3.1.77 \(\int \frac {(a+b x^2)^5}{x^6} \, dx\) [77]

Optimal. Leaf size=63 \[ -\frac {a^5}{5 x^5}-\frac {5 a^4 b}{3 x^3}-\frac {10 a^3 b^2}{x}+10 a^2 b^3 x+\frac {5}{3} a b^4 x^3+\frac {b^5 x^5}{5} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 63, 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 = {276} \begin {gather*} -\frac {a^5}{5 x^5}-\frac {5 a^4 b}{3 x^3}-\frac {10 a^3 b^2}{x}+10 a^2 b^3 x+\frac {5}{3} a b^4 x^3+\frac {b^5 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^6,x]

[Out]

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

Rule 276

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

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Mathematica [A]
time = 0.00, size = 63, normalized size = 1.00 \begin {gather*} -\frac {a^5}{5 x^5}-\frac {5 a^4 b}{3 x^3}-\frac {10 a^3 b^2}{x}+10 a^2 b^3 x+\frac {5}{3} a b^4 x^3+\frac {b^5 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^6,x]

[Out]

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

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Maple [A]
time = 0.03, size = 56, normalized size = 0.89

method	result	size
default	\(-\frac {a^{5}}{5 x^{5}}-\frac {5 a^{4} b}{3 x^{3}}-\frac {10 a^{3} b^{2}}{x}+10 a^{2} b^{3} x +\frac {5 a \,b^{4} x^{3}}{3}+\frac {b^{5} x^{5}}{5}\)	\(56\)
risch	\(\frac {b^{5} x^{5}}{5}+\frac {5 a \,b^{4} x^{3}}{3}+10 a^{2} b^{3} x +\frac {-10 a^{3} b^{2} x^{4}-\frac {5}{3} a^{4} b \,x^{2}-\frac {1}{5} a^{5}}{x^{5}}\)	\(58\)
norman	\(\frac {\frac {1}{5} b^{5} x^{10}+\frac {5}{3} a \,b^{4} x^{8}+10 a^{2} b^{3} x^{6}-10 a^{3} b^{2} x^{4}-\frac {5}{3} a^{4} b \,x^{2}-\frac {1}{5} a^{5}}{x^{5}}\)	\(59\)
gosper	\(-\frac {-3 b^{5} x^{10}-25 a \,b^{4} x^{8}-150 a^{2} b^{3} x^{6}+150 a^{3} b^{2} x^{4}+25 a^{4} b \,x^{2}+3 a^{5}}{15 x^{5}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^6,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 58, normalized size = 0.92 \begin {gather*} \frac {1}{5} \, b^{5} x^{5} + \frac {5}{3} \, a b^{4} x^{3} + 10 \, a^{2} b^{3} x - \frac {150 \, a^{3} b^{2} x^{4} + 25 \, a^{4} b x^{2} + 3 \, a^{5}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^6,x, algorithm="maxima")

[Out]

1/5*b^5*x^5 + 5/3*a*b^4*x^3 + 10*a^2*b^3*x - 1/15*(150*a^3*b^2*x^4 + 25*a^4*b*x^2 + 3*a^5)/x^5

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Fricas [A]
time = 1.10, size = 59, normalized size = 0.94 \begin {gather*} \frac {3 \, b^{5} x^{10} + 25 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} - 150 \, a^{3} b^{2} x^{4} - 25 \, a^{4} b x^{2} - 3 \, a^{5}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^6,x, algorithm="fricas")

[Out]

1/15*(3*b^5*x^10 + 25*a*b^4*x^8 + 150*a^2*b^3*x^6 - 150*a^3*b^2*x^4 - 25*a^4*b*x^2 - 3*a^5)/x^5

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Sympy [A]
time = 0.08, size = 63, normalized size = 1.00 \begin {gather*} 10 a^{2} b^{3} x + \frac {5 a b^{4} x^{3}}{3} + \frac {b^{5} x^{5}}{5} + \frac {- 3 a^{5} - 25 a^{4} b x^{2} - 150 a^{3} b^{2} x^{4}}{15 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**6,x)

[Out]

10*a**2*b**3*x + 5*a*b**4*x**3/3 + b**5*x**5/5 + (-3*a**5 - 25*a**4*b*x**2 - 150*a**3*b**2*x**4)/(15*x**5)

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Giac [A]
time = 1.57, size = 58, normalized size = 0.92 \begin {gather*} \frac {1}{5} \, b^{5} x^{5} + \frac {5}{3} \, a b^{4} x^{3} + 10 \, a^{2} b^{3} x - \frac {150 \, a^{3} b^{2} x^{4} + 25 \, a^{4} b x^{2} + 3 \, a^{5}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^6,x, algorithm="giac")

[Out]

1/5*b^5*x^5 + 5/3*a*b^4*x^3 + 10*a^2*b^3*x - 1/15*(150*a^3*b^2*x^4 + 25*a^4*b*x^2 + 3*a^5)/x^5

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Mupad [B]
time = 0.05, size = 58, normalized size = 0.92 \begin {gather*} \frac {b^5\,x^5}{5}-\frac {\frac {a^5}{5}+\frac {5\,a^4\,b\,x^2}{3}+10\,a^3\,b^2\,x^4}{x^5}+10\,a^2\,b^3\,x+\frac {5\,a\,b^4\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^6,x)

[Out]

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

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