3.247 \(\int \frac{x^2}{(a-b x^2)^5} \, dx\)

Optimal. Leaf size=109 \[ -\frac{5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2}}-\frac{5 x}{128 a^3 b \left (a-b x^2\right )}-\frac{5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac{x}{48 a b \left (a-b x^2\right )^3}+\frac{x}{8 b \left (a-b x^2\right )^4} \]

[Out]

x/(8*b*(a - b*x^2)^4) - x/(48*a*b*(a - b*x^2)^3) - (5*x)/(192*a^2*b*(a - b*x^2)^2) - (5*x)/(128*a^3*b*(a - b*x
^2)) - (5*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(7/2)*b^(3/2))

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Rubi [A]  time = 0.0368752, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {288, 199, 208} \[ -\frac{5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2}}-\frac{5 x}{128 a^3 b \left (a-b x^2\right )}-\frac{5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac{x}{48 a b \left (a-b x^2\right )^3}+\frac{x}{8 b \left (a-b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a - b*x^2)^5,x]

[Out]

x/(8*b*(a - b*x^2)^4) - x/(48*a*b*(a - b*x^2)^3) - (5*x)/(192*a^2*b*(a - b*x^2)^2) - (5*x)/(128*a^3*b*(a - b*x
^2)) - (5*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(7/2)*b^(3/2))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^2}{\left (a-b x^2\right )^5} \, dx &=\frac{x}{8 b \left (a-b x^2\right )^4}-\frac{\int \frac{1}{\left (a-b x^2\right )^4} \, dx}{8 b}\\ &=\frac{x}{8 b \left (a-b x^2\right )^4}-\frac{x}{48 a b \left (a-b x^2\right )^3}-\frac{5 \int \frac{1}{\left (a-b x^2\right )^3} \, dx}{48 a b}\\ &=\frac{x}{8 b \left (a-b x^2\right )^4}-\frac{x}{48 a b \left (a-b x^2\right )^3}-\frac{5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac{5 \int \frac{1}{\left (a-b x^2\right )^2} \, dx}{64 a^2 b}\\ &=\frac{x}{8 b \left (a-b x^2\right )^4}-\frac{x}{48 a b \left (a-b x^2\right )^3}-\frac{5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac{5 x}{128 a^3 b \left (a-b x^2\right )}-\frac{5 \int \frac{1}{a-b x^2} \, dx}{128 a^3 b}\\ &=\frac{x}{8 b \left (a-b x^2\right )^4}-\frac{x}{48 a b \left (a-b x^2\right )^3}-\frac{5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac{5 x}{128 a^3 b \left (a-b x^2\right )}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0477683, size = 81, normalized size = 0.74 \[ \frac{73 a^2 b x^3+15 a^3 x-55 a b^2 x^5+15 b^3 x^7}{384 a^3 b \left (a-b x^2\right )^4}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a - b*x^2)^5,x]

[Out]

(15*a^3*x + 73*a^2*b*x^3 - 55*a*b^2*x^5 + 15*b^3*x^7)/(384*a^3*b*(a - b*x^2)^4) - (5*ArcTanh[(Sqrt[b]*x)/Sqrt[
a]])/(128*a^(7/2)*b^(3/2))

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Maple [A]  time = 0.009, size = 72, normalized size = 0.7 \begin{align*} -{\frac{1}{ \left ( b{x}^{2}-a \right ) ^{4}} \left ( -{\frac{5\,{b}^{2}{x}^{7}}{128\,{a}^{3}}}+{\frac{55\,b{x}^{5}}{384\,{a}^{2}}}-{\frac{73\,{x}^{3}}{384\,a}}-{\frac{5\,x}{128\,b}} \right ) }-{\frac{5}{128\,{a}^{3}b}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-5/128*b^2/a^3*x^7+55/384*b/a^2*x^5-73/384/a*x^3-5/128*x/b)/(b*x^2-a)^4-5/128/a^3/b/(a*b)^(1/2)*arctanh(b*x/
(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28953, size = 684, normalized size = 6.28 \begin{align*} \left [\frac{30 \, a b^{4} x^{7} - 110 \, a^{2} b^{3} x^{5} + 146 \, a^{3} b^{2} x^{3} + 30 \, a^{4} b x + 15 \,{\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{a b} x + a}{b x^{2} - a}\right )}{768 \,{\left (a^{4} b^{6} x^{8} - 4 \, a^{5} b^{5} x^{6} + 6 \, a^{6} b^{4} x^{4} - 4 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}}, \frac{15 \, a b^{4} x^{7} - 55 \, a^{2} b^{3} x^{5} + 73 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x + 15 \,{\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b} x}{a}\right )}{384 \,{\left (a^{4} b^{6} x^{8} - 4 \, a^{5} b^{5} x^{6} + 6 \, a^{6} b^{4} x^{4} - 4 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(30*a*b^4*x^7 - 110*a^2*b^3*x^5 + 146*a^3*b^2*x^3 + 30*a^4*b*x + 15*(b^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2*
x^4 - 4*a^3*b*x^2 + a^4)*sqrt(a*b)*log((b*x^2 - 2*sqrt(a*b)*x + a)/(b*x^2 - a)))/(a^4*b^6*x^8 - 4*a^5*b^5*x^6
+ 6*a^6*b^4*x^4 - 4*a^7*b^3*x^2 + a^8*b^2), 1/384*(15*a*b^4*x^7 - 55*a^2*b^3*x^5 + 73*a^3*b^2*x^3 + 15*a^4*b*x
+ 15*(b^4*x^8 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4 - 4*a^3*b*x^2 + a^4)*sqrt(-a*b)*arctan(sqrt(-a*b)*x/a))/(a^4*b^6*
x^8 - 4*a^5*b^5*x^6 + 6*a^6*b^4*x^4 - 4*a^7*b^3*x^2 + a^8*b^2)]

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Sympy [A]  time = 0.851353, size = 160, normalized size = 1.47 \begin{align*} \frac{5 \sqrt{\frac{1}{a^{7} b^{3}}} \log{\left (- a^{4} b \sqrt{\frac{1}{a^{7} b^{3}}} + x \right )}}{256} - \frac{5 \sqrt{\frac{1}{a^{7} b^{3}}} \log{\left (a^{4} b \sqrt{\frac{1}{a^{7} b^{3}}} + x \right )}}{256} + \frac{15 a^{3} x + 73 a^{2} b x^{3} - 55 a b^{2} x^{5} + 15 b^{3} x^{7}}{384 a^{7} b - 1536 a^{6} b^{2} x^{2} + 2304 a^{5} b^{3} x^{4} - 1536 a^{4} b^{4} x^{6} + 384 a^{3} b^{5} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*sqrt(1/(a**7*b**3))*log(-a**4*b*sqrt(1/(a**7*b**3)) + x)/256 - 5*sqrt(1/(a**7*b**3))*log(a**4*b*sqrt(1/(a**7
*b**3)) + x)/256 + (15*a**3*x + 73*a**2*b*x**3 - 55*a*b**2*x**5 + 15*b**3*x**7)/(384*a**7*b - 1536*a**6*b**2*x
**2 + 2304*a**5*b**3*x**4 - 1536*a**4*b**4*x**6 + 384*a**3*b**5*x**8)

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Giac [A]  time = 2.57477, size = 104, normalized size = 0.95 \begin{align*} \frac{5 \, \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{128 \, \sqrt{-a b} a^{3} b} + \frac{15 \, b^{3} x^{7} - 55 \, a b^{2} x^{5} + 73 \, a^{2} b x^{3} + 15 \, a^{3} x}{384 \,{\left (b x^{2} - a\right )}^{4} a^{3} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/128*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a^3*b) + 1/384*(15*b^3*x^7 - 55*a*b^2*x^5 + 73*a^2*b*x^3 + 15*a^3*x)/
((b*x^2 - a)^4*a^3*b)