3.1.24 \(\int x^8 (a+b x^3)^5 (A+B x^3) \, dx\)

Optimal. Leaf size=95 \[ \frac {a^2 \left (a+b x^3\right )^6 (A b-a B)}{18 b^4}+\frac {\left (a+b x^3\right )^8 (A b-3 a B)}{24 b^4}-\frac {a \left (a+b x^3\right )^7 (2 A b-3 a B)}{21 b^4}+\frac {B \left (a+b x^3\right )^9}{27 b^4} \]

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Rubi [A]  time = 0.24, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 76} \begin {gather*} \frac {a^2 \left (a+b x^3\right )^6 (A b-a B)}{18 b^4}+\frac {\left (a+b x^3\right )^8 (A b-3 a B)}{24 b^4}-\frac {a \left (a+b x^3\right )^7 (2 A b-3 a B)}{21 b^4}+\frac {B \left (a+b x^3\right )^9}{27 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^2*(A*b - a*B)*(a + b*x^3)^6)/(18*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x^3)^7)/(21*b^4) + ((A*b - 3*a*B)*(a + b*
x^3)^8)/(24*b^4) + (B*(a + b*x^3)^9)/(27*b^4)

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^8 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^2 (a+b x)^5 (A+B x) \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a^2 (-A b+a B) (a+b x)^5}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^6}{b^3}+\frac {(A b-3 a B) (a+b x)^7}{b^3}+\frac {B (a+b x)^8}{b^3}\right ) \, dx,x,x^3\right )\\ &=\frac {a^2 (A b-a B) \left (a+b x^3\right )^6}{18 b^4}-\frac {a (2 A b-3 a B) \left (a+b x^3\right )^7}{21 b^4}+\frac {(A b-3 a B) \left (a+b x^3\right )^8}{24 b^4}+\frac {B \left (a+b x^3\right )^9}{27 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 107, normalized size = 1.13 \begin {gather*} \frac {x^9 \left (168 a^5 A+126 a^4 x^3 (a B+5 A b)+504 a^3 b x^6 (a B+2 A b)+840 a^2 b^2 x^9 (a B+A b)+63 b^4 x^{15} (5 a B+A b)+360 a b^3 x^{12} (2 a B+A b)+56 b^5 B x^{18}\right )}{1512} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(x^9*(168*a^5*A + 126*a^4*(5*A*b + a*B)*x^3 + 504*a^3*b*(2*A*b + a*B)*x^6 + 840*a^2*b^2*(A*b + a*B)*x^9 + 360*
a*b^3*(A*b + 2*a*B)*x^12 + 63*b^4*(A*b + 5*a*B)*x^15 + 56*b^5*B*x^18))/1512

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^8 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^8*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

IntegrateAlgebraic[x^8*(a + b*x^3)^5*(A + B*x^3), x]

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fricas [A]  time = 0.73, size = 125, normalized size = 1.32 \begin {gather*} \frac {1}{27} x^{27} b^{5} B + \frac {5}{24} x^{24} b^{4} a B + \frac {1}{24} x^{24} b^{5} A + \frac {10}{21} x^{21} b^{3} a^{2} B + \frac {5}{21} x^{21} b^{4} a A + \frac {5}{9} x^{18} b^{2} a^{3} B + \frac {5}{9} x^{18} b^{3} a^{2} A + \frac {1}{3} x^{15} b a^{4} B + \frac {2}{3} x^{15} b^{2} a^{3} A + \frac {1}{12} x^{12} a^{5} B + \frac {5}{12} x^{12} b a^{4} A + \frac {1}{9} x^{9} a^{5} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^5*(B*x^3+A),x, algorithm="fricas")

[Out]

1/27*x^27*b^5*B + 5/24*x^24*b^4*a*B + 1/24*x^24*b^5*A + 10/21*x^21*b^3*a^2*B + 5/21*x^21*b^4*a*A + 5/9*x^18*b^
2*a^3*B + 5/9*x^18*b^3*a^2*A + 1/3*x^15*b*a^4*B + 2/3*x^15*b^2*a^3*A + 1/12*x^12*a^5*B + 5/12*x^12*b*a^4*A + 1
/9*x^9*a^5*A

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giac [A]  time = 0.17, size = 125, normalized size = 1.32 \begin {gather*} \frac {1}{27} \, B b^{5} x^{27} + \frac {5}{24} \, B a b^{4} x^{24} + \frac {1}{24} \, A b^{5} x^{24} + \frac {10}{21} \, B a^{2} b^{3} x^{21} + \frac {5}{21} \, A a b^{4} x^{21} + \frac {5}{9} \, B a^{3} b^{2} x^{18} + \frac {5}{9} \, A a^{2} b^{3} x^{18} + \frac {1}{3} \, B a^{4} b x^{15} + \frac {2}{3} \, A a^{3} b^{2} x^{15} + \frac {1}{12} \, B a^{5} x^{12} + \frac {5}{12} \, A a^{4} b x^{12} + \frac {1}{9} \, A a^{5} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^5*(B*x^3+A),x, algorithm="giac")

[Out]

1/27*B*b^5*x^27 + 5/24*B*a*b^4*x^24 + 1/24*A*b^5*x^24 + 10/21*B*a^2*b^3*x^21 + 5/21*A*a*b^4*x^21 + 5/9*B*a^3*b
^2*x^18 + 5/9*A*a^2*b^3*x^18 + 1/3*B*a^4*b*x^15 + 2/3*A*a^3*b^2*x^15 + 1/12*B*a^5*x^12 + 5/12*A*a^4*b*x^12 + 1
/9*A*a^5*x^9

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maple [A]  time = 0.04, size = 124, normalized size = 1.31 \begin {gather*} \frac {B \,b^{5} x^{27}}{27}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{24}}{24}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{21}}{21}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{18}}{18}+\frac {A \,a^{5} x^{9}}{9}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{15}}{15}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{12}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x^3+a)^5*(B*x^3+A),x)

[Out]

1/27*b^5*B*x^27+1/24*(A*b^5+5*B*a*b^4)*x^24+1/21*(5*A*a*b^4+10*B*a^2*b^3)*x^21+1/18*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^18+1/15*(10*A*a^3*b^2+5*B*a^4*b)*x^15+1/12*(5*A*a^4*b+B*a^5)*x^12+1/9*a^5*A*x^9

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maxima [A]  time = 0.47, size = 119, normalized size = 1.25 \begin {gather*} \frac {1}{27} \, B b^{5} x^{27} + \frac {1}{24} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{24} + \frac {5}{21} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{21} + \frac {5}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{18} + \frac {1}{3} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{15} + \frac {1}{9} \, A a^{5} x^{9} + \frac {1}{12} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^5*(B*x^3+A),x, algorithm="maxima")

[Out]

1/27*B*b^5*x^27 + 1/24*(5*B*a*b^4 + A*b^5)*x^24 + 5/21*(2*B*a^2*b^3 + A*a*b^4)*x^21 + 5/9*(B*a^3*b^2 + A*a^2*b
^3)*x^18 + 1/3*(B*a^4*b + 2*A*a^3*b^2)*x^15 + 1/9*A*a^5*x^9 + 1/12*(B*a^5 + 5*A*a^4*b)*x^12

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mupad [B]  time = 2.34, size = 107, normalized size = 1.13 \begin {gather*} x^{12}\,\left (\frac {B\,a^5}{12}+\frac {5\,A\,b\,a^4}{12}\right )+x^{24}\,\left (\frac {A\,b^5}{24}+\frac {5\,B\,a\,b^4}{24}\right )+\frac {A\,a^5\,x^9}{9}+\frac {B\,b^5\,x^{27}}{27}+\frac {5\,a^2\,b^2\,x^{18}\,\left (A\,b+B\,a\right )}{9}+\frac {a^3\,b\,x^{15}\,\left (2\,A\,b+B\,a\right )}{3}+\frac {5\,a\,b^3\,x^{21}\,\left (A\,b+2\,B\,a\right )}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(A + B*x^3)*(a + b*x^3)^5,x)

[Out]

x^12*((B*a^5)/12 + (5*A*a^4*b)/12) + x^24*((A*b^5)/24 + (5*B*a*b^4)/24) + (A*a^5*x^9)/9 + (B*b^5*x^27)/27 + (5
*a^2*b^2*x^18*(A*b + B*a))/9 + (a^3*b*x^15*(2*A*b + B*a))/3 + (5*a*b^3*x^21*(A*b + 2*B*a))/21

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sympy [A]  time = 0.10, size = 136, normalized size = 1.43 \begin {gather*} \frac {A a^{5} x^{9}}{9} + \frac {B b^{5} x^{27}}{27} + x^{24} \left (\frac {A b^{5}}{24} + \frac {5 B a b^{4}}{24}\right ) + x^{21} \left (\frac {5 A a b^{4}}{21} + \frac {10 B a^{2} b^{3}}{21}\right ) + x^{18} \left (\frac {5 A a^{2} b^{3}}{9} + \frac {5 B a^{3} b^{2}}{9}\right ) + x^{15} \left (\frac {2 A a^{3} b^{2}}{3} + \frac {B a^{4} b}{3}\right ) + x^{12} \left (\frac {5 A a^{4} b}{12} + \frac {B a^{5}}{12}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x**3+a)**5*(B*x**3+A),x)

[Out]

A*a**5*x**9/9 + B*b**5*x**27/27 + x**24*(A*b**5/24 + 5*B*a*b**4/24) + x**21*(5*A*a*b**4/21 + 10*B*a**2*b**3/21
) + x**18*(5*A*a**2*b**3/9 + 5*B*a**3*b**2/9) + x**15*(2*A*a**3*b**2/3 + B*a**4*b/3) + x**12*(5*A*a**4*b/12 +
B*a**5/12)

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