3.1.68 \(\int \frac {(a+b x^2)^5}{x^{19}} \, dx\)

Optimal. Leaf size=69 \[ -\frac {a^5}{18 x^{18}}-\frac {5 a^4 b}{16 x^{16}}-\frac {5 a^3 b^2}{7 x^{14}}-\frac {5 a^2 b^3}{6 x^{12}}-\frac {a b^4}{2 x^{10}}-\frac {b^5}{8 x^8} \]

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Rubi [A]  time = 0.03, antiderivative size = 69, 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}{7 x^{14}}-\frac {5 a^2 b^3}{6 x^{12}}-\frac {5 a^4 b}{16 x^{16}}-\frac {a^5}{18 x^{18}}-\frac {a b^4}{2 x^{10}}-\frac {b^5}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^19,x]

[Out]

-a^5/(18*x^18) - (5*a^4*b)/(16*x^16) - (5*a^3*b^2)/(7*x^14) - (5*a^2*b^3)/(6*x^12) - (a*b^4)/(2*x^10) - b^5/(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 )^5}{x^{19}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^5}{x^{10}}+\frac {5 a^4 b}{x^9}+\frac {10 a^3 b^2}{x^8}+\frac {10 a^2 b^3}{x^7}+\frac {5 a b^4}{x^6}+\frac {b^5}{x^5}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^5}{18 x^{18}}-\frac {5 a^4 b}{16 x^{16}}-\frac {5 a^3 b^2}{7 x^{14}}-\frac {5 a^2 b^3}{6 x^{12}}-\frac {a b^4}{2 x^{10}}-\frac {b^5}{8 x^8}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.00 \begin {gather*} -\frac {a^5}{18 x^{18}}-\frac {5 a^4 b}{16 x^{16}}-\frac {5 a^3 b^2}{7 x^{14}}-\frac {5 a^2 b^3}{6 x^{12}}-\frac {a b^4}{2 x^{10}}-\frac {b^5}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^19,x]

[Out]

-1/18*a^5/x^18 - (5*a^4*b)/(16*x^16) - (5*a^3*b^2)/(7*x^14) - (5*a^2*b^3)/(6*x^12) - (a*b^4)/(2*x^10) - b^5/(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 )^5}{x^{19}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^19,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^19, x]

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fricas [A]  time = 1.12, size = 59, normalized size = 0.86 \begin {gather*} -\frac {126 \, b^{5} x^{10} + 504 \, a b^{4} x^{8} + 840 \, a^{2} b^{3} x^{6} + 720 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} + 56 \, a^{5}}{1008 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^19,x, algorithm="fricas")

[Out]

-1/1008*(126*b^5*x^10 + 504*a*b^4*x^8 + 840*a^2*b^3*x^6 + 720*a^3*b^2*x^4 + 315*a^4*b*x^2 + 56*a^5)/x^18

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giac [A]  time = 0.94, size = 59, normalized size = 0.86 \begin {gather*} -\frac {126 \, b^{5} x^{10} + 504 \, a b^{4} x^{8} + 840 \, a^{2} b^{3} x^{6} + 720 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} + 56 \, a^{5}}{1008 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^19,x, algorithm="giac")

[Out]

-1/1008*(126*b^5*x^10 + 504*a*b^4*x^8 + 840*a^2*b^3*x^6 + 720*a^3*b^2*x^4 + 315*a^4*b*x^2 + 56*a^5)/x^18

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maple [A]  time = 0.01, size = 58, normalized size = 0.84 \begin {gather*} -\frac {b^{5}}{8 x^{8}}-\frac {a \,b^{4}}{2 x^{10}}-\frac {5 a^{2} b^{3}}{6 x^{12}}-\frac {5 a^{3} b^{2}}{7 x^{14}}-\frac {5 a^{4} b}{16 x^{16}}-\frac {a^{5}}{18 x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^19,x)

[Out]

-1/18*a^5/x^18-5/16*a^4*b/x^16-5/7*a^3*b^2/x^14-5/6*a^2*b^3/x^12-1/2*a*b^4/x^10-1/8*b^5/x^8

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maxima [A]  time = 1.36, size = 59, normalized size = 0.86 \begin {gather*} -\frac {126 \, b^{5} x^{10} + 504 \, a b^{4} x^{8} + 840 \, a^{2} b^{3} x^{6} + 720 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} + 56 \, a^{5}}{1008 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^19,x, algorithm="maxima")

[Out]

-1/1008*(126*b^5*x^10 + 504*a*b^4*x^8 + 840*a^2*b^3*x^6 + 720*a^3*b^2*x^4 + 315*a^4*b*x^2 + 56*a^5)/x^18

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mupad [B]  time = 0.04, size = 59, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^5}{18}+\frac {5\,a^4\,b\,x^2}{16}+\frac {5\,a^3\,b^2\,x^4}{7}+\frac {5\,a^2\,b^3\,x^6}{6}+\frac {a\,b^4\,x^8}{2}+\frac {b^5\,x^{10}}{8}}{x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^19,x)

[Out]

-(a^5/18 + (b^5*x^10)/8 + (5*a^4*b*x^2)/16 + (a*b^4*x^8)/2 + (5*a^3*b^2*x^4)/7 + (5*a^2*b^3*x^6)/6)/x^18

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sympy [A]  time = 0.59, size = 63, normalized size = 0.91 \begin {gather*} \frac {- 56 a^{5} - 315 a^{4} b x^{2} - 720 a^{3} b^{2} x^{4} - 840 a^{2} b^{3} x^{6} - 504 a b^{4} x^{8} - 126 b^{5} x^{10}}{1008 x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**19,x)

[Out]

(-56*a**5 - 315*a**4*b*x**2 - 720*a**3*b**2*x**4 - 840*a**2*b**3*x**6 - 504*a*b**4*x**8 - 126*b**5*x**10)/(100
8*x**18)

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