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

Optimal. Leaf size=85 \[ \frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{7} a^3 A x^{7/2}+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2} \]

[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

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Rubi [A]  time = 0.040575, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{7} a^3 A x^{7/2}+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2} \]

Antiderivative was successfully verified.

[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 448

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*} \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx &=\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]  time = 0.0381048, size = 85, normalized size = 1. \[ \frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{7} a^3 A x^{7/2}+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2} \]

Antiderivative was successfully verified.

[In]

Integrate[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

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Maple [A]  time = 0.005, size = 80, normalized size = 0.9 \begin{align*}{\frac{14630\,B{b}^{3}{x}^{8}+17710\,{x}^{6}A{b}^{3}+53130\,{x}^{6}Ba{b}^{2}+67298\,{x}^{4}Aa{b}^{2}+67298\,{x}^{4}B{a}^{2}b+91770\,{x}^{2}A{a}^{2}b+30590\,{x}^{2}B{a}^{3}+48070\,A{a}^{3}}{168245}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/168245*x^(7/2)*(7315*B*b^3*x^8+8855*A*b^3*x^6+26565*B*a*b^2*x^6+33649*A*a*b^2*x^4+33649*B*a^2*b*x^4+45885*A*
a^2*b*x^2+15295*B*a^3*x^2+24035*A*a^3)

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Maxima [A]  time = 1.05957, size = 99, normalized size = 1.16 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0.870319, size = 201, normalized size = 2.36 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 20.6412, size = 114, normalized size = 1.34 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.15107, size = 104, normalized size = 1.22 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)