\(\int x^{7/2} (a+b x^2) (A+B x^2) \, dx\) [343]

Optimal result

Integrand size = 20, antiderivative size = 39 \[ \int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{9} a A x^{9/2}+\frac {2}{13} (A b+a B) x^{13/2}+\frac {2}{17} b B x^{17/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{13} x^{13/2} (a B+A b)+\frac {2}{9} a A x^{9/2}+\frac {2}{17} b B x^{17/2} \]

[In]

Int[x^(7/2)*(a + b*x^2)*(A + B*x^2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.05 \[ \int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2 \left (221 a A x^{9/2}+153 A b x^{13/2}+153 a B x^{13/2}+117 b B x^{17/2}\right )}{1989} \]

[In]

Integrate[x^(7/2)*(a + b*x^2)*(A + B*x^2),x]

[Out]

(2*(221*a*A*x^(9/2) + 153*A*b*x^(13/2) + 153*a*B*x^(13/2) + 117*b*B*x^(17/2)))/1989

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {2 a A \,x^{\frac {9}{2}}}{9}+\frac {2 \left (A b +B a \right ) x^{\frac {13}{2}}}{13}+\frac {2 b B \,x^{\frac {17}{2}}}{17}\)	\(28\)
default	\(\frac {2 a A \,x^{\frac {9}{2}}}{9}+\frac {2 \left (A b +B a \right ) x^{\frac {13}{2}}}{13}+\frac {2 b B \,x^{\frac {17}{2}}}{17}\)	\(28\)
gosper	\(\frac {2 x^{\frac {9}{2}} \left (117 b B \,x^{4}+153 A b \,x^{2}+153 B a \,x^{2}+221 A a \right )}{1989}\)	\(32\)
trager	\(\frac {2 x^{\frac {9}{2}} \left (117 b B \,x^{4}+153 A b \,x^{2}+153 B a \,x^{2}+221 A a \right )}{1989}\)	\(32\)
risch	\(\frac {2 x^{\frac {9}{2}} \left (117 b B \,x^{4}+153 A b \,x^{2}+153 B a \,x^{2}+221 A a \right )}{1989}\)	\(32\)

[In]

int(x^(7/2)*(b*x^2+a)*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{1989} \, {\left (117 \, B b x^{8} + 153 \, {\left (B a + A b\right )} x^{6} + 221 \, A a x^{4}\right )} \sqrt {x} \]

[In]

integrate(x^(7/2)*(b*x^2+a)*(B*x^2+A),x, algorithm="fricas")

[Out]

2/1989*(117*B*b*x^8 + 153*(B*a + A*b)*x^6 + 221*A*a*x^4)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.18 \[ \int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2 A a x^{\frac {9}{2}}}{9} + \frac {2 A b x^{\frac {13}{2}}}{13} + \frac {2 B a x^{\frac {13}{2}}}{13} + \frac {2 B b x^{\frac {17}{2}}}{17} \]

[In]

integrate(x**(7/2)*(b*x**2+a)*(B*x**2+A),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{17} \, B b x^{\frac {17}{2}} + \frac {2}{13} \, {\left (B a + A b\right )} x^{\frac {13}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \]

[In]

integrate(x^(7/2)*(b*x^2+a)*(B*x^2+A),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2}{17} \, B b x^{\frac {17}{2}} + \frac {2}{13} \, B a x^{\frac {13}{2}} + \frac {2}{13} \, A b x^{\frac {13}{2}} + \frac {2}{9} \, A a x^{\frac {9}{2}} \]

[In]

integrate(x^(7/2)*(b*x^2+a)*(B*x^2+A),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int x^{7/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {2\,x^{9/2}\,\left (221\,A\,a+153\,A\,b\,x^2+153\,B\,a\,x^2+117\,B\,b\,x^4\right )}{1989} \]

[In]

int(x^(7/2)*(A + B*x^2)*(a + b*x^2),x)

[Out]

(2*x^(9/2)*(221*A*a + 153*A*b*x^2 + 153*B*a*x^2 + 117*B*b*x^4))/1989