3.3.3 \(\int \frac {x^3}{(a+b x^2)^{10}} \, dx\) [203]

Optimal. Leaf size=34 \[ \frac {a}{18 b^2 \left (a+b x^2\right )^9}-\frac {1}{16 b^2 \left (a+b x^2\right )^8} \]

[Out]

1/18*a/b^2/(b*x^2+a)^9-1/16/b^2/(b*x^2+a)^8

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 34, 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}{18 b^2 \left (a+b x^2\right )^9}-\frac {1}{16 b^2 \left (a+b x^2\right )^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^10,x]

[Out]

a/(18*b^2*(a + b*x^2)^9) - 1/(16*b^2*(a + b*x^2)^8)

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^3}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{10}}+\frac {1}{b (a+b x)^9}\right ) \, dx,x,x^2\right )\\ &=\frac {a}{18 b^2 \left (a+b x^2\right )^9}-\frac {1}{16 b^2 \left (a+b x^2\right )^8}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 24, normalized size = 0.71 \begin {gather*} -\frac {a+9 b x^2}{144 b^2 \left (a+b x^2\right )^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^10,x]

[Out]

-1/144*(a + 9*b*x^2)/(b^2*(a + b*x^2)^9)

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 31, normalized size = 0.91

method	result	size
gosper	\(-\frac {9 b \,x^{2}+a}{144 \left (b \,x^{2}+a \right )^{9} b^{2}}\)	\(23\)
norman	\(\frac {-\frac {a}{144 b^{2}}-\frac {x^{2}}{16 b}}{\left (b \,x^{2}+a \right )^{9}}\)	\(26\)
risch	\(\frac {-\frac {a}{144 b^{2}}-\frac {x^{2}}{16 b}}{\left (b \,x^{2}+a \right )^{9}}\)	\(26\)
default	\(\frac {a}{18 b^{2} \left (b \,x^{2}+a \right )^{9}}-\frac {1}{16 b^{2} \left (b \,x^{2}+a \right )^{8}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2+a)^10,x,method=_RETURNVERBOSE)

[Out]

1/18*a/b^2/(b*x^2+a)^9-1/16/b^2/(b*x^2+a)^8

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (30) = 60\).
time = 0.30, size = 113, normalized size = 3.32 \begin {gather*} -\frac {9 \, b x^{2} + a}{144 \, {\left (b^{11} x^{18} + 9 \, a b^{10} x^{16} + 36 \, a^{2} b^{9} x^{14} + 84 \, a^{3} b^{8} x^{12} + 126 \, a^{4} b^{7} x^{10} + 126 \, a^{5} b^{6} x^{8} + 84 \, a^{6} b^{5} x^{6} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{2} + a^{9} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/144*(9*b*x^2 + a)/(b^11*x^18 + 9*a*b^10*x^16 + 36*a^2*b^9*x^14 + 84*a^3*b^8*x^12 + 126*a^4*b^7*x^10 + 126*a
^5*b^6*x^8 + 84*a^6*b^5*x^6 + 36*a^7*b^4*x^4 + 9*a^8*b^3*x^2 + a^9*b^2)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (30) = 60\).
time = 1.53, size = 113, normalized size = 3.32 \begin {gather*} -\frac {9 \, b x^{2} + a}{144 \, {\left (b^{11} x^{18} + 9 \, a b^{10} x^{16} + 36 \, a^{2} b^{9} x^{14} + 84 \, a^{3} b^{8} x^{12} + 126 \, a^{4} b^{7} x^{10} + 126 \, a^{5} b^{6} x^{8} + 84 \, a^{6} b^{5} x^{6} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{2} + a^{9} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/144*(9*b*x^2 + a)/(b^11*x^18 + 9*a*b^10*x^16 + 36*a^2*b^9*x^14 + 84*a^3*b^8*x^12 + 126*a^4*b^7*x^10 + 126*a
^5*b^6*x^8 + 84*a^6*b^5*x^6 + 36*a^7*b^4*x^4 + 9*a^8*b^3*x^2 + a^9*b^2)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (29) = 58\).
time = 0.41, size = 119, normalized size = 3.50 \begin {gather*} \frac {- a - 9 b x^{2}}{144 a^{9} b^{2} + 1296 a^{8} b^{3} x^{2} + 5184 a^{7} b^{4} x^{4} + 12096 a^{6} b^{5} x^{6} + 18144 a^{5} b^{6} x^{8} + 18144 a^{4} b^{7} x^{10} + 12096 a^{3} b^{8} x^{12} + 5184 a^{2} b^{9} x^{14} + 1296 a b^{10} x^{16} + 144 b^{11} x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**2+a)**10,x)

[Out]

(-a - 9*b*x**2)/(144*a**9*b**2 + 1296*a**8*b**3*x**2 + 5184*a**7*b**4*x**4 + 12096*a**6*b**5*x**6 + 18144*a**5
*b**6*x**8 + 18144*a**4*b**7*x**10 + 12096*a**3*b**8*x**12 + 5184*a**2*b**9*x**14 + 1296*a*b**10*x**16 + 144*b
**11*x**18)

________________________________________________________________________________________

Giac [A]
time = 1.25, size = 22, normalized size = 0.65 \begin {gather*} -\frac {9 \, b x^{2} + a}{144 \, {\left (b x^{2} + a\right )}^{9} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-1/144*(9*b*x^2 + a)/((b*x^2 + a)^9*b^2)

________________________________________________________________________________________

Mupad [B]
time = 0.11, size = 114, normalized size = 3.35 \begin {gather*} -\frac {\frac {a}{144\,b^2}+\frac {x^2}{16\,b}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^2)^10,x)

[Out]

-(a/(144*b^2) + x^2/(16*b))/(a^9 + b^9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 36*a^7*b^2*x^4 + 84*a^6*b^3*x^6 + 1
26*a^5*b^4*x^8 + 126*a^4*b^5*x^10 + 84*a^3*b^6*x^12 + 36*a^2*b^7*x^14)

________________________________________________________________________________________