3.1.41 \(\int \frac {(a+b x^2)^3}{x^{15}} \, dx\)

Optimal. Leaf size=43 \[ -\frac {a^3}{14 x^{14}}-\frac {a^2 b}{4 x^{12}}-\frac {3 a b^2}{10 x^{10}}-\frac {b^3}{8 x^8} \]

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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} \begin {gather*} -\frac {a^2 b}{4 x^{12}}-\frac {a^3}{14 x^{14}}-\frac {3 a b^2}{10 x^{10}}-\frac {b^3}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^15,x]

[Out]

-a^3/(14*x^14) - (a^2*b)/(4*x^12) - (3*a*b^2)/(10*x^10) - b^3/(8*x^8)

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^{15}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^3}{x^8} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^3}{x^8}+\frac {3 a^2 b}{x^7}+\frac {3 a b^2}{x^6}+\frac {b^3}{x^5}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^3}{14 x^{14}}-\frac {a^2 b}{4 x^{12}}-\frac {3 a b^2}{10 x^{10}}-\frac {b^3}{8 x^8}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} -\frac {a^3}{14 x^{14}}-\frac {a^2 b}{4 x^{12}}-\frac {3 a b^2}{10 x^{10}}-\frac {b^3}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^15,x]

[Out]

-1/14*a^3/x^14 - (a^2*b)/(4*x^12) - (3*a*b^2)/(10*x^10) - b^3/(8*x^8)

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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 )^3}{x^{15}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^3/x^15,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^3/x^15, x]

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fricas [A]  time = 0.71, size = 37, normalized size = 0.86 \begin {gather*} -\frac {35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^15,x, algorithm="fricas")

[Out]

-1/280*(35*b^3*x^6 + 84*a*b^2*x^4 + 70*a^2*b*x^2 + 20*a^3)/x^14

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giac [A]  time = 1.04, size = 37, normalized size = 0.86 \begin {gather*} -\frac {35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^15,x, algorithm="giac")

[Out]

-1/280*(35*b^3*x^6 + 84*a*b^2*x^4 + 70*a^2*b*x^2 + 20*a^3)/x^14

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maple [A]  time = 0.01, size = 36, normalized size = 0.84 \begin {gather*} -\frac {b^{3}}{8 x^{8}}-\frac {3 a \,b^{2}}{10 x^{10}}-\frac {a^{2} b}{4 x^{12}}-\frac {a^{3}}{14 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^15,x)

[Out]

-1/14*a^3/x^14-1/4*a^2*b/x^12-3/10*a*b^2/x^10-1/8*b^3/x^8

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maxima [A]  time = 1.32, size = 37, normalized size = 0.86 \begin {gather*} -\frac {35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^15,x, algorithm="maxima")

[Out]

-1/280*(35*b^3*x^6 + 84*a*b^2*x^4 + 70*a^2*b*x^2 + 20*a^3)/x^14

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mupad [B]  time = 0.03, size = 37, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^3}{14}+\frac {a^2\,b\,x^2}{4}+\frac {3\,a\,b^2\,x^4}{10}+\frac {b^3\,x^6}{8}}{x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^15,x)

[Out]

-(a^3/14 + (b^3*x^6)/8 + (a^2*b*x^2)/4 + (3*a*b^2*x^4)/10)/x^14

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sympy [A]  time = 0.35, size = 39, normalized size = 0.91 \begin {gather*} \frac {- 20 a^{3} - 70 a^{2} b x^{2} - 84 a b^{2} x^{4} - 35 b^{3} x^{6}}{280 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**15,x)

[Out]

(-20*a**3 - 70*a**2*b*x**2 - 84*a*b**2*x**4 - 35*b**3*x**6)/(280*x**14)

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