3.1.66 \(\int \frac {(a+b x^2)^5}{x^{15}} \, dx\)

Optimal. Leaf size=40 \[ \frac {b \left (a+b x^2\right )^6}{84 a^2 x^{12}}-\frac {\left (a+b x^2\right )^6}{14 a x^{14}} \]

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 45, 37} \begin {gather*} \frac {b \left (a+b x^2\right )^6}{84 a^2 x^{12}}-\frac {\left (a+b x^2\right )^6}{14 a x^{14}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^15,x]

[Out]

-(a + b*x^2)^6/(14*a*x^14) + (b*(a + b*x^2)^6)/(84*a^2*x^12)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^{15}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^6}{14 a x^{14}}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^2\right )}{14 a}\\ &=-\frac {\left (a+b x^2\right )^6}{14 a x^{14}}+\frac {b \left (a+b x^2\right )^6}{84 a^2 x^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 67, normalized size = 1.68 \begin {gather*} -\frac {a^5}{14 x^{14}}-\frac {5 a^4 b}{12 x^{12}}-\frac {a^3 b^2}{x^{10}}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{6 x^6}-\frac {b^5}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^15,x]

[Out]

-1/14*a^5/x^14 - (5*a^4*b)/(12*x^12) - (a^3*b^2)/x^10 - (5*a^2*b^3)/(4*x^8) - (5*a*b^4)/(6*x^6) - b^5/(4*x^4)

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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^{15}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^15,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^15, x]

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fricas [A]  time = 0.88, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^15,x, algorithm="fricas")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*x^2 + 6*a^5)/x^14

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giac [A]  time = 1.06, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^15,x, algorithm="giac")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*x^2 + 6*a^5)/x^14

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maple [A]  time = 0.00, size = 58, normalized size = 1.45 \begin {gather*} -\frac {b^{5}}{4 x^{4}}-\frac {5 a \,b^{4}}{6 x^{6}}-\frac {5 a^{2} b^{3}}{4 x^{8}}-\frac {a^{3} b^{2}}{x^{10}}-\frac {5 a^{4} b}{12 x^{12}}-\frac {a^{5}}{14 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^15,x)

[Out]

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

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maxima [A]  time = 1.34, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^15,x, algorithm="maxima")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*x^2 + 6*a^5)/x^14

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mupad [B]  time = 4.74, size = 58, normalized size = 1.45 \begin {gather*} -\frac {\frac {a^5}{14}+\frac {5\,a^4\,b\,x^2}{12}+a^3\,b^2\,x^4+\frac {5\,a^2\,b^3\,x^6}{4}+\frac {5\,a\,b^4\,x^8}{6}+\frac {b^5\,x^{10}}{4}}{x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^15,x)

[Out]

-(a^5/14 + (b^5*x^10)/4 + (5*a^4*b*x^2)/12 + (5*a*b^4*x^8)/6 + a^3*b^2*x^4 + (5*a^2*b^3*x^6)/4)/x^14

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sympy [A]  time = 0.51, size = 63, normalized size = 1.58 \begin {gather*} \frac {- 6 a^{5} - 35 a^{4} b x^{2} - 84 a^{3} b^{2} x^{4} - 105 a^{2} b^{3} x^{6} - 70 a b^{4} x^{8} - 21 b^{5} x^{10}}{84 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**15,x)

[Out]

(-6*a**5 - 35*a**4*b*x**2 - 84*a**3*b**2*x**4 - 105*a**2*b**3*x**6 - 70*a*b**4*x**8 - 21*b**5*x**10)/(84*x**14
)

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