3.40 \(\int \frac {(a+b x^2)^3}{x^{13}} \, dx\)

Optimal. Leaf size=43 \[ -\frac {a^3}{12 x^{12}}-\frac {3 a^2 b}{10 x^{10}}-\frac {3 a b^2}{8 x^8}-\frac {b^3}{6 x^6} \]

[Out]

-1/12*a^3/x^12-3/10*a^2*b/x^10-3/8*a*b^2/x^8-1/6*b^3/x^6

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Rubi [A]  time = 0.02, antiderivative size = 43, 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} \[ -\frac {3 a^2 b}{10 x^{10}}-\frac {a^3}{12 x^{12}}-\frac {3 a b^2}{8 x^8}-\frac {b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^13,x]

[Out]

-a^3/(12*x^12) - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(8*x^8) - b^3/(6*x^6)

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 )^3}{x^{13}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^3}{x^7} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^3}{x^7}+\frac {3 a^2 b}{x^6}+\frac {3 a b^2}{x^5}+\frac {b^3}{x^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3}{12 x^{12}}-\frac {3 a^2 b}{10 x^{10}}-\frac {3 a b^2}{8 x^8}-\frac {b^3}{6 x^6}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \[ -\frac {a^3}{12 x^{12}}-\frac {3 a^2 b}{10 x^{10}}-\frac {3 a b^2}{8 x^8}-\frac {b^3}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^13,x]

[Out]

-1/12*a^3/x^12 - (3*a^2*b)/(10*x^10) - (3*a*b^2)/(8*x^8) - b^3/(6*x^6)

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fricas [A]  time = 0.80, size = 37, normalized size = 0.86 \[ -\frac {20 \, b^{3} x^{6} + 45 \, a b^{2} x^{4} + 36 \, a^{2} b x^{2} + 10 \, a^{3}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^13,x, algorithm="fricas")

[Out]

-1/120*(20*b^3*x^6 + 45*a*b^2*x^4 + 36*a^2*b*x^2 + 10*a^3)/x^12

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giac [A]  time = 0.83, size = 37, normalized size = 0.86 \[ -\frac {20 \, b^{3} x^{6} + 45 \, a b^{2} x^{4} + 36 \, a^{2} b x^{2} + 10 \, a^{3}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^13,x, algorithm="giac")

[Out]

-1/120*(20*b^3*x^6 + 45*a*b^2*x^4 + 36*a^2*b*x^2 + 10*a^3)/x^12

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \[ -\frac {b^{3}}{6 x^{6}}-\frac {3 a \,b^{2}}{8 x^{8}}-\frac {3 a^{2} b}{10 x^{10}}-\frac {a^{3}}{12 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^13,x)

[Out]

-1/12*a^3/x^12-3/10*a^2*b/x^10-3/8*a*b^2/x^8-1/6*b^3/x^6

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maxima [A]  time = 1.29, size = 37, normalized size = 0.86 \[ -\frac {20 \, b^{3} x^{6} + 45 \, a b^{2} x^{4} + 36 \, a^{2} b x^{2} + 10 \, a^{3}}{120 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^13,x, algorithm="maxima")

[Out]

-1/120*(20*b^3*x^6 + 45*a*b^2*x^4 + 36*a^2*b*x^2 + 10*a^3)/x^12

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mupad [B]  time = 0.05, size = 37, normalized size = 0.86 \[ -\frac {\frac {a^3}{12}+\frac {3\,a^2\,b\,x^2}{10}+\frac {3\,a\,b^2\,x^4}{8}+\frac {b^3\,x^6}{6}}{x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^13,x)

[Out]

-(a^3/12 + (b^3*x^6)/6 + (3*a^2*b*x^2)/10 + (3*a*b^2*x^4)/8)/x^12

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sympy [A]  time = 0.32, size = 39, normalized size = 0.91 \[ \frac {- 10 a^{3} - 36 a^{2} b x^{2} - 45 a b^{2} x^{4} - 20 b^{3} x^{6}}{120 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**13,x)

[Out]

(-10*a**3 - 36*a**2*b*x**2 - 45*a*b**2*x**4 - 20*b**3*x**6)/(120*x**12)

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