3.2.24 \(\int \frac {x^{11}}{a+b x^2} \, dx\) [124]

Optimal. Leaf size=79 \[ \frac {a^4 x^2}{2 b^5}-\frac {a^3 x^4}{4 b^4}+\frac {a^2 x^6}{6 b^3}-\frac {a x^8}{8 b^2}+\frac {x^{10}}{10 b}-\frac {a^5 \log \left (a+b x^2\right )}{2 b^6} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 79, 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 = {272, 45} \begin {gather*} -\frac {a^5 \log \left (a+b x^2\right )}{2 b^6}+\frac {a^4 x^2}{2 b^5}-\frac {a^3 x^4}{4 b^4}+\frac {a^2 x^6}{6 b^3}-\frac {a x^8}{8 b^2}+\frac {x^{10}}{10 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^2),x]

[Out]

(a^4*x^2)/(2*b^5) - (a^3*x^4)/(4*b^4) + (a^2*x^6)/(6*b^3) - (a*x^8)/(8*b^2) + x^10/(10*b) - (a^5*Log[a + b*x^2
])/(2*b^6)

Rule 45

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 272

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 {x^{11}}{a+b x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^5}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^4}{b^5}-\frac {a^3 x}{b^4}+\frac {a^2 x^2}{b^3}-\frac {a x^3}{b^2}+\frac {x^4}{b}-\frac {a^5}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a^4 x^2}{2 b^5}-\frac {a^3 x^4}{4 b^4}+\frac {a^2 x^6}{6 b^3}-\frac {a x^8}{8 b^2}+\frac {x^{10}}{10 b}-\frac {a^5 \log \left (a+b x^2\right )}{2 b^6}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 79, normalized size = 1.00 \begin {gather*} \frac {a^4 x^2}{2 b^5}-\frac {a^3 x^4}{4 b^4}+\frac {a^2 x^6}{6 b^3}-\frac {a x^8}{8 b^2}+\frac {x^{10}}{10 b}-\frac {a^5 \log \left (a+b x^2\right )}{2 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^2),x]

[Out]

(a^4*x^2)/(2*b^5) - (a^3*x^4)/(4*b^4) + (a^2*x^6)/(6*b^3) - (a*x^8)/(8*b^2) + x^10/(10*b) - (a^5*Log[a + b*x^2
])/(2*b^6)

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Maple [A]
time = 0.05, size = 68, normalized size = 0.86

method	result	size
default	\(\frac {\frac {1}{5} b^{4} x^{10}-\frac {1}{4} a \,b^{3} x^{8}+\frac {1}{3} a^{2} b^{2} x^{6}-\frac {1}{2} a^{3} b \,x^{4}+a^{4} x^{2}}{2 b^{5}}-\frac {a^{5} \ln \left (b \,x^{2}+a \right )}{2 b^{6}}\)	\(68\)
norman	\(\frac {a^{4} x^{2}}{2 b^{5}}-\frac {a^{3} x^{4}}{4 b^{4}}+\frac {a^{2} x^{6}}{6 b^{3}}-\frac {a \,x^{8}}{8 b^{2}}+\frac {x^{10}}{10 b}-\frac {a^{5} \ln \left (b \,x^{2}+a \right )}{2 b^{6}}\)	\(68\)
risch	\(\frac {a^{4} x^{2}}{2 b^{5}}-\frac {a^{3} x^{4}}{4 b^{4}}+\frac {a^{2} x^{6}}{6 b^{3}}-\frac {a \,x^{8}}{8 b^{2}}+\frac {x^{10}}{10 b}-\frac {a^{5} \ln \left (b \,x^{2}+a \right )}{2 b^{6}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 68, normalized size = 0.86 \begin {gather*} -\frac {a^{5} \log \left (b x^{2} + a\right )}{2 \, b^{6}} + \frac {12 \, b^{4} x^{10} - 15 \, a b^{3} x^{8} + 20 \, a^{2} b^{2} x^{6} - 30 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a),x, algorithm="maxima")

[Out]

-1/2*a^5*log(b*x^2 + a)/b^6 + 1/120*(12*b^4*x^10 - 15*a*b^3*x^8 + 20*a^2*b^2*x^6 - 30*a^3*b*x^4 + 60*a^4*x^2)/
b^5

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Fricas [A]
time = 0.87, size = 67, normalized size = 0.85 \begin {gather*} \frac {12 \, b^{5} x^{10} - 15 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} - 30 \, a^{3} b^{2} x^{4} + 60 \, a^{4} b x^{2} - 60 \, a^{5} \log \left (b x^{2} + a\right )}{120 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a),x, algorithm="fricas")

[Out]

1/120*(12*b^5*x^10 - 15*a*b^4*x^8 + 20*a^2*b^3*x^6 - 30*a^3*b^2*x^4 + 60*a^4*b*x^2 - 60*a^5*log(b*x^2 + a))/b^
6

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Sympy [A]
time = 0.06, size = 68, normalized size = 0.86 \begin {gather*} - \frac {a^{5} \log {\left (a + b x^{2} \right )}}{2 b^{6}} + \frac {a^{4} x^{2}}{2 b^{5}} - \frac {a^{3} x^{4}}{4 b^{4}} + \frac {a^{2} x^{6}}{6 b^{3}} - \frac {a x^{8}}{8 b^{2}} + \frac {x^{10}}{10 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**2+a),x)

[Out]

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

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Giac [A]
time = 1.19, size = 69, normalized size = 0.87 \begin {gather*} -\frac {a^{5} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} + \frac {12 \, b^{4} x^{10} - 15 \, a b^{3} x^{8} + 20 \, a^{2} b^{2} x^{6} - 30 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a),x, algorithm="giac")

[Out]

-1/2*a^5*log(abs(b*x^2 + a))/b^6 + 1/120*(12*b^4*x^10 - 15*a*b^3*x^8 + 20*a^2*b^2*x^6 - 30*a^3*b*x^4 + 60*a^4*
x^2)/b^5

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Mupad [B]
time = 0.06, size = 67, normalized size = 0.85 \begin {gather*} \frac {x^{10}}{10\,b}-\frac {a\,x^8}{8\,b^2}-\frac {a^5\,\ln \left (b\,x^2+a\right )}{2\,b^6}+\frac {a^2\,x^6}{6\,b^3}-\frac {a^3\,x^4}{4\,b^4}+\frac {a^4\,x^2}{2\,b^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(a + b*x^2),x)

[Out]

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

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