3.1.53 \(\int \frac {(a+b x^3)^5 (A+B x^3)}{x^{21}} \, dx\)

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

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Rubi [A]  time = 0.06, 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 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac {a^4 (a B+5 A b)}{17 x^{17}}-\frac {5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac {a^5 A}{20 x^{20}}-\frac {5 a b^3 (2 a B+A b)}{8 x^8}-\frac {b^4 (5 a B+A b)}{5 x^5}-\frac {b^5 B}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^21,x]

[Out]

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

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^{21}} \, dx &=\int \left (\frac {a^5 A}{x^{21}}+\frac {a^4 (5 A b+a B)}{x^{18}}+\frac {5 a^3 b (2 A b+a B)}{x^{15}}+\frac {10 a^2 b^2 (A b+a B)}{x^{12}}+\frac {5 a b^3 (A b+2 a B)}{x^9}+\frac {b^4 (A b+5 a B)}{x^6}+\frac {b^5 B}{x^3}\right ) \, dx\\ &=-\frac {a^5 A}{20 x^{20}}-\frac {a^4 (5 A b+a B)}{17 x^{17}}-\frac {5 a^3 b (2 A b+a B)}{14 x^{14}}-\frac {10 a^2 b^2 (A b+a B)}{11 x^{11}}-\frac {5 a b^3 (A b+2 a B)}{8 x^8}-\frac {b^4 (A b+5 a B)}{5 x^5}-\frac {b^5 B}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 121, normalized size = 1.03 \begin {gather*} -\frac {154 a^5 \left (17 A+20 B x^3\right )+1100 a^4 b x^3 \left (14 A+17 B x^3\right )+3400 a^3 b^2 x^6 \left (11 A+14 B x^3\right )+5950 a^2 b^3 x^9 \left (8 A+11 B x^3\right )+6545 a b^4 x^{12} \left (5 A+8 B x^3\right )+5236 b^5 x^{15} \left (2 A+5 B x^3\right )}{52360 x^{20}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^21,x]

[Out]

-1/52360*(5236*b^5*x^15*(2*A + 5*B*x^3) + 6545*a*b^4*x^12*(5*A + 8*B*x^3) + 5950*a^2*b^3*x^9*(8*A + 11*B*x^3)
+ 3400*a^3*b^2*x^6*(11*A + 14*B*x^3) + 1100*a^4*b*x^3*(14*A + 17*B*x^3) + 154*a^5*(17*A + 20*B*x^3))/x^20

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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^{21}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^21,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^21, x]

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fricas [A]  time = 0.53, size = 121, normalized size = 1.03 \begin {gather*} -\frac {26180 \, B b^{5} x^{18} + 10472 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 32725 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 47600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 18700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 2618 \, A a^{5} + 3080 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{52360 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^21,x, algorithm="fricas")

[Out]

-1/52360*(26180*B*b^5*x^18 + 10472*(5*B*a*b^4 + A*b^5)*x^15 + 32725*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 47600*(B*a^
3*b^2 + A*a^2*b^3)*x^9 + 18700*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 2618*A*a^5 + 3080*(B*a^5 + 5*A*a^4*b)*x^3)/x^20

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giac [A]  time = 0.15, size = 127, normalized size = 1.09 \begin {gather*} -\frac {26180 \, B b^{5} x^{18} + 52360 \, B a b^{4} x^{15} + 10472 \, A b^{5} x^{15} + 65450 \, B a^{2} b^{3} x^{12} + 32725 \, A a b^{4} x^{12} + 47600 \, B a^{3} b^{2} x^{9} + 47600 \, A a^{2} b^{3} x^{9} + 18700 \, B a^{4} b x^{6} + 37400 \, A a^{3} b^{2} x^{6} + 3080 \, B a^{5} x^{3} + 15400 \, A a^{4} b x^{3} + 2618 \, A a^{5}}{52360 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^21,x, algorithm="giac")

[Out]

-1/52360*(26180*B*b^5*x^18 + 52360*B*a*b^4*x^15 + 10472*A*b^5*x^15 + 65450*B*a^2*b^3*x^12 + 32725*A*a*b^4*x^12
+ 47600*B*a^3*b^2*x^9 + 47600*A*a^2*b^3*x^9 + 18700*B*a^4*b*x^6 + 37400*A*a^3*b^2*x^6 + 3080*B*a^5*x^3 + 1540
0*A*a^4*b*x^3 + 2618*A*a^5)/x^20

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maple [A]  time = 0.05, size = 104, normalized size = 0.89 \begin {gather*} -\frac {B \,b^{5}}{2 x^{2}}-\frac {\left (A b +5 B a \right ) b^{4}}{5 x^{5}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{8 x^{8}}-\frac {10 \left (A b +B a \right ) a^{2} b^{2}}{11 x^{11}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{14 x^{14}}-\frac {\left (5 A b +B a \right ) a^{4}}{17 x^{17}}-\frac {A \,a^{5}}{20 x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^21,x)

[Out]

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

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maxima [A]  time = 0.55, size = 121, normalized size = 1.03 \begin {gather*} -\frac {26180 \, B b^{5} x^{18} + 10472 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 32725 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 47600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 18700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 2618 \, A a^{5} + 3080 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{52360 \, x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^21,x, algorithm="maxima")

[Out]

-1/52360*(26180*B*b^5*x^18 + 10472*(5*B*a*b^4 + A*b^5)*x^15 + 32725*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 47600*(B*a^
3*b^2 + A*a^2*b^3)*x^9 + 18700*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 2618*A*a^5 + 3080*(B*a^5 + 5*A*a^4*b)*x^3)/x^20

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mupad [B]  time = 2.35, size = 121, normalized size = 1.03 \begin {gather*} -\frac {\frac {A\,a^5}{20}+x^{12}\,\left (\frac {5\,B\,a^2\,b^3}{4}+\frac {5\,A\,a\,b^4}{8}\right )+x^6\,\left (\frac {5\,B\,a^4\,b}{14}+\frac {5\,A\,a^3\,b^2}{7}\right )+x^3\,\left (\frac {B\,a^5}{17}+\frac {5\,A\,b\,a^4}{17}\right )+x^{15}\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )+x^9\,\left (\frac {10\,B\,a^3\,b^2}{11}+\frac {10\,A\,a^2\,b^3}{11}\right )+\frac {B\,b^5\,x^{18}}{2}}{x^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^5)/x^21,x)

[Out]

-((A*a^5)/20 + x^12*((5*B*a^2*b^3)/4 + (5*A*a*b^4)/8) + x^6*((5*A*a^3*b^2)/7 + (5*B*a^4*b)/14) + x^3*((B*a^5)/
17 + (5*A*a^4*b)/17) + x^15*((A*b^5)/5 + B*a*b^4) + x^9*((10*A*a^2*b^3)/11 + (10*B*a^3*b^2)/11) + (B*b^5*x^18)
/2)/x^20

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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**21,x)

[Out]

Timed out

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