3.1.78 \(\int \frac {(a+b x^2)^5}{x^8} \, dx\)

Optimal. Leaf size=61 \[ -\frac {a^5}{7 x^7}-\frac {a^4 b}{x^5}-\frac {10 a^3 b^2}{3 x^3}-\frac {10 a^2 b^3}{x}+5 a b^4 x+\frac {b^5 x^3}{3} \]

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Rubi [A]  time = 0.02, antiderivative size = 61, 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 {10 a^3 b^2}{3 x^3}-\frac {10 a^2 b^3}{x}-\frac {a^4 b}{x^5}-\frac {a^5}{7 x^7}+5 a b^4 x+\frac {b^5 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^8,x]

[Out]

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

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

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Mathematica [A]  time = 0.00, size = 61, normalized size = 1.00 \begin {gather*} -\frac {a^5}{7 x^7}-\frac {a^4 b}{x^5}-\frac {10 a^3 b^2}{3 x^3}-\frac {10 a^2 b^3}{x}+5 a b^4 x+\frac {b^5 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^8,x]

[Out]

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

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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^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^8,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^8, x]

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fricas [A]  time = 1.04, size = 59, normalized size = 0.97 \begin {gather*} \frac {7 \, b^{5} x^{10} + 105 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} - 70 \, a^{3} b^{2} x^{4} - 21 \, a^{4} b x^{2} - 3 \, a^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^8,x, algorithm="fricas")

[Out]

1/21*(7*b^5*x^10 + 105*a*b^4*x^8 - 210*a^2*b^3*x^6 - 70*a^3*b^2*x^4 - 21*a^4*b*x^2 - 3*a^5)/x^7

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giac [A]  time = 1.12, size = 58, normalized size = 0.95 \begin {gather*} \frac {1}{3} \, b^{5} x^{3} + 5 \, a b^{4} x - \frac {210 \, a^{2} b^{3} x^{6} + 70 \, a^{3} b^{2} x^{4} + 21 \, a^{4} b x^{2} + 3 \, a^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^8,x, algorithm="giac")

[Out]

1/3*b^5*x^3 + 5*a*b^4*x - 1/21*(210*a^2*b^3*x^6 + 70*a^3*b^2*x^4 + 21*a^4*b*x^2 + 3*a^5)/x^7

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maple [A]  time = 0.01, size = 56, normalized size = 0.92 \begin {gather*} \frac {b^{5} x^{3}}{3}+5 a \,b^{4} x -\frac {10 a^{2} b^{3}}{x}-\frac {10 a^{3} b^{2}}{3 x^{3}}-\frac {a^{4} b}{x^{5}}-\frac {a^{5}}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^8,x)

[Out]

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

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maxima [A]  time = 1.44, size = 58, normalized size = 0.95 \begin {gather*} \frac {1}{3} \, b^{5} x^{3} + 5 \, a b^{4} x - \frac {210 \, a^{2} b^{3} x^{6} + 70 \, a^{3} b^{2} x^{4} + 21 \, a^{4} b x^{2} + 3 \, a^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^8,x, algorithm="maxima")

[Out]

1/3*b^5*x^3 + 5*a*b^4*x - 1/21*(210*a^2*b^3*x^6 + 70*a^3*b^2*x^4 + 21*a^4*b*x^2 + 3*a^5)/x^7

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mupad [B]  time = 4.79, size = 59, normalized size = 0.97 \begin {gather*} -\frac {3\,a^5+21\,a^4\,b\,x^2+70\,a^3\,b^2\,x^4+210\,a^2\,b^3\,x^6-105\,a\,b^4\,x^8-7\,b^5\,x^{10}}{21\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^8,x)

[Out]

-(3*a^5 - 7*b^5*x^10 + 21*a^4*b*x^2 - 105*a*b^4*x^8 + 70*a^3*b^2*x^4 + 210*a^2*b^3*x^6)/(21*x^7)

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sympy [A]  time = 0.29, size = 61, normalized size = 1.00 \begin {gather*} 5 a b^{4} x + \frac {b^{5} x^{3}}{3} + \frac {- 3 a^{5} - 21 a^{4} b x^{2} - 70 a^{3} b^{2} x^{4} - 210 a^{2} b^{3} x^{6}}{21 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**8,x)

[Out]

5*a*b**4*x + b**5*x**3/3 + (-3*a**5 - 21*a**4*b*x**2 - 70*a**3*b**2*x**4 - 210*a**2*b**3*x**6)/(21*x**7)

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