3.1.81 \(\int \frac {(a+b x^2)^5}{x^{14}} \, dx\)

Optimal. Leaf size=67 \[ -\frac {a^5}{13 x^{13}}-\frac {5 a^4 b}{11 x^{11}}-\frac {10 a^3 b^2}{9 x^9}-\frac {10 a^2 b^3}{7 x^7}-\frac {a b^4}{x^5}-\frac {b^5}{3 x^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} -\frac {10 a^3 b^2}{9 x^9}-\frac {10 a^2 b^3}{7 x^7}-\frac {5 a^4 b}{11 x^{11}}-\frac {a^5}{13 x^{13}}-\frac {a b^4}{x^5}-\frac {b^5}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^14,x]

[Out]

-a^5/(13*x^13) - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^3)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^5}{x^{14}} \, dx &=\int \left (\frac {a^5}{x^{14}}+\frac {5 a^4 b}{x^{12}}+\frac {10 a^3 b^2}{x^{10}}+\frac {10 a^2 b^3}{x^8}+\frac {5 a b^4}{x^6}+\frac {b^5}{x^4}\right ) \, dx\\ &=-\frac {a^5}{13 x^{13}}-\frac {5 a^4 b}{11 x^{11}}-\frac {10 a^3 b^2}{9 x^9}-\frac {10 a^2 b^3}{7 x^7}-\frac {a b^4}{x^5}-\frac {b^5}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 67, normalized size = 1.00 \begin {gather*} -\frac {a^5}{13 x^{13}}-\frac {5 a^4 b}{11 x^{11}}-\frac {10 a^3 b^2}{9 x^9}-\frac {10 a^2 b^3}{7 x^7}-\frac {a b^4}{x^5}-\frac {b^5}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^14,x]

[Out]

-1/13*a^5/x^13 - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^3)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^5}{x^{14}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^14,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^14, x]

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 59, normalized size = 0.88 \begin {gather*} -\frac {3003 \, b^{5} x^{10} + 9009 \, a b^{4} x^{8} + 12870 \, a^{2} b^{3} x^{6} + 10010 \, a^{3} b^{2} x^{4} + 4095 \, a^{4} b x^{2} + 693 \, a^{5}}{9009 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^14,x, algorithm="fricas")

[Out]

-1/9009*(3003*b^5*x^10 + 9009*a*b^4*x^8 + 12870*a^2*b^3*x^6 + 10010*a^3*b^2*x^4 + 4095*a^4*b*x^2 + 693*a^5)/x^
13

________________________________________________________________________________________

giac [A]  time = 1.09, size = 59, normalized size = 0.88 \begin {gather*} -\frac {3003 \, b^{5} x^{10} + 9009 \, a b^{4} x^{8} + 12870 \, a^{2} b^{3} x^{6} + 10010 \, a^{3} b^{2} x^{4} + 4095 \, a^{4} b x^{2} + 693 \, a^{5}}{9009 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^14,x, algorithm="giac")

[Out]

-1/9009*(3003*b^5*x^10 + 9009*a*b^4*x^8 + 12870*a^2*b^3*x^6 + 10010*a^3*b^2*x^4 + 4095*a^4*b*x^2 + 693*a^5)/x^
13

________________________________________________________________________________________

maple [A]  time = 0.01, size = 58, normalized size = 0.87 \begin {gather*} -\frac {b^{5}}{3 x^{3}}-\frac {a \,b^{4}}{x^{5}}-\frac {10 a^{2} b^{3}}{7 x^{7}}-\frac {10 a^{3} b^{2}}{9 x^{9}}-\frac {5 a^{4} b}{11 x^{11}}-\frac {a^{5}}{13 x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^14,x)

[Out]

-1/13*a^5/x^13-5/11*a^4*b/x^11-10/9*a^3*b^2/x^9-10/7*a^2*b^3/x^7-a*b^4/x^5-1/3*b^5/x^3

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 59, normalized size = 0.88 \begin {gather*} -\frac {3003 \, b^{5} x^{10} + 9009 \, a b^{4} x^{8} + 12870 \, a^{2} b^{3} x^{6} + 10010 \, a^{3} b^{2} x^{4} + 4095 \, a^{4} b x^{2} + 693 \, a^{5}}{9009 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^14,x, algorithm="maxima")

[Out]

-1/9009*(3003*b^5*x^10 + 9009*a*b^4*x^8 + 12870*a^2*b^3*x^6 + 10010*a^3*b^2*x^4 + 4095*a^4*b*x^2 + 693*a^5)/x^
13

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 58, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^5}{13}+\frac {5\,a^4\,b\,x^2}{11}+\frac {10\,a^3\,b^2\,x^4}{9}+\frac {10\,a^2\,b^3\,x^6}{7}+a\,b^4\,x^8+\frac {b^5\,x^{10}}{3}}{x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^14,x)

[Out]

-(a^5/13 + (b^5*x^10)/3 + (5*a^4*b*x^2)/11 + a*b^4*x^8 + (10*a^3*b^2*x^4)/9 + (10*a^2*b^3*x^6)/7)/x^13

________________________________________________________________________________________

sympy [A]  time = 0.47, size = 63, normalized size = 0.94 \begin {gather*} \frac {- 693 a^{5} - 4095 a^{4} b x^{2} - 10010 a^{3} b^{2} x^{4} - 12870 a^{2} b^{3} x^{6} - 9009 a b^{4} x^{8} - 3003 b^{5} x^{10}}{9009 x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**14,x)

[Out]

(-693*a**5 - 4095*a**4*b*x**2 - 10010*a**3*b**2*x**4 - 12870*a**2*b**3*x**6 - 9009*a*b**4*x**8 - 3003*b**5*x**
10)/(9009*x**13)

________________________________________________________________________________________