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

Optimal. Leaf size=39 \[ \frac{2}{15} x^{15/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{21} b B x^{21/2} \]

[Out]

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

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Rubi [A]  time = 0.0154295, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{2}{15} x^{15/2} (a B+A b)+\frac{2}{9} a A x^{9/2}+\frac{2}{21} b B x^{21/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0150327, size = 33, normalized size = 0.85 \[ \frac{2}{315} x^{9/2} \left (21 x^3 (a B+A b)+35 a A+15 b B x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(35*a*A + 21*(A*b + a*B)*x^3 + 15*b*B*x^6))/315

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \begin{align*}{\frac{30\,bB{x}^{6}+42\,A{x}^{3}b+42\,B{x}^{3}a+70\,Aa}{315}{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)*(B*x^3+A),x)

[Out]

2/315*x^(9/2)*(15*B*b*x^6+21*A*b*x^3+21*B*a*x^3+35*A*a)

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Maxima [A]  time = 0.956209, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{21} \, B b x^{\frac{21}{2}} + \frac{2}{15} \,{\left (B a + A b\right )} x^{\frac{15}{2}} + \frac{2}{9} \, A a 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)*(B*x^3+A),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.50193, size = 85, normalized size = 2.18 \begin{align*} \frac{2}{315} \,{\left (15 \, B b x^{10} + 21 \,{\left (B a + A b\right )} x^{7} + 35 \, A a 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)*(B*x^3+A),x, algorithm="fricas")

[Out]

2/315*(15*B*b*x^10 + 21*(B*a + A*b)*x^7 + 35*A*a*x^4)*sqrt(x)

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Sympy [A]  time = 35.8397, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A a x^{\frac{9}{2}}}{9} + \frac{2 A b x^{\frac{15}{2}}}{15} + \frac{2 B a x^{\frac{15}{2}}}{15} + \frac{2 B b x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11723, size = 39, normalized size = 1. \begin{align*} \frac{2}{21} \, B b x^{\frac{21}{2}} + \frac{2}{15} \, B a x^{\frac{15}{2}} + \frac{2}{15} \, A b x^{\frac{15}{2}} + \frac{2}{9} \, A a 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)*(B*x^3+A),x, algorithm="giac")

[Out]

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