3.1.62 \(\int \frac {(a+b x^2)^5}{x^7} \, dx\)

Optimal. Leaf size=64 \[ -\frac {a^5}{6 x^6}-\frac {5 a^4 b}{4 x^4}-\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)+\frac {5}{2} a b^4 x^2+\frac {b^5 x^4}{4} \]

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Rubi [A]  time = 0.03, antiderivative size = 64, 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 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)-\frac {5 a^4 b}{4 x^4}-\frac {a^5}{6 x^6}+\frac {5}{2} a b^4 x^2+\frac {b^5 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^7,x]

[Out]

-a^5/(6*x^6) - (5*a^4*b)/(4*x^4) - (5*a^3*b^2)/x^2 + (5*a*b^4*x^2)/2 + (b^5*x^4)/4 + 10*a^2*b^3*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^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (5 a b^4+\frac {a^5}{x^4}+\frac {5 a^4 b}{x^3}+\frac {10 a^3 b^2}{x^2}+\frac {10 a^2 b^3}{x}+b^5 x\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5}{6 x^6}-\frac {5 a^4 b}{4 x^4}-\frac {5 a^3 b^2}{x^2}+\frac {5}{2} a b^4 x^2+\frac {b^5 x^4}{4}+10 a^2 b^3 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 64, normalized size = 1.00 \begin {gather*} -\frac {a^5}{6 x^6}-\frac {5 a^4 b}{4 x^4}-\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 \log (x)+\frac {5}{2} a b^4 x^2+\frac {b^5 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^7,x]

[Out]

-1/6*a^5/x^6 - (5*a^4*b)/(4*x^4) - (5*a^3*b^2)/x^2 + (5*a*b^4*x^2)/2 + (b^5*x^4)/4 + 10*a^2*b^3*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^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^7,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^7, x]

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fricas [A]  time = 1.15, size = 61, normalized size = 0.95 \begin {gather*} \frac {3 \, b^{5} x^{10} + 30 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} \log \relax (x) - 60 \, a^{3} b^{2} x^{4} - 15 \, a^{4} b x^{2} - 2 \, a^{5}}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^7,x, algorithm="fricas")

[Out]

1/12*(3*b^5*x^10 + 30*a*b^4*x^8 + 120*a^2*b^3*x^6*log(x) - 60*a^3*b^2*x^4 - 15*a^4*b*x^2 - 2*a^5)/x^6

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giac [A]  time = 1.19, size = 72, normalized size = 1.12 \begin {gather*} \frac {1}{4} \, b^{5} x^{4} + \frac {5}{2} \, a b^{4} x^{2} + 5 \, a^{2} b^{3} \log \left (x^{2}\right ) - \frac {110 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} + 2 \, a^{5}}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^7,x, algorithm="giac")

[Out]

1/4*b^5*x^4 + 5/2*a*b^4*x^2 + 5*a^2*b^3*log(x^2) - 1/12*(110*a^2*b^3*x^6 + 60*a^3*b^2*x^4 + 15*a^4*b*x^2 + 2*a
^5)/x^6

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maple [A]  time = 0.01, size = 57, normalized size = 0.89 \begin {gather*} \frac {b^{5} x^{4}}{4}+\frac {5 a \,b^{4} x^{2}}{2}+10 a^{2} b^{3} \ln \relax (x )-\frac {5 a^{3} b^{2}}{x^{2}}-\frac {5 a^{4} b}{4 x^{4}}-\frac {a^{5}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^7,x)

[Out]

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

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maxima [A]  time = 1.40, size = 61, normalized size = 0.95 \begin {gather*} \frac {1}{4} \, b^{5} x^{4} + \frac {5}{2} \, a b^{4} x^{2} + 5 \, a^{2} b^{3} \log \left (x^{2}\right ) - \frac {60 \, a^{3} b^{2} x^{4} + 15 \, a^{4} b x^{2} + 2 \, a^{5}}{12 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^7,x, algorithm="maxima")

[Out]

1/4*b^5*x^4 + 5/2*a*b^4*x^2 + 5*a^2*b^3*log(x^2) - 1/12*(60*a^3*b^2*x^4 + 15*a^4*b*x^2 + 2*a^5)/x^6

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mupad [B]  time = 0.04, size = 59, normalized size = 0.92 \begin {gather*} \frac {b^5\,x^4}{4}-\frac {\frac {a^5}{6}+\frac {5\,a^4\,b\,x^2}{4}+5\,a^3\,b^2\,x^4}{x^6}+\frac {5\,a\,b^4\,x^2}{2}+10\,a^2\,b^3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^7,x)

[Out]

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

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sympy [A]  time = 0.28, size = 65, normalized size = 1.02 \begin {gather*} 10 a^{2} b^{3} \log {\relax (x )} + \frac {5 a b^{4} x^{2}}{2} + \frac {b^{5} x^{4}}{4} + \frac {- 2 a^{5} - 15 a^{4} b x^{2} - 60 a^{3} b^{2} x^{4}}{12 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**7,x)

[Out]

10*a**2*b**3*log(x) + 5*a*b**4*x**2/2 + b**5*x**4/4 + (-2*a**5 - 15*a**4*b*x**2 - 60*a**3*b**2*x**4)/(12*x**6)

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