3.1.50 \(\int \frac {(a+b x^3)^5 (A+B x^3)}{x^{18}} \, dx\)

Optimal. Leaf size=110 \[ -\frac {a^5 A}{17 x^{17}}-\frac {a^4 (a B+5 A b)}{14 x^{14}}-\frac {5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac {5 a^2 b^2 (a B+A b)}{4 x^8}-\frac {b^4 (5 a B+A b)}{2 x^2}-\frac {a b^3 (2 a B+A b)}{x^5}+b^5 B x \]

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Rubi [A]  time = 0.06, 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 {5 a^2 b^2 (a B+A b)}{4 x^8}-\frac {a^4 (a B+5 A b)}{14 x^{14}}-\frac {5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac {a^5 A}{17 x^{17}}-\frac {a b^3 (2 a B+A b)}{x^5}-\frac {b^4 (5 a B+A b)}{2 x^2}+b^5 B x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B))/(11*x^11) - (5*a^2*b^2*(A*b + a*B
))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(2*x^2) + b^5*B*x

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^{18}} \, dx &=\int \left (b^5 B+\frac {a^5 A}{x^{18}}+\frac {a^4 (5 A b+a B)}{x^{15}}+\frac {5 a^3 b (2 A b+a B)}{x^{12}}+\frac {10 a^2 b^2 (A b+a B)}{x^9}+\frac {5 a b^3 (A b+2 a B)}{x^6}+\frac {b^4 (A b+5 a B)}{x^3}\right ) \, dx\\ &=-\frac {a^5 A}{17 x^{17}}-\frac {a^4 (5 A b+a B)}{14 x^{14}}-\frac {5 a^3 b (2 A b+a B)}{11 x^{11}}-\frac {5 a^2 b^2 (A b+a B)}{4 x^8}-\frac {a b^3 (A b+2 a B)}{x^5}-\frac {b^4 (A b+5 a B)}{2 x^2}+b^5 B x\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 110, normalized size = 1.00 \begin {gather*} -\frac {a^5 A}{17 x^{17}}-\frac {a^4 (a B+5 A b)}{14 x^{14}}-\frac {5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac {5 a^2 b^2 (a B+A b)}{4 x^8}-\frac {b^4 (5 a B+A b)}{2 x^2}-\frac {a b^3 (2 a B+A b)}{x^5}+b^5 B x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^18,x]

[Out]

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

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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^{18}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^18,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^18, x]

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fricas [A]  time = 0.67, size = 121, normalized size = 1.10 \begin {gather*} \frac {5236 \, B b^{5} x^{18} - 2618 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 5236 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 6545 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 2380 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 308 \, A a^{5} - 374 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="fricas")

[Out]

1/5236*(5236*B*b^5*x^18 - 2618*(5*B*a*b^4 + A*b^5)*x^15 - 5236*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 6545*(B*a^3*b^2
+ A*a^2*b^3)*x^9 - 2380*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 308*A*a^5 - 374*(B*a^5 + 5*A*a^4*b)*x^3)/x^17

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giac [A]  time = 0.17, size = 125, normalized size = 1.14 \begin {gather*} B b^{5} x - \frac {13090 \, B a b^{4} x^{15} + 2618 \, A b^{5} x^{15} + 10472 \, B a^{2} b^{3} x^{12} + 5236 \, A a b^{4} x^{12} + 6545 \, B a^{3} b^{2} x^{9} + 6545 \, A a^{2} b^{3} x^{9} + 2380 \, B a^{4} b x^{6} + 4760 \, A a^{3} b^{2} x^{6} + 374 \, B a^{5} x^{3} + 1870 \, A a^{4} b x^{3} + 308 \, A a^{5}}{5236 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="giac")

[Out]

B*b^5*x - 1/5236*(13090*B*a*b^4*x^15 + 2618*A*b^5*x^15 + 10472*B*a^2*b^3*x^12 + 5236*A*a*b^4*x^12 + 6545*B*a^3
*b^2*x^9 + 6545*A*a^2*b^3*x^9 + 2380*B*a^4*b*x^6 + 4760*A*a^3*b^2*x^6 + 374*B*a^5*x^3 + 1870*A*a^4*b*x^3 + 308
*A*a^5)/x^17

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maple [A]  time = 0.05, size = 101, normalized size = 0.92 \begin {gather*} B \,b^{5} x -\frac {\left (A b +5 B a \right ) b^{4}}{2 x^{2}}-\frac {\left (A b +2 B a \right ) a \,b^{3}}{x^{5}}-\frac {5 \left (A b +B a \right ) a^{2} b^{2}}{4 x^{8}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{11 x^{11}}-\frac {\left (5 A b +B a \right ) a^{4}}{14 x^{14}}-\frac {A \,a^{5}}{17 x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^18,x)

[Out]

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

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maxima [A]  time = 0.52, size = 119, normalized size = 1.08 \begin {gather*} B b^{5} x - \frac {2618 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 5236 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 6545 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 2380 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 308 \, A a^{5} + 374 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="maxima")

[Out]

B*b^5*x - 1/5236*(2618*(5*B*a*b^4 + A*b^5)*x^15 + 5236*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 6545*(B*a^3*b^2 + A*a^2*
b^3)*x^9 + 2380*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 308*A*a^5 + 374*(B*a^5 + 5*A*a^4*b)*x^3)/x^17

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mupad [B]  time = 0.08, size = 119, normalized size = 1.08 \begin {gather*} B\,b^5\,x-\frac {\frac {A\,a^5}{17}+x^{12}\,\left (2\,B\,a^2\,b^3+A\,a\,b^4\right )+x^6\,\left (\frac {5\,B\,a^4\,b}{11}+\frac {10\,A\,a^3\,b^2}{11}\right )+x^3\,\left (\frac {B\,a^5}{14}+\frac {5\,A\,b\,a^4}{14}\right )+x^{15}\,\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )+x^9\,\left (\frac {5\,B\,a^3\,b^2}{4}+\frac {5\,A\,a^2\,b^3}{4}\right )}{x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^5)/x^18,x)

[Out]

B*b^5*x - ((A*a^5)/17 + x^12*(2*B*a^2*b^3 + A*a*b^4) + x^6*((10*A*a^3*b^2)/11 + (5*B*a^4*b)/11) + x^3*((B*a^5)
/14 + (5*A*a^4*b)/14) + x^15*((A*b^5)/2 + (5*B*a*b^4)/2) + x^9*((5*A*a^2*b^3)/4 + (5*B*a^3*b^2)/4))/x^17

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**18,x)

[Out]

Timed out

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