3.1.43 \(\int \frac {(a+b x^3)^5 (A+B x^3)}{x^{11}} \, dx\)

Optimal. Leaf size=115 \[ -\frac {a^5 A}{10 x^{10}}-\frac {a^4 (a B+5 A b)}{7 x^7}-\frac {5 a^3 b (a B+2 A b)}{4 x^4}-\frac {10 a^2 b^2 (a B+A b)}{x}+\frac {1}{5} b^4 x^5 (5 a B+A b)+\frac {5}{2} a b^3 x^2 (2 a B+A b)+\frac {1}{8} b^5 B x^8 \]

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Rubi [A]  time = 0.06, antiderivative size = 115, 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 a^2 b^2 (a B+A b)}{x}-\frac {5 a^3 b (a B+2 A b)}{4 x^4}-\frac {a^4 (a B+5 A b)}{7 x^7}-\frac {a^5 A}{10 x^{10}}+\frac {1}{5} b^4 x^5 (5 a B+A b)+\frac {5}{2} a b^3 x^2 (2 a B+A b)+\frac {1}{8} b^5 B x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^11,x]

[Out]

-(a^5*A)/(10*x^10) - (a^4*(5*A*b + a*B))/(7*x^7) - (5*a^3*b*(2*A*b + a*B))/(4*x^4) - (10*a^2*b^2*(A*b + a*B))/
x + (5*a*b^3*(A*b + 2*a*B)*x^2)/2 + (b^4*(A*b + 5*a*B)*x^5)/5 + (b^5*B*x^8)/8

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 \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx &=\int \left (\frac {a^5 A}{x^{11}}+\frac {a^4 (5 A b+a B)}{x^8}+\frac {5 a^3 b (2 A b+a B)}{x^5}+\frac {10 a^2 b^2 (A b+a B)}{x^2}+5 a b^3 (A b+2 a B) x+b^4 (A b+5 a B) x^4+b^5 B x^7\right ) \, dx\\ &=-\frac {a^5 A}{10 x^{10}}-\frac {a^4 (5 A b+a B)}{7 x^7}-\frac {5 a^3 b (2 A b+a B)}{4 x^4}-\frac {10 a^2 b^2 (A b+a B)}{x}+\frac {5}{2} a b^3 (A b+2 a B) x^2+\frac {1}{5} b^4 (A b+5 a B) x^5+\frac {1}{8} b^5 B x^8\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 118, normalized size = 1.03 \begin {gather*} \frac {-4 a^5 \left (7 A+10 B x^3\right )-50 a^4 b x^3 \left (4 A+7 B x^3\right )-700 a^3 b^2 x^6 \left (A+4 B x^3\right )+1400 a^2 b^3 x^9 \left (B x^3-2 A\right )+140 a b^4 x^{12} \left (5 A+2 B x^3\right )+7 b^5 x^{15} \left (8 A+5 B x^3\right )}{280 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^11,x]

[Out]

(1400*a^2*b^3*x^9*(-2*A + B*x^3) + 140*a*b^4*x^12*(5*A + 2*B*x^3) - 700*a^3*b^2*x^6*(A + 4*B*x^3) + 7*b^5*x^15
*(8*A + 5*B*x^3) - 50*a^4*b*x^3*(4*A + 7*B*x^3) - 4*a^5*(7*A + 10*B*x^3))/(280*x^10)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^11,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^11, x]

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fricas [A]  time = 1.05, size = 121, normalized size = 1.05 \begin {gather*} \frac {35 \, B b^{5} x^{18} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 700 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 2800 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 350 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 28 \, A a^{5} - 40 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{280 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^11,x, algorithm="fricas")

[Out]

1/280*(35*B*b^5*x^18 + 56*(5*B*a*b^4 + A*b^5)*x^15 + 700*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 2800*(B*a^3*b^2 + A*a^
2*b^3)*x^9 - 350*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 28*A*a^5 - 40*(B*a^5 + 5*A*a^4*b)*x^3)/x^10

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giac [A]  time = 0.15, size = 127, normalized size = 1.10 \begin {gather*} \frac {1}{8} \, B b^{5} x^{8} + B a b^{4} x^{5} + \frac {1}{5} \, A b^{5} x^{5} + 5 \, B a^{2} b^{3} x^{2} + \frac {5}{2} \, A a b^{4} x^{2} - \frac {1400 \, B a^{3} b^{2} x^{9} + 1400 \, A a^{2} b^{3} x^{9} + 175 \, B a^{4} b x^{6} + 350 \, A a^{3} b^{2} x^{6} + 20 \, B a^{5} x^{3} + 100 \, A a^{4} b x^{3} + 14 \, A a^{5}}{140 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^11,x, algorithm="giac")

[Out]

1/8*B*b^5*x^8 + B*a*b^4*x^5 + 1/5*A*b^5*x^5 + 5*B*a^2*b^3*x^2 + 5/2*A*a*b^4*x^2 - 1/140*(1400*B*a^3*b^2*x^9 +
1400*A*a^2*b^3*x^9 + 175*B*a^4*b*x^6 + 350*A*a^3*b^2*x^6 + 20*B*a^5*x^3 + 100*A*a^4*b*x^3 + 14*A*a^5)/x^10

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maple [A]  time = 0.05, size = 111, normalized size = 0.97 \begin {gather*} \frac {B \,b^{5} x^{8}}{8}+\frac {A \,b^{5} x^{5}}{5}+B a \,b^{4} x^{5}+\frac {5 A a \,b^{4} x^{2}}{2}+5 B \,a^{2} b^{3} x^{2}-\frac {10 \left (A b +B a \right ) a^{2} b^{2}}{x}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{4 x^{4}}-\frac {\left (5 A b +B a \right ) a^{4}}{7 x^{7}}-\frac {A \,a^{5}}{10 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^11,x)

[Out]

1/8*b^5*B*x^8+1/5*A*x^5*b^5+B*x^5*a*b^4+5/2*A*x^2*a*b^4+5*B*x^2*a^2*b^3-5/4*a^3*b*(2*A*b+B*a)/x^4-1/7*a^4*(5*A
*b+B*a)/x^7-10*a^2*b^2*(A*b+B*a)/x-1/10*a^5*A/x^10

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maxima [A]  time = 0.47, size = 122, normalized size = 1.06 \begin {gather*} \frac {1}{8} \, B b^{5} x^{8} + \frac {1}{5} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{2} - \frac {1400 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 175 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 14 \, A a^{5} + 20 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{140 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^11,x, algorithm="maxima")

[Out]

1/8*B*b^5*x^8 + 1/5*(5*B*a*b^4 + A*b^5)*x^5 + 5/2*(2*B*a^2*b^3 + A*a*b^4)*x^2 - 1/140*(1400*(B*a^3*b^2 + A*a^2
*b^3)*x^9 + 175*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 14*A*a^5 + 20*(B*a^5 + 5*A*a^4*b)*x^3)/x^10

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mupad [B]  time = 2.36, size = 118, normalized size = 1.03 \begin {gather*} x^5\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )-\frac {\frac {A\,a^5}{10}+x^6\,\left (\frac {5\,B\,a^4\,b}{4}+\frac {5\,A\,a^3\,b^2}{2}\right )+x^3\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+x^9\,\left (10\,B\,a^3\,b^2+10\,A\,a^2\,b^3\right )}{x^{10}}+\frac {B\,b^5\,x^8}{8}+\frac {5\,a\,b^3\,x^2\,\left (A\,b+2\,B\,a\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^5)/x^11,x)

[Out]

x^5*((A*b^5)/5 + B*a*b^4) - ((A*a^5)/10 + x^6*((5*A*a^3*b^2)/2 + (5*B*a^4*b)/4) + x^3*((B*a^5)/7 + (5*A*a^4*b)
/7) + x^9*(10*A*a^2*b^3 + 10*B*a^3*b^2))/x^10 + (B*b^5*x^8)/8 + (5*a*b^3*x^2*(A*b + 2*B*a))/2

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sympy [A]  time = 2.93, size = 131, normalized size = 1.14 \begin {gather*} \frac {B b^{5} x^{8}}{8} + x^{5} \left (\frac {A b^{5}}{5} + B a b^{4}\right ) + x^{2} \left (\frac {5 A a b^{4}}{2} + 5 B a^{2} b^{3}\right ) + \frac {- 14 A a^{5} + x^{9} \left (- 1400 A a^{2} b^{3} - 1400 B a^{3} b^{2}\right ) + x^{6} \left (- 350 A a^{3} b^{2} - 175 B a^{4} b\right ) + x^{3} \left (- 100 A a^{4} b - 20 B a^{5}\right )}{140 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**11,x)

[Out]

B*b**5*x**8/8 + x**5*(A*b**5/5 + B*a*b**4) + x**2*(5*A*a*b**4/2 + 5*B*a**2*b**3) + (-14*A*a**5 + x**9*(-1400*A
*a**2*b**3 - 1400*B*a**3*b**2) + x**6*(-350*A*a**3*b**2 - 175*B*a**4*b) + x**3*(-100*A*a**4*b - 20*B*a**5))/(1
40*x**10)

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