3.1.44 \(\int \frac {(a+b x^3)^5 (A+B x^3)}{x^{12}} \, dx\)

Optimal. Leaf size=109 \[ -\frac {a^5 A}{11 x^{11}}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {a^3 b (a B+2 A b)}{x^5}-\frac {5 a^2 b^2 (a B+A b)}{x^2}+\frac {1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac {1}{7} b^5 B x^7 \]

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Rubi [A]  time = 0.06, antiderivative size = 109, 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 {5 a^2 b^2 (a B+A b)}{x^2}-\frac {a^3 b (a B+2 A b)}{x^5}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {a^5 A}{11 x^{11}}+\frac {1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac {1}{7} b^5 B x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^12,x]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (5*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)/4 + (b^5*B*x^7)/7

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^{12}} \, dx &=\int \left (5 a b^3 (A b+2 a B)+\frac {a^5 A}{x^{12}}+\frac {a^4 (5 A b+a B)}{x^9}+\frac {5 a^3 b (2 A b+a B)}{x^6}+\frac {10 a^2 b^2 (A b+a B)}{x^3}+b^4 (A b+5 a B) x^3+b^5 B x^6\right ) \, dx\\ &=-\frac {a^5 A}{11 x^{11}}-\frac {a^4 (5 A b+a B)}{8 x^8}-\frac {a^3 b (2 A b+a B)}{x^5}-\frac {5 a^2 b^2 (A b+a B)}{x^2}+5 a b^3 (A b+2 a B) x+\frac {1}{4} b^4 (A b+5 a B) x^4+\frac {1}{7} b^5 B x^7\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 109, normalized size = 1.00 \begin {gather*} -\frac {a^5 A}{11 x^{11}}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {a^3 b (a B+2 A b)}{x^5}-\frac {5 a^2 b^2 (a B+A b)}{x^2}+\frac {1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac {1}{7} b^5 B x^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^12,x]

[Out]

-1/11*(a^5*A)/x^11 - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (5*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)/4 + (b^5*B*x^7)/7

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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^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^12,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^12, x]

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fricas [A]  time = 0.79, size = 121, normalized size = 1.11 \begin {gather*} \frac {88 \, B b^{5} x^{18} + 154 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 3080 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 3080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 616 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 56 \, A a^{5} - 77 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="fricas")

[Out]

1/616*(88*B*b^5*x^18 + 154*(5*B*a*b^4 + A*b^5)*x^15 + 3080*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 3080*(B*a^3*b^2 + A*
a^2*b^3)*x^9 - 616*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 56*A*a^5 - 77*(B*a^5 + 5*A*a^4*b)*x^3)/x^11

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giac [A]  time = 0.18, size = 124, normalized size = 1.14 \begin {gather*} \frac {1}{7} \, B b^{5} x^{7} + \frac {5}{4} \, B a b^{4} x^{4} + \frac {1}{4} \, A b^{5} x^{4} + 10 \, B a^{2} b^{3} x + 5 \, A a b^{4} x - \frac {440 \, B a^{3} b^{2} x^{9} + 440 \, A a^{2} b^{3} x^{9} + 88 \, B a^{4} b x^{6} + 176 \, A a^{3} b^{2} x^{6} + 11 \, B a^{5} x^{3} + 55 \, A a^{4} b x^{3} + 8 \, A a^{5}}{88 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="giac")

[Out]

1/7*B*b^5*x^7 + 5/4*B*a*b^4*x^4 + 1/4*A*b^5*x^4 + 10*B*a^2*b^3*x + 5*A*a*b^4*x - 1/88*(440*B*a^3*b^2*x^9 + 440
*A*a^2*b^3*x^9 + 88*B*a^4*b*x^6 + 176*A*a^3*b^2*x^6 + 11*B*a^5*x^3 + 55*A*a^4*b*x^3 + 8*A*a^5)/x^11

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maple [A]  time = 0.05, size = 108, normalized size = 0.99 \begin {gather*} \frac {B \,b^{5} x^{7}}{7}+\frac {A \,b^{5} x^{4}}{4}+\frac {5 B a \,b^{4} x^{4}}{4}+5 A a \,b^{4} x +10 B \,a^{2} b^{3} x -\frac {5 \left (A b +B a \right ) a^{2} b^{2}}{x^{2}}-\frac {\left (2 A b +B a \right ) a^{3} b}{x^{5}}-\frac {\left (5 A b +B a \right ) a^{4}}{8 x^{8}}-\frac {A \,a^{5}}{11 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^12,x)

[Out]

1/7*b^5*B*x^7+1/4*A*x^4*b^5+5/4*B*x^4*a*b^4+5*a*b^4*A*x+10*a^2*b^3*B*x-a^3*b*(2*A*b+B*a)/x^5-1/8*a^4*(5*A*b+B*
a)/x^8-5*a^2*b^2*(A*b+B*a)/x^2-1/11*a^5*A/x^11

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maxima [A]  time = 0.53, size = 120, normalized size = 1.10 \begin {gather*} \frac {1}{7} \, B b^{5} x^{7} + \frac {1}{4} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{4} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x - \frac {440 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 88 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 8 \, A a^{5} + 11 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{88 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="maxima")

[Out]

1/7*B*b^5*x^7 + 1/4*(5*B*a*b^4 + A*b^5)*x^4 + 5*(2*B*a^2*b^3 + A*a*b^4)*x - 1/88*(440*(B*a^3*b^2 + A*a^2*b^3)*
x^9 + 88*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 8*A*a^5 + 11*(B*a^5 + 5*A*a^4*b)*x^3)/x^11

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mupad [B]  time = 0.07, size = 116, normalized size = 1.06 \begin {gather*} x^4\,\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )-\frac {\frac {A\,a^5}{11}+x^6\,\left (B\,a^4\,b+2\,A\,a^3\,b^2\right )+x^3\,\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )+x^9\,\left (5\,B\,a^3\,b^2+5\,A\,a^2\,b^3\right )}{x^{11}}+\frac {B\,b^5\,x^7}{7}+5\,a\,b^3\,x\,\left (A\,b+2\,B\,a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^5)/x^12,x)

[Out]

x^4*((A*b^5)/4 + (5*B*a*b^4)/4) - ((A*a^5)/11 + x^6*(2*A*a^3*b^2 + B*a^4*b) + x^3*((B*a^5)/8 + (5*A*a^4*b)/8)
+ x^9*(5*A*a^2*b^3 + 5*B*a^3*b^2))/x^11 + (B*b^5*x^7)/7 + 5*a*b^3*x*(A*b + 2*B*a)

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sympy [A]  time = 3.73, size = 131, normalized size = 1.20 \begin {gather*} \frac {B b^{5} x^{7}}{7} + x^{4} \left (\frac {A b^{5}}{4} + \frac {5 B a b^{4}}{4}\right ) + x \left (5 A a b^{4} + 10 B a^{2} b^{3}\right ) + \frac {- 8 A a^{5} + x^{9} \left (- 440 A a^{2} b^{3} - 440 B a^{3} b^{2}\right ) + x^{6} \left (- 176 A a^{3} b^{2} - 88 B a^{4} b\right ) + x^{3} \left (- 55 A a^{4} b - 11 B a^{5}\right )}{88 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**12,x)

[Out]

B*b**5*x**7/7 + x**4*(A*b**5/4 + 5*B*a*b**4/4) + x*(5*A*a*b**4 + 10*B*a**2*b**3) + (-8*A*a**5 + x**9*(-440*A*a
**2*b**3 - 440*B*a**3*b**2) + x**6*(-176*A*a**3*b**2 - 88*B*a**4*b) + x**3*(-55*A*a**4*b - 11*B*a**5))/(88*x**
11)

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