\(\int x^{7/2} (a+b x^2)^3 (A+B x^2) \, dx\) [359]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{9} a^3 A x^{9/2}+\frac {2}{13} a^2 (3 A b+a B) x^{13/2}+\frac {6}{17} a b (A b+a B) x^{17/2}+\frac {2}{21} b^2 (A b+3 a B) x^{21/2}+\frac {2}{25} b^3 B x^{25/2} \]

[Out]

2/9*a^3*A*x^(9/2)+2/13*a^2*(3*A*b+B*a)*x^(13/2)+6/17*a*b*(A*b+B*a)*x^(17/2)+2/21*b^2*(A*b+3*B*a)*x^(21/2)+2/25
*b^3*B*x^(25/2)

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} \[ \int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{9} a^3 A x^{9/2}+\frac {2}{13} a^2 x^{13/2} (a B+3 A b)+\frac {2}{21} b^2 x^{21/2} (3 a B+A b)+\frac {6}{17} a b x^{17/2} (a B+A b)+\frac {2}{25} b^3 B x^{25/2} \]

[In]

Int[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(17/2))/17 + (2*b^2*(A*b + 3*a*
B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25

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^{7/2}+a^2 (3 A b+a B) x^{11/2}+3 a b (A b+a B) x^{15/2}+b^2 (A b+3 a B) x^{19/2}+b^3 B x^{23/2}\right ) \, dx \\ & = \frac {2}{9} a^3 A x^{9/2}+\frac {2}{13} a^2 (3 A b+a B) x^{13/2}+\frac {6}{17} a b (A b+a B) x^{17/2}+\frac {2}{21} b^2 (A b+3 a B) x^{21/2}+\frac {2}{25} b^3 B x^{25/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2 x^{9/2} \left (2975 a^3 \left (13 A+9 B x^2\right )+4725 a^2 b x^2 \left (17 A+13 B x^2\right )+2925 a b^2 x^4 \left (21 A+17 B x^2\right )+663 b^3 x^6 \left (25 A+21 B x^2\right )\right )}{348075} \]

[In]

Integrate[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*x^(9/2)*(2975*a^3*(13*A + 9*B*x^2) + 4725*a^2*b*x^2*(17*A + 13*B*x^2) + 2925*a*b^2*x^4*(21*A + 17*B*x^2) +
663*b^3*x^6*(25*A + 21*B*x^2)))/348075

Maple [A] (verified)

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

method	result	size
derivativedivides	\(\frac {2 b^{3} B \,x^{\frac {25}{2}}}{25}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {13}{2}}}{13}+\frac {2 a^{3} A \,x^{\frac {9}{2}}}{9}\)	\(76\)
default	\(\frac {2 b^{3} B \,x^{\frac {25}{2}}}{25}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {13}{2}}}{13}+\frac {2 a^{3} A \,x^{\frac {9}{2}}}{9}\)	\(76\)
gosper	\(\frac {2 x^{\frac {9}{2}} \left (13923 b^{3} B \,x^{8}+16575 x^{6} b^{3} A +49725 x^{6} a \,b^{2} B +61425 A a \,b^{2} x^{4}+61425 B \,a^{2} b \,x^{4}+80325 x^{2} a^{2} b A +26775 B \,a^{3} x^{2}+38675 a^{3} A \right )}{348075}\)	\(80\)
trager	\(\frac {2 x^{\frac {9}{2}} \left (13923 b^{3} B \,x^{8}+16575 x^{6} b^{3} A +49725 x^{6} a \,b^{2} B +61425 A a \,b^{2} x^{4}+61425 B \,a^{2} b \,x^{4}+80325 x^{2} a^{2} b A +26775 B \,a^{3} x^{2}+38675 a^{3} A \right )}{348075}\)	\(80\)
risch	\(\frac {2 x^{\frac {9}{2}} \left (13923 b^{3} B \,x^{8}+16575 x^{6} b^{3} A +49725 x^{6} a \,b^{2} B +61425 A a \,b^{2} x^{4}+61425 B \,a^{2} b \,x^{4}+80325 x^{2} a^{2} b A +26775 B \,a^{3} x^{2}+38675 a^{3} A \right )}{348075}\)	\(80\)

[In]

int(x^(7/2)*(b*x^2+a)^3*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

2/25*b^3*B*x^(25/2)+2/21*(A*b^3+3*B*a*b^2)*x^(21/2)+2/17*(3*A*a*b^2+3*B*a^2*b)*x^(17/2)+2/13*(3*A*a^2*b+B*a^3)
*x^(13/2)+2/9*a^3*A*x^(9/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{348075} \, {\left (13923 \, B b^{3} x^{12} + 16575 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{10} + 61425 \, {\left (B a^{2} b + A a b^{2}\right )} x^{8} + 38675 \, A a^{3} x^{4} + 26775 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6}\right )} \sqrt {x} \]

[In]

integrate(x^(7/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="fricas")

[Out]

2/348075*(13923*B*b^3*x^12 + 16575*(3*B*a*b^2 + A*b^3)*x^10 + 61425*(B*a^2*b + A*a*b^2)*x^8 + 38675*A*a^3*x^4
+ 26775*(B*a^3 + 3*A*a^2*b)*x^6)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 1.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2 A a^{3} x^{\frac {9}{2}}}{9} + \frac {6 A a^{2} b x^{\frac {13}{2}}}{13} + \frac {6 A a b^{2} x^{\frac {17}{2}}}{17} + \frac {2 A b^{3} x^{\frac {21}{2}}}{21} + \frac {2 B a^{3} x^{\frac {13}{2}}}{13} + \frac {6 B a^{2} b x^{\frac {17}{2}}}{17} + \frac {2 B a b^{2} x^{\frac {21}{2}}}{7} + \frac {2 B b^{3} x^{\frac {25}{2}}}{25} \]

[In]

integrate(x**(7/2)*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

2*A*a**3*x**(9/2)/9 + 6*A*a**2*b*x**(13/2)/13 + 6*A*a*b**2*x**(17/2)/17 + 2*A*b**3*x**(21/2)/21 + 2*B*a**3*x**
(13/2)/13 + 6*B*a**2*b*x**(17/2)/17 + 2*B*a*b**2*x**(21/2)/7 + 2*B*b**3*x**(25/2)/25

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {2}{21} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {21}{2}} + \frac {6}{17} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {17}{2}} + \frac {2}{9} \, A a^{3} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {13}{2}} \]

[In]

integrate(x^(7/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="maxima")

[Out]

2/25*B*b^3*x^(25/2) + 2/21*(3*B*a*b^2 + A*b^3)*x^(21/2) + 6/17*(B*a^2*b + A*a*b^2)*x^(17/2) + 2/9*A*a^3*x^(9/2
) + 2/13*(B*a^3 + 3*A*a^2*b)*x^(13/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {2}{7} \, B a b^{2} x^{\frac {21}{2}} + \frac {2}{21} \, A b^{3} x^{\frac {21}{2}} + \frac {6}{17} \, B a^{2} b x^{\frac {17}{2}} + \frac {6}{17} \, A a b^{2} x^{\frac {17}{2}} + \frac {2}{13} \, B a^{3} x^{\frac {13}{2}} + \frac {6}{13} \, A a^{2} b x^{\frac {13}{2}} + \frac {2}{9} \, A a^{3} x^{\frac {9}{2}} \]

[In]

integrate(x^(7/2)*(b*x^2+a)^3*(B*x^2+A),x, algorithm="giac")

[Out]

2/25*B*b^3*x^(25/2) + 2/7*B*a*b^2*x^(21/2) + 2/21*A*b^3*x^(21/2) + 6/17*B*a^2*b*x^(17/2) + 6/17*A*a*b^2*x^(17/
2) + 2/13*B*a^3*x^(13/2) + 6/13*A*a^2*b*x^(13/2) + 2/9*A*a^3*x^(9/2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=x^{13/2}\,\left (\frac {2\,B\,a^3}{13}+\frac {6\,A\,b\,a^2}{13}\right )+x^{21/2}\,\left (\frac {2\,A\,b^3}{21}+\frac {2\,B\,a\,b^2}{7}\right )+\frac {2\,A\,a^3\,x^{9/2}}{9}+\frac {2\,B\,b^3\,x^{25/2}}{25}+\frac {6\,a\,b\,x^{17/2}\,\left (A\,b+B\,a\right )}{17} \]

[In]

int(x^(7/2)*(A + B*x^2)*(a + b*x^2)^3,x)

[Out]

x^(13/2)*((2*B*a^3)/13 + (6*A*a^2*b)/13) + x^(21/2)*((2*A*b^3)/21 + (2*B*a*b^2)/7) + (2*A*a^3*x^(9/2))/9 + (2*
B*b^3*x^(25/2))/25 + (6*a*b*x^(17/2)*(A*b + B*a))/17