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

Optimal. Leaf size=85 \[ \frac{2}{15} a^2 x^{15/2} (a B+3 A b)+\frac{2}{9} a^3 A x^{9/2}+\frac{2}{27} b^2 x^{27/2} (3 a B+A b)+\frac{2}{7} a b x^{21/2} (a B+A b)+\frac{2}{33} b^3 B x^{33/2} \]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(15/2))/15 + (2*a*b*(A*b + a*B)*x^(21/2))/7 + (2*b^2*(A*b + 3*a*B
)*x^(27/2))/27 + (2*b^3*B*x^(33/2))/33

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Rubi [A]  time = 0.0436307, 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} \[ \frac{2}{15} a^2 x^{15/2} (a B+3 A b)+\frac{2}{9} a^3 A x^{9/2}+\frac{2}{27} b^2 x^{27/2} (3 a B+A b)+\frac{2}{7} a b x^{21/2} (a B+A b)+\frac{2}{33} b^3 B x^{33/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(15/2))/15 + (2*a*b*(A*b + a*B)*x^(21/2))/7 + (2*b^2*(A*b + 3*a*B
)*x^(27/2))/27 + (2*b^3*B*x^(33/2))/33

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 )^3 \left (A+B x^3\right ) \, dx &=\int \left (a^3 A x^{7/2}+a^2 (3 A b+a B) x^{13/2}+3 a b (A b+a B) x^{19/2}+b^2 (A b+3 a B) x^{25/2}+b^3 B x^{31/2}\right ) \, dx\\ &=\frac{2}{9} a^3 A x^{9/2}+\frac{2}{15} a^2 (3 A b+a B) x^{15/2}+\frac{2}{7} a b (A b+a B) x^{21/2}+\frac{2}{27} b^2 (A b+3 a B) x^{27/2}+\frac{2}{33} b^3 B x^{33/2}\\ \end{align*}

Mathematica [A]  time = 0.0414161, size = 71, normalized size = 0.84 \[ \frac{2 x^{9/2} \left (693 a^2 x^3 (a B+3 A b)+1155 a^3 A+385 b^2 x^9 (3 a B+A b)+1485 a b x^6 (a B+A b)+315 b^3 B x^{12}\right )}{10395} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(1155*a^3*A + 693*a^2*(3*A*b + a*B)*x^3 + 1485*a*b*(A*b + a*B)*x^6 + 385*b^2*(A*b + 3*a*B)*x^9 + 31
5*b^3*B*x^12))/10395

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Maple [A]  time = 0.004, size = 80, normalized size = 0.9 \begin{align*}{\frac{630\,B{b}^{3}{x}^{12}+770\,{x}^{9}A{b}^{3}+2310\,{x}^{9}Ba{b}^{2}+2970\,{x}^{6}Aa{b}^{2}+2970\,{x}^{6}B{a}^{2}b+4158\,{x}^{3}A{a}^{2}b+1386\,{x}^{3}B{a}^{3}+2310\,A{a}^{3}}{10395}{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)^3*(B*x^3+A),x)

[Out]

2/10395*x^(9/2)*(315*B*b^3*x^12+385*A*b^3*x^9+1155*B*a*b^2*x^9+1485*A*a*b^2*x^6+1485*B*a^2*b*x^6+2079*A*a^2*b*
x^3+693*B*a^3*x^3+1155*A*a^3)

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Maxima [A]  time = 0.945578, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{33} \, B b^{3} x^{\frac{33}{2}} + \frac{2}{27} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{27}{2}} + \frac{2}{7} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{21}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} + \frac{2}{15} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{15}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/33*B*b^3*x^(33/2) + 2/27*(3*B*a*b^2 + A*b^3)*x^(27/2) + 2/7*(B*a^2*b + A*a*b^2)*x^(21/2) + 2/9*A*a^3*x^(9/2)
+ 2/15*(B*a^3 + 3*A*a^2*b)*x^(15/2)

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Fricas [A]  time = 1.62614, size = 194, normalized size = 2.28 \begin{align*} \frac{2}{10395} \,{\left (315 \, B b^{3} x^{16} + 385 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{13} + 1485 \,{\left (B a^{2} b + A a b^{2}\right )} x^{10} + 1155 \, A a^{3} x^{4} + 693 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{7}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/10395*(315*B*b^3*x^16 + 385*(3*B*a*b^2 + A*b^3)*x^13 + 1485*(B*a^2*b + A*a*b^2)*x^10 + 1155*A*a^3*x^4 + 693*
(B*a^3 + 3*A*a^2*b)*x^7)*sqrt(x)

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Sympy [A]  time = 114.892, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A a^{3} x^{\frac{9}{2}}}{9} + \frac{2 A a^{2} b x^{\frac{15}{2}}}{5} + \frac{2 A a b^{2} x^{\frac{21}{2}}}{7} + \frac{2 A b^{3} x^{\frac{27}{2}}}{27} + \frac{2 B a^{3} x^{\frac{15}{2}}}{15} + \frac{2 B a^{2} b x^{\frac{21}{2}}}{7} + \frac{2 B a b^{2} x^{\frac{27}{2}}}{9} + \frac{2 B b^{3} x^{\frac{33}{2}}}{33} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**3*x**(9/2)/9 + 2*A*a**2*b*x**(15/2)/5 + 2*A*a*b**2*x**(21/2)/7 + 2*A*b**3*x**(27/2)/27 + 2*B*a**3*x**(1
5/2)/15 + 2*B*a**2*b*x**(21/2)/7 + 2*B*a*b**2*x**(27/2)/9 + 2*B*b**3*x**(33/2)/33

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Giac [A]  time = 1.13504, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{33} \, B b^{3} x^{\frac{33}{2}} + \frac{2}{9} \, B a b^{2} x^{\frac{27}{2}} + \frac{2}{27} \, A b^{3} x^{\frac{27}{2}} + \frac{2}{7} \, B a^{2} b x^{\frac{21}{2}} + \frac{2}{7} \, A a b^{2} x^{\frac{21}{2}} + \frac{2}{15} \, B a^{3} x^{\frac{15}{2}} + \frac{2}{5} \, A a^{2} b x^{\frac{15}{2}} + \frac{2}{9} \, A a^{3} 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)^3*(B*x^3+A),x, algorithm="giac")

[Out]

2/33*B*b^3*x^(33/2) + 2/9*B*a*b^2*x^(27/2) + 2/27*A*b^3*x^(27/2) + 2/7*B*a^2*b*x^(21/2) + 2/7*A*a*b^2*x^(21/2)
+ 2/15*B*a^3*x^(15/2) + 2/5*A*a^2*b*x^(15/2) + 2/9*A*a^3*x^(9/2)