3.6.24 \(\int \frac {x^3}{(a+b x^2)^{9/2}} \, dx\) [524]

Optimal. Leaf size=38 \[ \frac {a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac {1}{5 b^2 \left (a+b x^2\right )^{5/2}} \]

[Out]

1/7*a/b^2/(b*x^2+a)^(7/2)-1/5/b^2/(b*x^2+a)^(5/2)

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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \begin {gather*} \frac {a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac {1}{5 b^2 \left (a+b x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^2)^(9/2),x]

[Out]

a/(7*b^2*(a + b*x^2)^(7/2)) - 1/(5*b^2*(a + b*x^2)^(5/2))

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 )^{9/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^{9/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{9/2}}+\frac {1}{b (a+b x)^{7/2}}\right ) \, dx,x,x^2\right )\\ &=\frac {a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac {1}{5 b^2 \left (a+b x^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 28, normalized size = 0.74 \begin {gather*} \frac {-2 a-7 b x^2}{35 b^2 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^2)^(9/2),x]

[Out]

(-2*a - 7*b*x^2)/(35*b^2*(a + b*x^2)^(7/2))

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Maple [A]
time = 0.05, size = 34, normalized size = 0.89

method	result	size
gosper	\(-\frac {7 b \,x^{2}+2 a}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}\)	\(25\)
trager	\(-\frac {7 b \,x^{2}+2 a}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b^{2}}\)	\(25\)
default	\(-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*x^2/b/(b*x^2+a)^(7/2)-2/35*a/b^2/(b*x^2+a)^(7/2)

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Maxima [A]
time = 0.27, size = 33, normalized size = 0.87 \begin {gather*} -\frac {x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {2 \, a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-1/5*x^2/((b*x^2 + a)^(7/2)*b) - 2/35*a/((b*x^2 + a)^(7/2)*b^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (30) = 60\).
time = 1.01, size = 69, normalized size = 1.82 \begin {gather*} -\frac {{\left (7 \, b x^{2} + 2 \, a\right )} \sqrt {b x^{2} + a}}{35 \, {\left (b^{6} x^{8} + 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

-1/35*(7*b*x^2 + 2*a)*sqrt(b*x^2 + a)/(b^6*x^8 + 4*a*b^5*x^6 + 6*a^2*b^4*x^4 + 4*a^3*b^3*x^2 + a^4*b^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (32) = 64\).
time = 0.86, size = 180, normalized size = 4.74 \begin {gather*} \begin {cases} - \frac {2 a}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} - \frac {7 b x^{2}}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {9}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x**2+a)**(9/2),x)

[Out]

Piecewise((-2*a/(35*a**3*b**2*sqrt(a + b*x**2) + 105*a**2*b**3*x**2*sqrt(a + b*x**2) + 105*a*b**4*x**4*sqrt(a
+ b*x**2) + 35*b**5*x**6*sqrt(a + b*x**2)) - 7*b*x**2/(35*a**3*b**2*sqrt(a + b*x**2) + 105*a**2*b**3*x**2*sqrt
(a + b*x**2) + 105*a*b**4*x**4*sqrt(a + b*x**2) + 35*b**5*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**4/(4*a**(9/2)
), True))

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Giac [A]
time = 0.62, size = 24, normalized size = 0.63 \begin {gather*} -\frac {7 \, b x^{2} + 2 \, a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-1/35*(7*b*x^2 + 2*a)/((b*x^2 + a)^(7/2)*b^2)

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Mupad [B]
time = 4.82, size = 24, normalized size = 0.63 \begin {gather*} -\frac {7\,b\,x^2+2\,a}{35\,b^2\,{\left (b\,x^2+a\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^2)^(9/2),x)

[Out]

-(2*a + 7*b*x^2)/(35*b^2*(a + b*x^2)^(7/2))

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