3.1.52 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{20}} \, dx\)

Optimal. Leaf size=117 \[ -\frac {a^5 A}{19 x^{19}}-\frac {a^4 (a B+5 A b)}{17 x^{17}}-\frac {a^3 b (a B+2 A b)}{3 x^{15}}-\frac {10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac {b^4 (5 a B+A b)}{9 x^9}-\frac {5 a b^3 (2 a B+A b)}{11 x^{11}}-\frac {b^5 B}{7 x^7} \]

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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)}{13 x^{13}}-\frac {a^4 (a B+5 A b)}{17 x^{17}}-\frac {a^3 b (a B+2 A b)}{3 x^{15}}-\frac {a^5 A}{19 x^{19}}-\frac {5 a b^3 (2 a B+A b)}{11 x^{11}}-\frac {b^4 (5 a B+A b)}{9 x^9}-\frac {b^5 B}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^20,x]

[Out]

-(a^5*A)/(19*x^19) - (a^4*(5*A*b + a*B))/(17*x^17) - (a^3*b*(2*A*b + a*B))/(3*x^15) - (10*a^2*b^2*(A*b + a*B))
/(13*x^13) - (5*a*b^3*(A*b + 2*a*B))/(11*x^11) - (b^4*(A*b + 5*a*B))/(9*x^9) - (b^5*B)/(7*x^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^2\right )^5 \left (A+B x^2\right )}{x^{20}} \, dx &=\int \left (\frac {a^5 A}{x^{20}}+\frac {a^4 (5 A b+a B)}{x^{18}}+\frac {5 a^3 b (2 A b+a B)}{x^{16}}+\frac {10 a^2 b^2 (A b+a B)}{x^{14}}+\frac {5 a b^3 (A b+2 a B)}{x^{12}}+\frac {b^4 (A b+5 a B)}{x^{10}}+\frac {b^5 B}{x^8}\right ) \, dx\\ &=-\frac {a^5 A}{19 x^{19}}-\frac {a^4 (5 A b+a B)}{17 x^{17}}-\frac {a^3 b (2 A b+a B)}{3 x^{15}}-\frac {10 a^2 b^2 (A b+a B)}{13 x^{13}}-\frac {5 a b^3 (A b+2 a B)}{11 x^{11}}-\frac {b^4 (A b+5 a B)}{9 x^9}-\frac {b^5 B}{7 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 117, normalized size = 1.00 \begin {gather*} -\frac {a^5 A}{19 x^{19}}-\frac {a^4 (a B+5 A b)}{17 x^{17}}-\frac {a^3 b (a B+2 A b)}{3 x^{15}}-\frac {10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac {b^4 (5 a B+A b)}{9 x^9}-\frac {5 a b^3 (2 a B+A b)}{11 x^{11}}-\frac {b^5 B}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^20,x]

[Out]

-1/19*(a^5*A)/x^19 - (a^4*(5*A*b + a*B))/(17*x^17) - (a^3*b*(2*A*b + a*B))/(3*x^15) - (10*a^2*b^2*(A*b + a*B))
/(13*x^13) - (5*a*b^3*(A*b + 2*a*B))/(11*x^11) - (b^4*(A*b + 5*a*B))/(9*x^9) - (b^5*B)/(7*x^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^{20}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^20,x]

[Out]

IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^20, x]

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fricas [A]  time = 0.42, size = 121, normalized size = 1.03 \begin {gather*} -\frac {415701 \, B b^{5} x^{12} + 323323 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1322685 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 2238390 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 153153 \, A a^{5} + 969969 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 171171 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{2909907 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^20,x, algorithm="fricas")

[Out]

-1/2909907*(415701*B*b^5*x^12 + 323323*(5*B*a*b^4 + A*b^5)*x^10 + 1322685*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 223839
0*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 153153*A*a^5 + 969969*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 171171*(B*a^5 + 5*A*a^4*b)
*x^2)/x^19

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giac [A]  time = 0.42, size = 127, normalized size = 1.09 \begin {gather*} -\frac {415701 \, B b^{5} x^{12} + 1616615 \, B a b^{4} x^{10} + 323323 \, A b^{5} x^{10} + 2645370 \, B a^{2} b^{3} x^{8} + 1322685 \, A a b^{4} x^{8} + 2238390 \, B a^{3} b^{2} x^{6} + 2238390 \, A a^{2} b^{3} x^{6} + 969969 \, B a^{4} b x^{4} + 1939938 \, A a^{3} b^{2} x^{4} + 171171 \, B a^{5} x^{2} + 855855 \, A a^{4} b x^{2} + 153153 \, A a^{5}}{2909907 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^20,x, algorithm="giac")

[Out]

-1/2909907*(415701*B*b^5*x^12 + 1616615*B*a*b^4*x^10 + 323323*A*b^5*x^10 + 2645370*B*a^2*b^3*x^8 + 1322685*A*a
*b^4*x^8 + 2238390*B*a^3*b^2*x^6 + 2238390*A*a^2*b^3*x^6 + 969969*B*a^4*b*x^4 + 1939938*A*a^3*b^2*x^4 + 171171
*B*a^5*x^2 + 855855*A*a^4*b*x^2 + 153153*A*a^5)/x^19

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maple [A]  time = 0.01, size = 104, normalized size = 0.89 \begin {gather*} -\frac {B \,b^{5}}{7 x^{7}}-\frac {\left (A b +5 B a \right ) b^{4}}{9 x^{9}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{11 x^{11}}-\frac {10 \left (A b +B a \right ) a^{2} b^{2}}{13 x^{13}}-\frac {\left (2 A b +B a \right ) a^{3} b}{3 x^{15}}-\frac {A \,a^{5}}{19 x^{19}}-\frac {\left (5 A b +B a \right ) a^{4}}{17 x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^20,x)

[Out]

-1/19*a^5*A/x^19-1/17*a^4*(5*A*b+B*a)/x^17-1/3*a^3*b*(2*A*b+B*a)/x^15-10/13*a^2*b^2*(A*b+B*a)/x^13-5/11*a*b^3*
(A*b+2*B*a)/x^11-1/9*b^4*(A*b+5*B*a)/x^9-1/7*b^5*B/x^7

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maxima [A]  time = 1.04, size = 121, normalized size = 1.03 \begin {gather*} -\frac {415701 \, B b^{5} x^{12} + 323323 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1322685 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 2238390 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 153153 \, A a^{5} + 969969 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 171171 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{2909907 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^20,x, algorithm="maxima")

[Out]

-1/2909907*(415701*B*b^5*x^12 + 323323*(5*B*a*b^4 + A*b^5)*x^10 + 1322685*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 223839
0*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 153153*A*a^5 + 969969*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 171171*(B*a^5 + 5*A*a^4*b)
*x^2)/x^19

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mupad [B]  time = 0.09, size = 122, normalized size = 1.04 \begin {gather*} -\frac {\frac {A\,a^5}{19}+x^4\,\left (\frac {B\,a^4\,b}{3}+\frac {2\,A\,a^3\,b^2}{3}\right )+x^8\,\left (\frac {10\,B\,a^2\,b^3}{11}+\frac {5\,A\,a\,b^4}{11}\right )+x^2\,\left (\frac {B\,a^5}{17}+\frac {5\,A\,b\,a^4}{17}\right )+x^{10}\,\left (\frac {A\,b^5}{9}+\frac {5\,B\,a\,b^4}{9}\right )+x^6\,\left (\frac {10\,B\,a^3\,b^2}{13}+\frac {10\,A\,a^2\,b^3}{13}\right )+\frac {B\,b^5\,x^{12}}{7}}{x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^20,x)

[Out]

-((A*a^5)/19 + x^4*((2*A*a^3*b^2)/3 + (B*a^4*b)/3) + x^8*((10*B*a^2*b^3)/11 + (5*A*a*b^4)/11) + x^2*((B*a^5)/1
7 + (5*A*a^4*b)/17) + x^10*((A*b^5)/9 + (5*B*a*b^4)/9) + x^6*((10*A*a^2*b^3)/13 + (10*B*a^3*b^2)/13) + (B*b^5*
x^12)/7)/x^19

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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**2+a)**5*(B*x**2+A)/x**20,x)

[Out]

Timed out

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