3.1.95 \(\int \frac {(a+b x^2)^8}{x^7} \, dx\)

Optimal. Leaf size=94 \[ -\frac {a^8}{6 x^6}-\frac {2 a^7 b}{x^4}-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac {14}{3} a^2 b^6 x^6+a b^7 x^8+\frac {b^8 x^{10}}{10} \]

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Rubi [A]  time = 0.06, antiderivative size = 94, 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 {14}{3} a^2 b^6 x^6+14 a^3 b^5 x^4+35 a^4 b^4 x^2-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)-\frac {2 a^7 b}{x^4}-\frac {a^8}{6 x^6}+a b^7 x^8+\frac {b^8 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^7,x]

[Out]

-a^8/(6*x^6) - (2*a^7*b)/x^4 - (14*a^6*b^2)/x^2 + 35*a^4*b^4*x^2 + 14*a^3*b^5*x^4 + (14*a^2*b^6*x^6)/3 + a*b^7
*x^8 + (b^8*x^10)/10 + 56*a^5*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 )^8}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (70 a^4 b^4+\frac {a^8}{x^4}+\frac {8 a^7 b}{x^3}+\frac {28 a^6 b^2}{x^2}+\frac {56 a^5 b^3}{x}+56 a^3 b^5 x+28 a^2 b^6 x^2+8 a b^7 x^3+b^8 x^4\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{6 x^6}-\frac {2 a^7 b}{x^4}-\frac {14 a^6 b^2}{x^2}+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac {14}{3} a^2 b^6 x^6+a b^7 x^8+\frac {b^8 x^{10}}{10}+56 a^5 b^3 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 94, normalized size = 1.00 \begin {gather*} -\frac {a^8}{6 x^6}-\frac {2 a^7 b}{x^4}-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac {14}{3} a^2 b^6 x^6+a b^7 x^8+\frac {b^8 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^7,x]

[Out]

-1/6*a^8/x^6 - (2*a^7*b)/x^4 - (14*a^6*b^2)/x^2 + 35*a^4*b^4*x^2 + 14*a^3*b^5*x^4 + (14*a^2*b^6*x^6)/3 + a*b^7
*x^8 + (b^8*x^10)/10 + 56*a^5*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 )^8}{x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^8/x^7,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^8/x^7, x]

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fricas [A]  time = 1.29, size = 94, normalized size = 1.00 \begin {gather*} \frac {3 \, b^{8} x^{16} + 30 \, a b^{7} x^{14} + 140 \, a^{2} b^{6} x^{12} + 420 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 1680 \, a^{5} b^{3} x^{6} \log \relax (x) - 420 \, a^{6} b^{2} x^{4} - 60 \, a^{7} b x^{2} - 5 \, a^{8}}{30 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^7,x, algorithm="fricas")

[Out]

1/30*(3*b^8*x^16 + 30*a*b^7*x^14 + 140*a^2*b^6*x^12 + 420*a^3*b^5*x^10 + 1050*a^4*b^4*x^8 + 1680*a^5*b^3*x^6*l
og(x) - 420*a^6*b^2*x^4 - 60*a^7*b*x^2 - 5*a^8)/x^6

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giac [A]  time = 1.05, size = 102, normalized size = 1.09 \begin {gather*} \frac {1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac {14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac {308 \, a^{5} b^{3} x^{6} + 84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^7,x, algorithm="giac")

[Out]

1/10*b^8*x^10 + a*b^7*x^8 + 14/3*a^2*b^6*x^6 + 14*a^3*b^5*x^4 + 35*a^4*b^4*x^2 + 28*a^5*b^3*log(x^2) - 1/6*(30
8*a^5*b^3*x^6 + 84*a^6*b^2*x^4 + 12*a^7*b*x^2 + a^8)/x^6

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maple [A]  time = 0.01, size = 89, normalized size = 0.95 \begin {gather*} \frac {b^{8} x^{10}}{10}+a \,b^{7} x^{8}+\frac {14 a^{2} b^{6} x^{6}}{3}+14 a^{3} b^{5} x^{4}+35 a^{4} b^{4} x^{2}+56 a^{5} b^{3} \ln \relax (x )-\frac {14 a^{6} b^{2}}{x^{2}}-\frac {2 a^{7} b}{x^{4}}-\frac {a^{8}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^7,x)

[Out]

-1/6*a^8/x^6-2*a^7*b/x^4-14*a^6*b^2/x^2+35*a^4*b^4*x^2+14*a^3*b^5*x^4+14/3*a^2*b^6*x^6+a*b^7*x^8+1/10*b^8*x^10
+56*a^5*b^3*ln(x)

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maxima [A]  time = 1.35, size = 91, normalized size = 0.97 \begin {gather*} \frac {1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac {14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac {84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^7,x, algorithm="maxima")

[Out]

1/10*b^8*x^10 + a*b^7*x^8 + 14/3*a^2*b^6*x^6 + 14*a^3*b^5*x^4 + 35*a^4*b^4*x^2 + 28*a^5*b^3*log(x^2) - 1/6*(84
*a^6*b^2*x^4 + 12*a^7*b*x^2 + a^8)/x^6

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mupad [B]  time = 0.05, size = 91, normalized size = 0.97 \begin {gather*} \frac {b^8\,x^{10}}{10}-\frac {\frac {a^8}{6}+2\,a^7\,b\,x^2+14\,a^6\,b^2\,x^4}{x^6}+a\,b^7\,x^8+35\,a^4\,b^4\,x^2+14\,a^3\,b^5\,x^4+\frac {14\,a^2\,b^6\,x^6}{3}+56\,a^5\,b^3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^7,x)

[Out]

(b^8*x^10)/10 - (a^8/6 + 2*a^7*b*x^2 + 14*a^6*b^2*x^4)/x^6 + a*b^7*x^8 + 35*a^4*b^4*x^2 + 14*a^3*b^5*x^4 + (14
*a^2*b^6*x^6)/3 + 56*a^5*b^3*log(x)

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sympy [A]  time = 0.33, size = 97, normalized size = 1.03 \begin {gather*} 56 a^{5} b^{3} \log {\relax (x )} + 35 a^{4} b^{4} x^{2} + 14 a^{3} b^{5} x^{4} + \frac {14 a^{2} b^{6} x^{6}}{3} + a b^{7} x^{8} + \frac {b^{8} x^{10}}{10} + \frac {- a^{8} - 12 a^{7} b x^{2} - 84 a^{6} b^{2} x^{4}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**7,x)

[Out]

56*a**5*b**3*log(x) + 35*a**4*b**4*x**2 + 14*a**3*b**5*x**4 + 14*a**2*b**6*x**6/3 + a*b**7*x**8 + b**8*x**10/1
0 + (-a**8 - 12*a**7*b*x**2 - 84*a**6*b**2*x**4)/(6*x**6)

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