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

Optimal. Leaf size=65 \[ a^5 \log (x)+\frac {5}{2} a^4 b x^2+\frac {5}{2} a^3 b^2 x^4+\frac {5}{3} a^2 b^3 x^6+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{10}}{10} \]

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Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {5}{3} a^2 b^3 x^6+\frac {5}{2} a^3 b^2 x^4+\frac {5}{2} a^4 b x^2+a^5 \log (x)+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x,x]

[Out]

(5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5*x^10)/10 + a^5*Log[x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^5}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (5 a^4 b+\frac {a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^2\right )\\ &=\frac {5}{2} a^4 b x^2+\frac {5}{2} a^3 b^2 x^4+\frac {5}{3} a^2 b^3 x^6+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{10}}{10}+a^5 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 65, normalized size = 1.00 \begin {gather*} a^5 \log (x)+\frac {5}{2} a^4 b x^2+\frac {5}{2} a^3 b^2 x^4+\frac {5}{3} a^2 b^3 x^6+\frac {5}{8} a b^4 x^8+\frac {b^5 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x,x]

[Out]

(5*a^4*b*x^2)/2 + (5*a^3*b^2*x^4)/2 + (5*a^2*b^3*x^6)/3 + (5*a*b^4*x^8)/8 + (b^5*x^10)/10 + a^5*Log[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} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x,x]

[Out]

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

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fricas [A]  time = 0.90, size = 55, normalized size = 0.85 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} + \frac {5}{8} \, a b^{4} x^{8} + \frac {5}{3} \, a^{2} b^{3} x^{6} + \frac {5}{2} \, a^{3} b^{2} x^{4} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.92, size = 58, normalized size = 0.89 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} + \frac {5}{8} \, a b^{4} x^{8} + \frac {5}{3} \, a^{2} b^{3} x^{6} + \frac {5}{2} \, a^{3} b^{2} x^{4} + \frac {5}{2} \, a^{4} b x^{2} + \frac {1}{2} \, a^{5} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 56, normalized size = 0.86 \begin {gather*} \frac {b^{5} x^{10}}{10}+\frac {5 a \,b^{4} x^{8}}{8}+\frac {5 a^{2} b^{3} x^{6}}{3}+\frac {5 a^{3} b^{2} x^{4}}{2}+\frac {5 a^{4} b \,x^{2}}{2}+a^{5} \ln \relax (x ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 58, normalized size = 0.89 \begin {gather*} \frac {1}{10} \, b^{5} x^{10} + \frac {5}{8} \, a b^{4} x^{8} + \frac {5}{3} \, a^{2} b^{3} x^{6} + \frac {5}{2} \, a^{3} b^{2} x^{4} + \frac {5}{2} \, a^{4} b x^{2} + \frac {1}{2} \, a^{5} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 55, normalized size = 0.85 \begin {gather*} a^5\,\ln \relax (x)+\frac {b^5\,x^{10}}{10}+\frac {5\,a^4\,b\,x^2}{2}+\frac {5\,a\,b^4\,x^8}{8}+\frac {5\,a^3\,b^2\,x^4}{2}+\frac {5\,a^2\,b^3\,x^6}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x,x)

[Out]

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

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sympy [A]  time = 0.14, size = 65, normalized size = 1.00 \begin {gather*} a^{5} \log {\relax (x )} + \frac {5 a^{4} b x^{2}}{2} + \frac {5 a^{3} b^{2} x^{4}}{2} + \frac {5 a^{2} b^{3} x^{6}}{3} + \frac {5 a b^{4} x^{8}}{8} + \frac {b^{5} x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*log(x) + 5*a**4*b*x**2/2 + 5*a**3*b**2*x**4/2 + 5*a**2*b**3*x**6/3 + 5*a*b**4*x**8/8 + b**5*x**10/10

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