\(\int x^{3/2} (a+b x^2)^2 (A+B x^2) \, dx\) [353]

Optimal result

Integrand size = 22, antiderivative size = 63 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{5} a^2 A x^{5/2}+\frac {2}{9} a (2 A b+a B) x^{9/2}+\frac {2}{13} b (A b+2 a B) x^{13/2}+\frac {2}{17} b^2 B x^{17/2} \]

[Out]

2/5*a^2*A*x^(5/2)+2/9*a*(2*A*b+B*a)*x^(9/2)+2/13*b*(A*b+2*B*a)*x^(13/2)+2/17*b^2*B*x^(17/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, 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} \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{5} a^2 A x^{5/2}+\frac {2}{13} b x^{13/2} (2 a B+A b)+\frac {2}{9} a x^{9/2} (a B+2 A b)+\frac {2}{17} b^2 B x^{17/2} \]

[In]

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

[Out]

(2*a^2*A*x^(5/2))/5 + (2*a*(2*A*b + a*B)*x^(9/2))/9 + (2*b*(A*b + 2*a*B)*x^(13/2))/13 + (2*b^2*B*x^(17/2))/17

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2 x^{5/2} \left (221 a^2 \left (9 A+5 B x^2\right )+170 a b x^2 \left (13 A+9 B x^2\right )+45 b^2 x^4 \left (17 A+13 B x^2\right )\right )}{9945} \]

[In]

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

[Out]

(2*x^(5/2)*(221*a^2*(9*A + 5*B*x^2) + 170*a*b*x^2*(13*A + 9*B*x^2) + 45*b^2*x^4*(17*A + 13*B*x^2)))/9945

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {2 b^{2} B \,x^{\frac {17}{2}}}{17}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} A \,x^{\frac {5}{2}}}{5}\)	\(52\)
default	\(\frac {2 b^{2} B \,x^{\frac {17}{2}}}{17}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} A \,x^{\frac {5}{2}}}{5}\)	\(52\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (585 b^{2} B \,x^{6}+765 A \,b^{2} x^{4}+1530 B a b \,x^{4}+2210 a A b \,x^{2}+1105 a^{2} B \,x^{2}+1989 a^{2} A \right )}{9945}\)	\(56\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (585 b^{2} B \,x^{6}+765 A \,b^{2} x^{4}+1530 B a b \,x^{4}+2210 a A b \,x^{2}+1105 a^{2} B \,x^{2}+1989 a^{2} A \right )}{9945}\)	\(56\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (585 b^{2} B \,x^{6}+765 A \,b^{2} x^{4}+1530 B a b \,x^{4}+2210 a A b \,x^{2}+1105 a^{2} B \,x^{2}+1989 a^{2} A \right )}{9945}\)	\(56\)

[In]

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

[Out]

2/17*b^2*B*x^(17/2)+2/13*(A*b^2+2*B*a*b)*x^(13/2)+2/9*(2*A*a*b+B*a^2)*x^(9/2)+2/5*a^2*A*x^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{9945} \, {\left (585 \, B b^{2} x^{8} + 765 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 1989 \, A a^{2} x^{2} + 1105 \, {\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} \sqrt {x} \]

[In]

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

[Out]

2/9945*(585*B*b^2*x^8 + 765*(2*B*a*b + A*b^2)*x^6 + 1989*A*a^2*x^2 + 1105*(B*a^2 + 2*A*a*b)*x^4)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.27 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2 A a^{2} x^{\frac {5}{2}}}{5} + \frac {4 A a b x^{\frac {9}{2}}}{9} + \frac {2 A b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B a^{2} x^{\frac {9}{2}}}{9} + \frac {4 B a b x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} x^{\frac {17}{2}}}{17} \]

[In]

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

[Out]

2*A*a**2*x**(5/2)/5 + 4*A*a*b*x**(9/2)/9 + 2*A*b**2*x**(13/2)/13 + 2*B*a**2*x**(9/2)/9 + 4*B*a*b*x**(13/2)/13
+ 2*B*b**2*x**(17/2)/17

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{17} \, B b^{2} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {13}{2}} + \frac {2}{5} \, A a^{2} x^{\frac {5}{2}} + \frac {2}{9} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {9}{2}} \]

[In]

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

[Out]

2/17*B*b^2*x^(17/2) + 2/13*(2*B*a*b + A*b^2)*x^(13/2) + 2/5*A*a^2*x^(5/2) + 2/9*(B*a^2 + 2*A*a*b)*x^(9/2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.84 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=\frac {2}{17} \, B b^{2} x^{\frac {17}{2}} + \frac {4}{13} \, B a b x^{\frac {13}{2}} + \frac {2}{13} \, A b^{2} x^{\frac {13}{2}} + \frac {2}{9} \, B a^{2} x^{\frac {9}{2}} + \frac {4}{9} \, A a b x^{\frac {9}{2}} + \frac {2}{5} \, A a^{2} x^{\frac {5}{2}} \]

[In]

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

[Out]

2/17*B*b^2*x^(17/2) + 4/13*B*a*b*x^(13/2) + 2/13*A*b^2*x^(13/2) + 2/9*B*a^2*x^(9/2) + 4/9*A*a*b*x^(9/2) + 2/5*
A*a^2*x^(5/2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.81 \[ \int x^{3/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx=x^{9/2}\,\left (\frac {2\,B\,a^2}{9}+\frac {4\,A\,b\,a}{9}\right )+x^{13/2}\,\left (\frac {2\,A\,b^2}{13}+\frac {4\,B\,a\,b}{13}\right )+\frac {2\,A\,a^2\,x^{5/2}}{5}+\frac {2\,B\,b^2\,x^{17/2}}{17} \]

[In]

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

[Out]

x^(9/2)*((2*B*a^2)/9 + (4*A*a*b)/9) + x^(13/2)*((2*A*b^2)/13 + (4*B*a*b)/13) + (2*A*a^2*x^(5/2))/5 + (2*B*b^2*
x^(17/2))/17