\(\int x^{5/2} (a+b x^2)^3 (A+B x^2) \, dx\) [360]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 (3 A b+a B) x^{11/2}+\frac {2}{5} a b (A b+a B) x^{15/2}+\frac {2}{19} b^2 (A b+3 a B) x^{19/2}+\frac {2}{23} b^3 B x^{23/2} \]

[Out]

2/7*a^3*A*x^(7/2)+2/11*a^2*(3*A*b+B*a)*x^(11/2)+2/5*a*b*(A*b+B*a)*x^(15/2)+2/19*b^2*(A*b+3*B*a)*x^(19/2)+2/23*
b^3*B*x^(23/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{19} b^2 x^{19/2} (3 a B+A b)+\frac {2}{5} a b x^{15/2} (a B+A b)+\frac {2}{23} b^3 B x^{23/2} \]

[In]

Int[x^(5/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (2*a*b*(A*b + a*B)*x^(15/2))/5 + (2*b^2*(A*b + 3*a*B
)*x^(19/2))/19 + (2*b^3*B*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^3 A x^{5/2}+a^2 (3 A b+a B) x^{9/2}+3 a b (A b+a B) x^{13/2}+b^2 (A b+3 a B) x^{17/2}+b^3 B x^{21/2}\right ) \, dx \\ & = \frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 (3 A b+a B) x^{11/2}+\frac {2}{5} a b (A b+a B) x^{15/2}+\frac {2}{19} b^2 (A b+3 a B) x^{19/2}+\frac {2}{23} b^3 B x^{23/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2 x^{7/2} \left (2185 a^3 \left (11 A+7 B x^2\right )+3059 a^2 b x^2 \left (15 A+11 B x^2\right )+1771 a b^2 x^4 \left (19 A+15 B x^2\right )+385 b^3 x^6 \left (23 A+19 B x^2\right )\right )}{168245} \]

[In]

Integrate[x^(5/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*x^(7/2)*(2185*a^3*(11*A + 7*B*x^2) + 3059*a^2*b*x^2*(15*A + 11*B*x^2) + 1771*a*b^2*x^4*(19*A + 15*B*x^2) +
385*b^3*x^6*(23*A + 19*B*x^2)))/168245

Maple [A] (verified)

Time = 2.73 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {2 b^{3} B \,x^{\frac {23}{2}}}{23}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}\)	\(76\)
default	\(\frac {2 b^{3} B \,x^{\frac {23}{2}}}{23}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}\)	\(76\)
gosper	\(\frac {2 x^{\frac {7}{2}} \left (7315 b^{3} B \,x^{8}+8855 x^{6} b^{3} A +26565 x^{6} a \,b^{2} B +33649 A a \,b^{2} x^{4}+33649 B \,a^{2} b \,x^{4}+45885 x^{2} a^{2} b A +15295 B \,a^{3} x^{2}+24035 a^{3} A \right )}{168245}\)	\(80\)
trager	\(\frac {2 x^{\frac {7}{2}} \left (7315 b^{3} B \,x^{8}+8855 x^{6} b^{3} A +26565 x^{6} a \,b^{2} B +33649 A a \,b^{2} x^{4}+33649 B \,a^{2} b \,x^{4}+45885 x^{2} a^{2} b A +15295 B \,a^{3} x^{2}+24035 a^{3} A \right )}{168245}\)	\(80\)
risch	\(\frac {2 x^{\frac {7}{2}} \left (7315 b^{3} B \,x^{8}+8855 x^{6} b^{3} A +26565 x^{6} a \,b^{2} B +33649 A a \,b^{2} x^{4}+33649 B \,a^{2} b \,x^{4}+45885 x^{2} a^{2} b A +15295 B \,a^{3} x^{2}+24035 a^{3} A \right )}{168245}\)	\(80\)

[In]

int(x^(5/2)*(b*x^2+a)^3*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

2/23*b^3*B*x^(23/2)+2/19*(A*b^3+3*B*a*b^2)*x^(19/2)+2/15*(3*A*a*b^2+3*B*a^2*b)*x^(15/2)+2/11*(3*A*a^2*b+B*a^3)
*x^(11/2)+2/7*a^3*A*x^(7/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{168245} \, {\left (7315 \, B b^{3} x^{11} + 8855 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 33649 \, {\left (B a^{2} b + A a b^{2}\right )} x^{7} + 24035 \, A a^{3} x^{3} + 15295 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt {x} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="fricas")

[Out]

2/168245*(7315*B*b^3*x^11 + 8855*(3*B*a*b^2 + A*b^3)*x^9 + 33649*(B*a^2*b + A*a*b^2)*x^7 + 24035*A*a^3*x^3 + 1
5295*(B*a^3 + 3*A*a^2*b)*x^5)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2 A a^{3} x^{\frac {7}{2}}}{7} + \frac {6 A a^{2} b x^{\frac {11}{2}}}{11} + \frac {2 A a b^{2} x^{\frac {15}{2}}}{5} + \frac {2 A b^{3} x^{\frac {19}{2}}}{19} + \frac {2 B a^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{2} b x^{\frac {15}{2}}}{5} + \frac {6 B a b^{2} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {23}{2}}}{23} \]

[In]

integrate(x**(5/2)*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

2*A*a**3*x**(7/2)/7 + 6*A*a**2*b*x**(11/2)/11 + 2*A*a*b**2*x**(15/2)/5 + 2*A*b**3*x**(19/2)/19 + 2*B*a**3*x**(
11/2)/11 + 2*B*a**2*b*x**(15/2)/5 + 6*B*a*b**2*x**(19/2)/19 + 2*B*b**3*x**(23/2)/23

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{23} \, B b^{3} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {19}{2}} + \frac {2}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {15}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {11}{2}} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="maxima")

[Out]

2/23*B*b^3*x^(23/2) + 2/19*(3*B*a*b^2 + A*b^3)*x^(19/2) + 2/5*(B*a^2*b + A*a*b^2)*x^(15/2) + 2/7*A*a^3*x^(7/2)
+ 2/11*(B*a^3 + 3*A*a^2*b)*x^(11/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{23} \, B b^{3} x^{\frac {23}{2}} + \frac {6}{19} \, B a b^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A b^{3} x^{\frac {19}{2}} + \frac {2}{5} \, B a^{2} b x^{\frac {15}{2}} + \frac {2}{5} \, A a b^{2} x^{\frac {15}{2}} + \frac {2}{11} \, B a^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A a^{2} b x^{\frac {11}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \]

[In]

integrate(x^(5/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="giac")

[Out]

2/23*B*b^3*x^(23/2) + 6/19*B*a*b^2*x^(19/2) + 2/19*A*b^3*x^(19/2) + 2/5*B*a^2*b*x^(15/2) + 2/5*A*a*b^2*x^(15/2
) + 2/11*B*a^3*x^(11/2) + 6/11*A*a^2*b*x^(11/2) + 2/7*A*a^3*x^(7/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=x^{11/2}\,\left (\frac {2\,B\,a^3}{11}+\frac {6\,A\,b\,a^2}{11}\right )+x^{19/2}\,\left (\frac {2\,A\,b^3}{19}+\frac {6\,B\,a\,b^2}{19}\right )+\frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {2\,B\,b^3\,x^{23/2}}{23}+\frac {2\,a\,b\,x^{15/2}\,\left (A\,b+B\,a\right )}{5} \]

[In]

int(x^(5/2)*(A + B*x^2)*(a + b*x^2)^3,x)

[Out]

x^(11/2)*((2*B*a^3)/11 + (6*A*a^2*b)/11) + x^(19/2)*((2*A*b^3)/19 + (6*B*a*b^2)/19) + (2*A*a^3*x^(7/2))/7 + (2
*B*b^3*x^(23/2))/23 + (2*a*b*x^(15/2)*(A*b + B*a))/5