3.1.79 \(\int \frac {(a+b x^2)^5}{x^{10}} \, dx\)

Optimal. Leaf size=60 \[ -\frac {a^5}{9 x^9}-\frac {5 a^4 b}{7 x^7}-\frac {2 a^3 b^2}{x^5}-\frac {10 a^2 b^3}{3 x^3}-\frac {5 a b^4}{x}+b^5 x \]

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Rubi [A]  time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} -\frac {2 a^3 b^2}{x^5}-\frac {10 a^2 b^3}{3 x^3}-\frac {5 a^4 b}{7 x^7}-\frac {a^5}{9 x^9}-\frac {5 a b^4}{x}+b^5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^10,x]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(7*x^7) - (2*a^3*b^2)/x^5 - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/x + b^5*x

Rule 270

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 60, normalized size = 1.00 \begin {gather*} -\frac {a^5}{9 x^9}-\frac {5 a^4 b}{7 x^7}-\frac {2 a^3 b^2}{x^5}-\frac {10 a^2 b^3}{3 x^3}-\frac {5 a b^4}{x}+b^5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^10,x]

[Out]

-1/9*a^5/x^9 - (5*a^4*b)/(7*x^7) - (2*a^3*b^2)/x^5 - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/x + b^5*x

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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^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^10,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^10, x]

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fricas [A]  time = 1.06, size = 59, normalized size = 0.98 \begin {gather*} \frac {63 \, b^{5} x^{10} - 315 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} - 126 \, a^{3} b^{2} x^{4} - 45 \, a^{4} b x^{2} - 7 \, a^{5}}{63 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^10,x, algorithm="fricas")

[Out]

1/63*(63*b^5*x^10 - 315*a*b^4*x^8 - 210*a^2*b^3*x^6 - 126*a^3*b^2*x^4 - 45*a^4*b*x^2 - 7*a^5)/x^9

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giac [A]  time = 1.04, size = 57, normalized size = 0.95 \begin {gather*} b^{5} x - \frac {315 \, a b^{4} x^{8} + 210 \, a^{2} b^{3} x^{6} + 126 \, a^{3} b^{2} x^{4} + 45 \, a^{4} b x^{2} + 7 \, a^{5}}{63 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^10,x, algorithm="giac")

[Out]

b^5*x - 1/63*(315*a*b^4*x^8 + 210*a^2*b^3*x^6 + 126*a^3*b^2*x^4 + 45*a^4*b*x^2 + 7*a^5)/x^9

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maple [A]  time = 0.01, size = 55, normalized size = 0.92 \begin {gather*} b^{5} x -\frac {5 a \,b^{4}}{x}-\frac {10 a^{2} b^{3}}{3 x^{3}}-\frac {2 a^{3} b^{2}}{x^{5}}-\frac {5 a^{4} b}{7 x^{7}}-\frac {a^{5}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^10,x)

[Out]

-1/9*a^5/x^9-5/7*a^4*b/x^7-2*a^3*b^2/x^5-10/3*a^2*b^3/x^3-5*a*b^4/x+b^5*x

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maxima [A]  time = 1.44, size = 57, normalized size = 0.95 \begin {gather*} b^{5} x - \frac {315 \, a b^{4} x^{8} + 210 \, a^{2} b^{3} x^{6} + 126 \, a^{3} b^{2} x^{4} + 45 \, a^{4} b x^{2} + 7 \, a^{5}}{63 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^10,x, algorithm="maxima")

[Out]

b^5*x - 1/63*(315*a*b^4*x^8 + 210*a^2*b^3*x^6 + 126*a^3*b^2*x^4 + 45*a^4*b*x^2 + 7*a^5)/x^9

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mupad [B]  time = 0.04, size = 57, normalized size = 0.95 \begin {gather*} b^5\,x-\frac {\frac {a^5}{9}+\frac {5\,a^4\,b\,x^2}{7}+2\,a^3\,b^2\,x^4+\frac {10\,a^2\,b^3\,x^6}{3}+5\,a\,b^4\,x^8}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^10,x)

[Out]

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

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sympy [A]  time = 0.36, size = 60, normalized size = 1.00 \begin {gather*} b^{5} x + \frac {- 7 a^{5} - 45 a^{4} b x^{2} - 126 a^{3} b^{2} x^{4} - 210 a^{2} b^{3} x^{6} - 315 a b^{4} x^{8}}{63 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**10,x)

[Out]

b**5*x + (-7*a**5 - 45*a**4*b*x**2 - 126*a**3*b**2*x**4 - 210*a**2*b**3*x**6 - 315*a*b**4*x**8)/(63*x**9)

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