3.1.65 \(\int \frac {(a+b x^2)^5}{x^{13}} \, dx\)

Optimal. Leaf size=19 \[ -\frac {\left (a+b x^2\right )^6}{12 a x^{12}} \]

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \begin {gather*} -\frac {\left (a+b x^2\right )^6}{12 a x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^13,x]

[Out]

-(a + b*x^2)^6/(12*a*x^12)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^5}{x^{13}} \, dx &=-\frac {\left (a+b x^2\right )^6}{12 a x^{12}}\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 69, normalized size = 3.63 \begin {gather*} -\frac {a^5}{12 x^{12}}-\frac {a^4 b}{2 x^{10}}-\frac {5 a^3 b^2}{4 x^8}-\frac {5 a^2 b^3}{3 x^6}-\frac {5 a b^4}{4 x^4}-\frac {b^5}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^13,x]

[Out]

-1/12*a^5/x^12 - (a^4*b)/(2*x^10) - (5*a^3*b^2)/(4*x^8) - (5*a^2*b^3)/(3*x^6) - (5*a*b^4)/(4*x^4) - b^5/(2*x^2
)

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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}{x^{13}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^13,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^13, x]

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fricas [B]  time = 0.60, size = 57, normalized size = 3.00 \begin {gather*} -\frac {6 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 15 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} + a^{5}}{12 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^13,x, algorithm="fricas")

[Out]

-1/12*(6*b^5*x^10 + 15*a*b^4*x^8 + 20*a^2*b^3*x^6 + 15*a^3*b^2*x^4 + 6*a^4*b*x^2 + a^5)/x^12

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giac [B]  time = 1.07, size = 57, normalized size = 3.00 \begin {gather*} -\frac {6 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 15 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} + a^{5}}{12 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^13,x, algorithm="giac")

[Out]

-1/12*(6*b^5*x^10 + 15*a*b^4*x^8 + 20*a^2*b^3*x^6 + 15*a^3*b^2*x^4 + 6*a^4*b*x^2 + a^5)/x^12

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maple [B]  time = 0.01, size = 58, normalized size = 3.05 \begin {gather*} -\frac {b^{5}}{2 x^{2}}-\frac {5 a \,b^{4}}{4 x^{4}}-\frac {5 a^{2} b^{3}}{3 x^{6}}-\frac {5 a^{3} b^{2}}{4 x^{8}}-\frac {a^{4} b}{2 x^{10}}-\frac {a^{5}}{12 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^13,x)

[Out]

-5/3*a^2*b^3/x^6-1/2*b^5/x^2-5/4*a*b^4/x^4-1/2*a^4*b/x^10-1/12*a^5/x^12-5/4*a^3*b^2/x^8

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maxima [B]  time = 1.37, size = 57, normalized size = 3.00 \begin {gather*} -\frac {6 \, b^{5} x^{10} + 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 15 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} + a^{5}}{12 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^13,x, algorithm="maxima")

[Out]

-1/12*(6*b^5*x^10 + 15*a*b^4*x^8 + 20*a^2*b^3*x^6 + 15*a^3*b^2*x^4 + 6*a^4*b*x^2 + a^5)/x^12

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mupad [B]  time = 4.75, size = 59, normalized size = 3.11 \begin {gather*} -\frac {\frac {a^5}{12}+\frac {a^4\,b\,x^2}{2}+\frac {5\,a^3\,b^2\,x^4}{4}+\frac {5\,a^2\,b^3\,x^6}{3}+\frac {5\,a\,b^4\,x^8}{4}+\frac {b^5\,x^{10}}{2}}{x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^13,x)

[Out]

-(a^5/12 + (b^5*x^10)/2 + (a^4*b*x^2)/2 + (5*a*b^4*x^8)/4 + (5*a^3*b^2*x^4)/4 + (5*a^2*b^3*x^6)/3)/x^12

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sympy [B]  time = 0.47, size = 61, normalized size = 3.21 \begin {gather*} \frac {- a^{5} - 6 a^{4} b x^{2} - 15 a^{3} b^{2} x^{4} - 20 a^{2} b^{3} x^{6} - 15 a b^{4} x^{8} - 6 b^{5} x^{10}}{12 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**13,x)

[Out]

(-a**5 - 6*a**4*b*x**2 - 15*a**3*b**2*x**4 - 20*a**2*b**3*x**6 - 15*a*b**4*x**8 - 6*b**5*x**10)/(12*x**12)

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