3.1.47 \(\int \frac {(a+b x^3)^5 (A+B x^3)}{x^{15}} \, dx\)

Optimal. Leaf size=110 \[ -\frac {a^5 A}{14 x^{14}}-\frac {a^4 (a B+5 A b)}{11 x^{11}}-\frac {5 a^3 b (a B+2 A b)}{8 x^8}-\frac {2 a^2 b^2 (a B+A b)}{x^5}+b^4 x (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{2 x^2}+\frac {1}{4} b^5 B x^4 \]

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Rubi [A]  time = 0.07, antiderivative size = 110, 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 {2 a^2 b^2 (a B+A b)}{x^5}-\frac {a^4 (a B+5 A b)}{11 x^{11}}-\frac {5 a^3 b (a B+2 A b)}{8 x^8}-\frac {a^5 A}{14 x^{14}}-\frac {5 a b^3 (2 a B+A b)}{2 x^2}+b^4 x (5 a B+A b)+\frac {1}{4} b^5 B x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^15,x]

[Out]

-(a^5*A)/(14*x^14) - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B))/(8*x^8) - (2*a^2*b^2*(A*b + a*B))
/x^5 - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + b^4*(A*b + 5*a*B)*x + (b^5*B*x^4)/4

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^{15}} \, dx &=\int \left (b^4 (A b+5 a B)+\frac {a^5 A}{x^{15}}+\frac {a^4 (5 A b+a B)}{x^{12}}+\frac {5 a^3 b (2 A b+a B)}{x^9}+\frac {10 a^2 b^2 (A b+a B)}{x^6}+\frac {5 a b^3 (A b+2 a B)}{x^3}+b^5 B x^3\right ) \, dx\\ &=-\frac {a^5 A}{14 x^{14}}-\frac {a^4 (5 A b+a B)}{11 x^{11}}-\frac {5 a^3 b (2 A b+a B)}{8 x^8}-\frac {2 a^2 b^2 (A b+a B)}{x^5}-\frac {5 a b^3 (A b+2 a B)}{2 x^2}+b^4 (A b+5 a B) x+\frac {1}{4} b^5 B x^4\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 110, normalized size = 1.00 \begin {gather*} -\frac {a^5 A}{14 x^{14}}-\frac {a^4 (a B+5 A b)}{11 x^{11}}-\frac {5 a^3 b (a B+2 A b)}{8 x^8}-\frac {2 a^2 b^2 (a B+A b)}{x^5}+b^4 x (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{2 x^2}+\frac {1}{4} b^5 B x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^15,x]

[Out]

-1/14*(a^5*A)/x^14 - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B))/(8*x^8) - (2*a^2*b^2*(A*b + a*B))
/x^5 - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + b^4*(A*b + 5*a*B)*x + (b^5*B*x^4)/4

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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^{15}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^15,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^15, x]

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fricas [A]  time = 0.56, size = 121, normalized size = 1.10 \begin {gather*} \frac {154 \, B b^{5} x^{18} + 616 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 1540 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 1232 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 385 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 44 \, A a^{5} - 56 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="fricas")

[Out]

1/616*(154*B*b^5*x^18 + 616*(5*B*a*b^4 + A*b^5)*x^15 - 1540*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 1232*(B*a^3*b^2 + A
*a^2*b^3)*x^9 - 385*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 44*A*a^5 - 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^14

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giac [A]  time = 0.16, size = 123, normalized size = 1.12 \begin {gather*} \frac {1}{4} \, B b^{5} x^{4} + 5 \, B a b^{4} x + A b^{5} x - \frac {3080 \, B a^{2} b^{3} x^{12} + 1540 \, A a b^{4} x^{12} + 1232 \, B a^{3} b^{2} x^{9} + 1232 \, A a^{2} b^{3} x^{9} + 385 \, B a^{4} b x^{6} + 770 \, A a^{3} b^{2} x^{6} + 56 \, B a^{5} x^{3} + 280 \, A a^{4} b x^{3} + 44 \, A a^{5}}{616 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="giac")

[Out]

1/4*B*b^5*x^4 + 5*B*a*b^4*x + A*b^5*x - 1/616*(3080*B*a^2*b^3*x^12 + 1540*A*a*b^4*x^12 + 1232*B*a^3*b^2*x^9 +
1232*A*a^2*b^3*x^9 + 385*B*a^4*b*x^6 + 770*A*a^3*b^2*x^6 + 56*B*a^5*x^3 + 280*A*a^4*b*x^3 + 44*A*a^5)/x^14

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maple [A]  time = 0.05, size = 102, normalized size = 0.93 \begin {gather*} \frac {B \,b^{5} x^{4}}{4}+A \,b^{5} x +5 B a \,b^{4} x -\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{2 x^{2}}-\frac {2 \left (A b +B a \right ) a^{2} b^{2}}{x^{5}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{8 x^{8}}-\frac {\left (5 A b +B a \right ) a^{4}}{11 x^{11}}-\frac {A \,a^{5}}{14 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^15,x)

[Out]

1/4*b^5*B*x^4+b^5*A*x+5*a*b^4*B*x-2*a^2*b^2*(A*b+B*a)/x^5-5/8*a^3*b*(2*A*b+B*a)/x^8-5/2*a*b^3*(A*b+2*B*a)/x^2-
1/14*a^5*A/x^14-1/11*a^4*(5*A*b+B*a)/x^11

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maxima [A]  time = 0.61, size = 119, normalized size = 1.08 \begin {gather*} \frac {1}{4} \, B b^{5} x^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} x - \frac {1540 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 1232 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 385 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 44 \, A a^{5} + 56 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="maxima")

[Out]

1/4*B*b^5*x^4 + (5*B*a*b^4 + A*b^5)*x - 1/616*(1540*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 1232*(B*a^3*b^2 + A*a^2*b^3
)*x^9 + 385*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 44*A*a^5 + 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^14

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mupad [B]  time = 2.39, size = 120, normalized size = 1.09 \begin {gather*} x\,\left (A\,b^5+5\,B\,a\,b^4\right )-\frac {\frac {A\,a^5}{14}+x^{12}\,\left (5\,B\,a^2\,b^3+\frac {5\,A\,a\,b^4}{2}\right )+x^6\,\left (\frac {5\,B\,a^4\,b}{8}+\frac {5\,A\,a^3\,b^2}{4}\right )+x^3\,\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )+x^9\,\left (2\,B\,a^3\,b^2+2\,A\,a^2\,b^3\right )}{x^{14}}+\frac {B\,b^5\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^5)/x^15,x)

[Out]

x*(A*b^5 + 5*B*a*b^4) - ((A*a^5)/14 + x^12*(5*B*a^2*b^3 + (5*A*a*b^4)/2) + x^6*((5*A*a^3*b^2)/4 + (5*B*a^4*b)/
8) + x^3*((B*a^5)/11 + (5*A*a^4*b)/11) + x^9*(2*A*a^2*b^3 + 2*B*a^3*b^2))/x^14 + (B*b^5*x^4)/4

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sympy [A]  time = 22.98, size = 129, normalized size = 1.17 \begin {gather*} \frac {B b^{5} x^{4}}{4} + x \left (A b^{5} + 5 B a b^{4}\right ) + \frac {- 44 A a^{5} + x^{12} \left (- 1540 A a b^{4} - 3080 B a^{2} b^{3}\right ) + x^{9} \left (- 1232 A a^{2} b^{3} - 1232 B a^{3} b^{2}\right ) + x^{6} \left (- 770 A a^{3} b^{2} - 385 B a^{4} b\right ) + x^{3} \left (- 280 A a^{4} b - 56 B a^{5}\right )}{616 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**15,x)

[Out]

B*b**5*x**4/4 + x*(A*b**5 + 5*B*a*b**4) + (-44*A*a**5 + x**12*(-1540*A*a*b**4 - 3080*B*a**2*b**3) + x**9*(-123
2*A*a**2*b**3 - 1232*B*a**3*b**2) + x**6*(-770*A*a**3*b**2 - 385*B*a**4*b) + x**3*(-280*A*a**4*b - 56*B*a**5))
/(616*x**14)

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