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

Optimal. Leaf size=63 \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{21} b x^{21/2} (2 a B+A b)+\frac{2}{15} a x^{15/2} (a B+2 A b)+\frac{2}{27} b^2 B x^{27/2} \]

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(15/2))/15 + (2*b*(A*b + 2*a*B)*x^(21/2))/21 + (2*b^2*B*x^(27/2))/2
7

________________________________________________________________________________________

Rubi [A]  time = 0.0306103, antiderivative size = 63, 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} \[ \frac{2}{9} a^2 A x^{9/2}+\frac{2}{21} b x^{21/2} (2 a B+A b)+\frac{2}{15} a x^{15/2} (a B+2 A b)+\frac{2}{27} b^2 B x^{27/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*A*x^(9/2))/9 + (2*a*(2*A*b + a*B)*x^(15/2))/15 + (2*b*(A*b + 2*a*B)*x^(21/2))/21 + (2*b^2*B*x^(27/2))/2
7

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^{7/2} \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx &=\int \left (a^2 A x^{7/2}+a (2 A b+a B) x^{13/2}+b (A b+2 a B) x^{19/2}+b^2 B x^{25/2}\right ) \, dx\\ &=\frac{2}{9} a^2 A x^{9/2}+\frac{2}{15} a (2 A b+a B) x^{15/2}+\frac{2}{21} b (A b+2 a B) x^{21/2}+\frac{2}{27} b^2 B x^{27/2}\\ \end{align*}

Mathematica [A]  time = 0.030353, size = 53, normalized size = 0.84 \[ \frac{2}{945} x^{9/2} \left (105 a^2 A+45 b x^6 (2 a B+A b)+63 a x^3 (a B+2 A b)+35 b^2 B x^9\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(105*a^2*A + 63*a*(2*A*b + a*B)*x^3 + 45*b*(A*b + 2*a*B)*x^6 + 35*b^2*B*x^9))/945

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*}{\frac{70\,B{b}^{2}{x}^{9}+90\,A{b}^{2}{x}^{6}+180\,B{x}^{6}ab+252\,aAb{x}^{3}+126\,B{x}^{3}{a}^{2}+210\,{a}^{2}A}{945}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/945*x^(9/2)*(35*B*b^2*x^9+45*A*b^2*x^6+90*B*a*b*x^6+126*A*a*b*x^3+63*B*a^2*x^3+105*A*a^2)

________________________________________________________________________________________

Maxima [A]  time = 0.945678, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{27} \, B b^{2} x^{\frac{27}{2}} + \frac{2}{21} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{21}{2}} + \frac{2}{15} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{15}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*B*b^2*x^(27/2) + 2/21*(2*B*a*b + A*b^2)*x^(21/2) + 2/15*(B*a^2 + 2*A*a*b)*x^(15/2) + 2/9*A*a^2*x^(9/2)

________________________________________________________________________________________

Fricas [A]  time = 1.67165, size = 138, normalized size = 2.19 \begin{align*} \frac{2}{945} \,{\left (35 \, B b^{2} x^{13} + 45 \,{\left (2 \, B a b + A b^{2}\right )} x^{10} + 63 \,{\left (B a^{2} + 2 \, A a b\right )} x^{7} + 105 \, A a^{2} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/945*(35*B*b^2*x^13 + 45*(2*B*a*b + A*b^2)*x^10 + 63*(B*a^2 + 2*A*a*b)*x^7 + 105*A*a^2*x^4)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 51.8472, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A a^{2} x^{\frac{9}{2}}}{9} + \frac{4 A a b x^{\frac{15}{2}}}{15} + \frac{2 A b^{2} x^{\frac{21}{2}}}{21} + \frac{2 B a^{2} x^{\frac{15}{2}}}{15} + \frac{4 B a b x^{\frac{21}{2}}}{21} + \frac{2 B b^{2} x^{\frac{27}{2}}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**2*x**(9/2)/9 + 4*A*a*b*x**(15/2)/15 + 2*A*b**2*x**(21/2)/21 + 2*B*a**2*x**(15/2)/15 + 4*B*a*b*x**(21/2)
/21 + 2*B*b**2*x**(27/2)/27

________________________________________________________________________________________

Giac [A]  time = 1.07986, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{27} \, B b^{2} x^{\frac{27}{2}} + \frac{4}{21} \, B a b x^{\frac{21}{2}} + \frac{2}{21} \, A b^{2} x^{\frac{21}{2}} + \frac{2}{15} \, B a^{2} x^{\frac{15}{2}} + \frac{4}{15} \, A a b x^{\frac{15}{2}} + \frac{2}{9} \, A a^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/27*B*b^2*x^(27/2) + 4/21*B*a*b*x^(21/2) + 2/21*A*b^2*x^(21/2) + 2/15*B*a^2*x^(15/2) + 4/15*A*a*b*x^(15/2) +
2/9*A*a^2*x^(9/2)