3.1.44 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{12}} \, dx\)

Optimal. Leaf size=108 \[ -\frac {a^5 A}{11 x^{11}}-\frac {a^4 (a B+5 A b)}{9 x^9}-\frac {5 a^3 b (a B+2 A b)}{7 x^7}-\frac {2 a^2 b^2 (a B+A b)}{x^5}-\frac {b^4 (5 a B+A b)}{x}-\frac {5 a b^3 (2 a B+A b)}{3 x^3}+b^5 B x \]

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Rubi [A]  time = 0.06, antiderivative size = 108, 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)}{9 x^9}-\frac {5 a^3 b (a B+2 A b)}{7 x^7}-\frac {a^5 A}{11 x^{11}}-\frac {5 a b^3 (2 a B+A b)}{3 x^3}-\frac {b^4 (5 a B+A b)}{x}+b^5 B x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^12,x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 122, normalized size = 1.13 \begin {gather*} -\frac {a^5 \left (9 A+11 B x^2\right )}{99 x^{11}}-\frac {5 a^4 b \left (7 A+9 B x^2\right )}{63 x^9}-\frac {2 a^3 b^2 \left (5 A+7 B x^2\right )}{7 x^7}-\frac {2 a^2 b^3 \left (3 A+5 B x^2\right )}{3 x^5}-\frac {5 a b^4 \left (A+3 B x^2\right )}{3 x^3}-\frac {A b^5}{x}+b^5 B x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^12,x]

[Out]

-((A*b^5)/x) + b^5*B*x - (5*a*b^4*(A + 3*B*x^2))/(3*x^3) - (2*a^2*b^3*(3*A + 5*B*x^2))/(3*x^5) - (2*a^3*b^2*(5
*A + 7*B*x^2))/(7*x^7) - (5*a^4*b*(7*A + 9*B*x^2))/(63*x^9) - (a^5*(9*A + 11*B*x^2))/(99*x^11)

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^12,x]

[Out]

IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^12, x]

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fricas [A]  time = 0.46, size = 121, normalized size = 1.12 \begin {gather*} \frac {693 \, B b^{5} x^{12} - 693 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} - 1155 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 1386 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 63 \, A a^{5} - 495 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 77 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{693 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^12,x, algorithm="fricas")

[Out]

1/693*(693*B*b^5*x^12 - 693*(5*B*a*b^4 + A*b^5)*x^10 - 1155*(2*B*a^2*b^3 + A*a*b^4)*x^8 - 1386*(B*a^3*b^2 + A*
a^2*b^3)*x^6 - 63*A*a^5 - 495*(B*a^4*b + 2*A*a^3*b^2)*x^4 - 77*(B*a^5 + 5*A*a^4*b)*x^2)/x^11

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giac [A]  time = 0.36, size = 125, normalized size = 1.16 \begin {gather*} B b^{5} x - \frac {3465 \, B a b^{4} x^{10} + 693 \, A b^{5} x^{10} + 2310 \, B a^{2} b^{3} x^{8} + 1155 \, A a b^{4} x^{8} + 1386 \, B a^{3} b^{2} x^{6} + 1386 \, A a^{2} b^{3} x^{6} + 495 \, B a^{4} b x^{4} + 990 \, A a^{3} b^{2} x^{4} + 77 \, B a^{5} x^{2} + 385 \, A a^{4} b x^{2} + 63 \, A a^{5}}{693 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^12,x, algorithm="giac")

[Out]

B*b^5*x - 1/693*(3465*B*a*b^4*x^10 + 693*A*b^5*x^10 + 2310*B*a^2*b^3*x^8 + 1155*A*a*b^4*x^8 + 1386*B*a^3*b^2*x
^6 + 1386*A*a^2*b^3*x^6 + 495*B*a^4*b*x^4 + 990*A*a^3*b^2*x^4 + 77*B*a^5*x^2 + 385*A*a^4*b*x^2 + 63*A*a^5)/x^1
1

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maple [A]  time = 0.01, size = 101, normalized size = 0.94 \begin {gather*} B \,b^{5} x -\frac {\left (A b +5 B a \right ) b^{4}}{x}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{3 x^{3}}-\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}{7 x^{7}}-\frac {A \,a^{5}}{11 x^{11}}-\frac {\left (5 A b +B a \right ) a^{4}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^12,x)

[Out]

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

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maxima [A]  time = 1.02, size = 119, normalized size = 1.10 \begin {gather*} B b^{5} x - \frac {693 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 1155 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1386 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 63 \, A a^{5} + 495 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 77 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{693 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^12,x, algorithm="maxima")

[Out]

B*b^5*x - 1/693*(693*(5*B*a*b^4 + A*b^5)*x^10 + 1155*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 1386*(B*a^3*b^2 + A*a^2*b^3
)*x^6 + 63*A*a^5 + 495*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 77*(B*a^5 + 5*A*a^4*b)*x^2)/x^11

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mupad [B]  time = 0.07, size = 119, normalized size = 1.10 \begin {gather*} B\,b^5\,x-\frac {\frac {A\,a^5}{11}+x^8\,\left (\frac {10\,B\,a^2\,b^3}{3}+\frac {5\,A\,a\,b^4}{3}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{7}+\frac {10\,A\,a^3\,b^2}{7}\right )+x^2\,\left (\frac {B\,a^5}{9}+\frac {5\,A\,b\,a^4}{9}\right )+x^{10}\,\left (A\,b^5+5\,B\,a\,b^4\right )+x^6\,\left (2\,B\,a^3\,b^2+2\,A\,a^2\,b^3\right )}{x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^12,x)

[Out]

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

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sympy [A]  time = 6.68, size = 131, normalized size = 1.21 \begin {gather*} B b^{5} x + \frac {- 63 A a^{5} + x^{10} \left (- 693 A b^{5} - 3465 B a b^{4}\right ) + x^{8} \left (- 1155 A a b^{4} - 2310 B a^{2} b^{3}\right ) + x^{6} \left (- 1386 A a^{2} b^{3} - 1386 B a^{3} b^{2}\right ) + x^{4} \left (- 990 A a^{3} b^{2} - 495 B a^{4} b\right ) + x^{2} \left (- 385 A a^{4} b - 77 B a^{5}\right )}{693 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**12,x)

[Out]

B*b**5*x + (-63*A*a**5 + x**10*(-693*A*b**5 - 3465*B*a*b**4) + x**8*(-1155*A*a*b**4 - 2310*B*a**2*b**3) + x**6
*(-1386*A*a**2*b**3 - 1386*B*a**3*b**2) + x**4*(-990*A*a**3*b**2 - 495*B*a**4*b) + x**2*(-385*A*a**4*b - 77*B*
a**5))/(693*x**11)

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