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

Optimal. Leaf size=61 \[ -\frac {a^5}{x}+5 a^4 b x+\frac {10}{3} a^3 b^2 x^3+2 a^2 b^3 x^5+\frac {5}{7} a b^4 x^7+\frac {b^5 x^9}{9} \]

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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*} 2 a^2 b^3 x^5+\frac {10}{3} a^3 b^2 x^3+5 a^4 b x-\frac {a^5}{x}+\frac {5}{7} a b^4 x^7+\frac {b^5 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.00, size = 61, normalized size = 1.00 \begin {gather*} -\frac {a^5}{x}+5 a^4 b x+\frac {10}{3} a^3 b^2 x^3+2 a^2 b^3 x^5+\frac {5}{7} a b^4 x^7+\frac {b^5 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^2,x]

[Out]

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

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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^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.71, size = 59, normalized size = 0.97 \begin {gather*} \frac {7 \, b^{5} x^{10} + 45 \, a b^{4} x^{8} + 126 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} - 63 \, a^{5}}{63 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.13, size = 55, normalized size = 0.90 \begin {gather*} \frac {1}{9} \, b^{5} x^{9} + \frac {5}{7} \, a b^{4} x^{7} + 2 \, a^{2} b^{3} x^{5} + \frac {10}{3} \, a^{3} b^{2} x^{3} + 5 \, a^{4} b x - \frac {a^{5}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 56, normalized size = 0.92 \begin {gather*} \frac {b^{5} x^{9}}{9}+\frac {5 a \,b^{4} x^{7}}{7}+2 a^{2} b^{3} x^{5}+\frac {10 a^{3} b^{2} x^{3}}{3}+5 a^{4} b x -\frac {a^{5}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.39, size = 55, normalized size = 0.90 \begin {gather*} \frac {1}{9} \, b^{5} x^{9} + \frac {5}{7} \, a b^{4} x^{7} + 2 \, a^{2} b^{3} x^{5} + \frac {10}{3} \, a^{3} b^{2} x^{3} + 5 \, a^{4} b x - \frac {a^{5}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 55, normalized size = 0.90 \begin {gather*} \frac {b^5\,x^9}{9}-\frac {a^5}{x}+\frac {5\,a\,b^4\,x^7}{7}+\frac {10\,a^3\,b^2\,x^3}{3}+2\,a^2\,b^3\,x^5+5\,a^4\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^2,x)

[Out]

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

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sympy [A]  time = 0.13, size = 58, normalized size = 0.95 \begin {gather*} - \frac {a^{5}}{x} + 5 a^{4} b x + \frac {10 a^{3} b^{2} x^{3}}{3} + 2 a^{2} b^{3} x^{5} + \frac {5 a b^{4} x^{7}}{7} + \frac {b^{5} x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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