^
2*c*d+2*a*b*c^2)*x^(11/2)+2/7*a^2*c^2*x^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{504735} \, {\left (21945 \, b^{2} d^{2} x^{11} + 53130 \, {\left (b^{2} c d + a b d^{2}\right )} x^{9} + 33649 \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{7} + 72105 \, a^{2} c^{2} x^{3} + 91770 \, {\left (a b c^{2} + a^{2} c d\right )} x^{5}\right )} \sqrt {x} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="fricas")

[Out]

2/504735*(21945*b^2*d^2*x^11 + 53130*(b^2*c*d + a*b*d^2)*x^9 + 33649*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7 + 721
05*a^2*c^2*x^3 + 91770*(a*b*c^2 + a^2*c*d)*x^5)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.40 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2 a^{2} c^{2} x^{\frac {7}{2}}}{7} + \frac {4 a^{2} c d x^{\frac {11}{2}}}{11} + \frac {2 a^{2} d^{2} x^{\frac {15}{2}}}{15} + \frac {4 a b c^{2} x^{\frac {11}{2}}}{11} + \frac {8 a b c d x^{\frac {15}{2}}}{15} + \frac {4 a b d^{2} x^{\frac {19}{2}}}{19} + \frac {2 b^{2} c^{2} x^{\frac {15}{2}}}{15} + \frac {4 b^{2} c d x^{\frac {19}{2}}}{19} + \frac {2 b^{2} d^{2} x^{\frac {23}{2}}}{23} \]

[In]

integrate(x**(5/2)*(b*x**2+a)**2*(d*x**2+c)**2,x)

[Out]

2*a**2*c**2*x**(7/2)/7 + 4*a**2*c*d*x**(11/2)/11 + 2*a**2*d**2*x**(15/2)/15 + 4*a*b*c**2*x**(11/2)/11 + 8*a*b*
c*d*x**(15/2)/15 + 4*a*b*d**2*x**(19/2)/19 + 2*b**2*c**2*x**(15/2)/15 + 4*b**2*c*d*x**(19/2)/19 + 2*b**2*d**2*
x**(23/2)/23

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.88 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{23} \, b^{2} d^{2} x^{\frac {23}{2}} + \frac {4}{19} \, {\left (b^{2} c d + a b d^{2}\right )} x^{\frac {19}{2}} + \frac {2}{15} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac {15}{2}} + \frac {2}{7} \, a^{2} c^{2} x^{\frac {7}{2}} + \frac {4}{11} \, {\left (a b c^{2} + a^{2} c d\right )} x^{\frac {11}{2}} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

2/23*b^2*d^2*x^(23/2) + 4/19*(b^2*c*d + a*b*d^2)*x^(19/2) + 2/15*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(15/2) + 2/
7*a^2*c^2*x^(7/2) + 4/11*(a*b*c^2 + a^2*c*d)*x^(11/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {2}{23} \, b^{2} d^{2} x^{\frac {23}{2}} + \frac {4}{19} \, b^{2} c d x^{\frac {19}{2}} + \frac {4}{19} \, a b d^{2} x^{\frac {19}{2}} + \frac {2}{15} \, b^{2} c^{2} x^{\frac {15}{2}} + \frac {8}{15} \, a b c d x^{\frac {15}{2}} + \frac {2}{15} \, a^{2} d^{2} x^{\frac {15}{2}} + \frac {4}{11} \, a b c^{2} x^{\frac {11}{2}} + \frac {4}{11} \, a^{2} c d x^{\frac {11}{2}} + \frac {2}{7} \, a^{2} c^{2} x^{\frac {7}{2}} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="giac")

[Out]

2/23*b^2*d^2*x^(23/2) + 4/19*b^2*c*d*x^(19/2) + 4/19*a*b*d^2*x^(19/2) + 2/15*b^2*c^2*x^(15/2) + 8/15*a*b*c*d*x
^(15/2) + 2/15*a^2*d^2*x^(15/2) + 4/11*a*b*c^2*x^(11/2) + 4/11*a^2*c*d*x^(11/2) + 2/7*a^2*c^2*x^(7/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^{15/2}\,\left (\frac {2\,a^2\,d^2}{15}+\frac {8\,a\,b\,c\,d}{15}+\frac {2\,b^2\,c^2}{15}\right )+\frac {2\,a^2\,c^2\,x^{7/2}}{7}+\frac {2\,b^2\,d^2\,x^{23/2}}{23}+\frac {4\,a\,c\,x^{11/2}\,\left (a\,d+b\,c\right )}{11}+\frac {4\,b\,d\,x^{19/2}\,\left (a\,d+b\,c\right )}{19} \]

[In]

int(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^2,x)

[Out]

x^(15/2)*((2*a^2*d^2)/15 + (2*b^2*c^2)/15 + (8*a*b*c*d)/15) + (2*a^2*c^2*x^(7/2))/7 + (2*b^2*d^2*x^(23/2))/23
+ (4*a*c*x^(11/2)*(a*d + b*c))/11 + (4*b*d*x^(19/2)*(a*d + b*c))/19