3.1.25 \(\int x^7 (a+b x^3)^5 (A+B x^3) \, dx\)

Optimal. Leaf size=117 \[ \frac {1}{8} a^5 A x^8+\frac {1}{11} a^4 x^{11} (a B+5 A b)+\frac {5}{14} a^3 b x^{14} (a B+2 A b)+\frac {10}{17} a^2 b^2 x^{17} (a B+A b)+\frac {1}{23} b^4 x^{23} (5 a B+A b)+\frac {1}{4} a b^3 x^{20} (2 a B+A b)+\frac {1}{26} b^5 B x^{26} \]

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Rubi [A]  time = 0.07, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \begin {gather*} \frac {10}{17} a^2 b^2 x^{17} (a B+A b)+\frac {5}{14} a^3 b x^{14} (a B+2 A b)+\frac {1}{11} a^4 x^{11} (a B+5 A b)+\frac {1}{8} a^5 A x^8+\frac {1}{23} b^4 x^{23} (5 a B+A b)+\frac {1}{4} a b^3 x^{20} (2 a B+A b)+\frac {1}{26} b^5 B x^{26} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (10*a^2*b^2*(A*b + a*B)*x^17)/
17 + (a*b^3*(A*b + 2*a*B)*x^20)/4 + (b^4*(A*b + 5*a*B)*x^23)/23 + (b^5*B*x^26)/26

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 \left (a+b x^3\right )^5 \left (A+B x^3\right ) \, dx &=\int \left (a^5 A x^7+a^4 (5 A b+a B) x^{10}+5 a^3 b (2 A b+a B) x^{13}+10 a^2 b^2 (A b+a B) x^{16}+5 a b^3 (A b+2 a B) x^{19}+b^4 (A b+5 a B) x^{22}+b^5 B x^{25}\right ) \, dx\\ &=\frac {1}{8} a^5 A x^8+\frac {1}{11} a^4 (5 A b+a B) x^{11}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {10}{17} a^2 b^2 (A b+a B) x^{17}+\frac {1}{4} a b^3 (A b+2 a B) x^{20}+\frac {1}{23} b^4 (A b+5 a B) x^{23}+\frac {1}{26} b^5 B x^{26}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 117, normalized size = 1.00 \begin {gather*} \frac {1}{8} a^5 A x^8+\frac {1}{11} a^4 x^{11} (a B+5 A b)+\frac {5}{14} a^3 b x^{14} (a B+2 A b)+\frac {10}{17} a^2 b^2 x^{17} (a B+A b)+\frac {1}{23} b^4 x^{23} (5 a B+A b)+\frac {1}{4} a b^3 x^{20} (2 a B+A b)+\frac {1}{26} b^5 B x^{26} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

(a^5*A*x^8)/8 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (10*a^2*b^2*(A*b + a*B)*x^17)/
17 + (a*b^3*(A*b + 2*a*B)*x^20)/4 + (b^4*(A*b + 5*a*B)*x^23)/23 + (b^5*B*x^26)/26

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^7 \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^7*(a + b*x^3)^5*(A + B*x^3),x]

[Out]

IntegrateAlgebraic[x^7*(a + b*x^3)^5*(A + B*x^3), x]

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fricas [A]  time = 0.69, size = 125, normalized size = 1.07 \begin {gather*} \frac {1}{26} x^{26} b^{5} B + \frac {5}{23} x^{23} b^{4} a B + \frac {1}{23} x^{23} b^{5} A + \frac {1}{2} x^{20} b^{3} a^{2} B + \frac {1}{4} x^{20} b^{4} a A + \frac {10}{17} x^{17} b^{2} a^{3} B + \frac {10}{17} x^{17} b^{3} a^{2} A + \frac {5}{14} x^{14} b a^{4} B + \frac {5}{7} x^{14} b^{2} a^{3} A + \frac {1}{11} x^{11} a^{5} B + \frac {5}{11} x^{11} b a^{4} A + \frac {1}{8} x^{8} a^{5} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^3+a)^5*(B*x^3+A),x, algorithm="fricas")

[Out]

1/26*x^26*b^5*B + 5/23*x^23*b^4*a*B + 1/23*x^23*b^5*A + 1/2*x^20*b^3*a^2*B + 1/4*x^20*b^4*a*A + 10/17*x^17*b^2
*a^3*B + 10/17*x^17*b^3*a^2*A + 5/14*x^14*b*a^4*B + 5/7*x^14*b^2*a^3*A + 1/11*x^11*a^5*B + 5/11*x^11*b*a^4*A +
1/8*x^8*a^5*A

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giac [A]  time = 0.15, size = 125, normalized size = 1.07 \begin {gather*} \frac {1}{26} \, B b^{5} x^{26} + \frac {5}{23} \, B a b^{4} x^{23} + \frac {1}{23} \, A b^{5} x^{23} + \frac {1}{2} \, B a^{2} b^{3} x^{20} + \frac {1}{4} \, A a b^{4} x^{20} + \frac {10}{17} \, B a^{3} b^{2} x^{17} + \frac {10}{17} \, A a^{2} b^{3} x^{17} + \frac {5}{14} \, B a^{4} b x^{14} + \frac {5}{7} \, A a^{3} b^{2} x^{14} + \frac {1}{11} \, B a^{5} x^{11} + \frac {5}{11} \, A a^{4} b x^{11} + \frac {1}{8} \, A a^{5} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^3+a)^5*(B*x^3+A),x, algorithm="giac")

[Out]

1/26*B*b^5*x^26 + 5/23*B*a*b^4*x^23 + 1/23*A*b^5*x^23 + 1/2*B*a^2*b^3*x^20 + 1/4*A*a*b^4*x^20 + 10/17*B*a^3*b^
2*x^17 + 10/17*A*a^2*b^3*x^17 + 5/14*B*a^4*b*x^14 + 5/7*A*a^3*b^2*x^14 + 1/11*B*a^5*x^11 + 5/11*A*a^4*b*x^11 +
1/8*A*a^5*x^8

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maple [A]  time = 0.04, size = 124, normalized size = 1.06 \begin {gather*} \frac {B \,b^{5} x^{26}}{26}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{23}}{23}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{20}}{20}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{17}}{17}+\frac {A \,a^{5} x^{8}}{8}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{14}}{14}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{11}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(b*x^3+a)^5*(B*x^3+A),x)

[Out]

1/26*b^5*B*x^26+1/23*(A*b^5+5*B*a*b^4)*x^23+1/20*(5*A*a*b^4+10*B*a^2*b^3)*x^20+1/17*(10*A*a^2*b^3+10*B*a^3*b^2
)*x^17+1/14*(10*A*a^3*b^2+5*B*a^4*b)*x^14+1/11*(5*A*a^4*b+B*a^5)*x^11+1/8*a^5*A*x^8

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maxima [A]  time = 0.47, size = 119, normalized size = 1.02 \begin {gather*} \frac {1}{26} \, B b^{5} x^{26} + \frac {1}{23} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{23} + \frac {1}{4} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{20} + \frac {10}{17} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{17} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{8} \, A a^{5} x^{8} + \frac {1}{11} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x^3+a)^5*(B*x^3+A),x, algorithm="maxima")

[Out]

1/26*B*b^5*x^26 + 1/23*(5*B*a*b^4 + A*b^5)*x^23 + 1/4*(2*B*a^2*b^3 + A*a*b^4)*x^20 + 10/17*(B*a^3*b^2 + A*a^2*
b^3)*x^17 + 5/14*(B*a^4*b + 2*A*a^3*b^2)*x^14 + 1/8*A*a^5*x^8 + 1/11*(B*a^5 + 5*A*a^4*b)*x^11

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mupad [B]  time = 0.04, size = 107, normalized size = 0.91 \begin {gather*} x^{11}\,\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )+x^{23}\,\left (\frac {A\,b^5}{23}+\frac {5\,B\,a\,b^4}{23}\right )+\frac {A\,a^5\,x^8}{8}+\frac {B\,b^5\,x^{26}}{26}+\frac {10\,a^2\,b^2\,x^{17}\,\left (A\,b+B\,a\right )}{17}+\frac {5\,a^3\,b\,x^{14}\,\left (2\,A\,b+B\,a\right )}{14}+\frac {a\,b^3\,x^{20}\,\left (A\,b+2\,B\,a\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(A + B*x^3)*(a + b*x^3)^5,x)

[Out]

x^11*((B*a^5)/11 + (5*A*a^4*b)/11) + x^23*((A*b^5)/23 + (5*B*a*b^4)/23) + (A*a^5*x^8)/8 + (B*b^5*x^26)/26 + (1
0*a^2*b^2*x^17*(A*b + B*a))/17 + (5*a^3*b*x^14*(2*A*b + B*a))/14 + (a*b^3*x^20*(A*b + 2*B*a))/4

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sympy [A]  time = 0.10, size = 134, normalized size = 1.15 \begin {gather*} \frac {A a^{5} x^{8}}{8} + \frac {B b^{5} x^{26}}{26} + x^{23} \left (\frac {A b^{5}}{23} + \frac {5 B a b^{4}}{23}\right ) + x^{20} \left (\frac {A a b^{4}}{4} + \frac {B a^{2} b^{3}}{2}\right ) + x^{17} \left (\frac {10 A a^{2} b^{3}}{17} + \frac {10 B a^{3} b^{2}}{17}\right ) + x^{14} \left (\frac {5 A a^{3} b^{2}}{7} + \frac {5 B a^{4} b}{14}\right ) + x^{11} \left (\frac {5 A a^{4} b}{11} + \frac {B a^{5}}{11}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(b*x**3+a)**5*(B*x**3+A),x)

[Out]

A*a**5*x**8/8 + B*b**5*x**26/26 + x**23*(A*b**5/23 + 5*B*a*b**4/23) + x**20*(A*a*b**4/4 + B*a**2*b**3/2) + x**
17*(10*A*a**2*b**3/17 + 10*B*a**3*b**2/17) + x**14*(5*A*a**3*b**2/7 + 5*B*a**4*b/14) + x**11*(5*A*a**4*b/11 +
B*a**5/11)

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