3.249 \(\int \frac{1}{(a-b x^2)^5} \, dx\)

Optimal. Leaf size=100 \[ \frac{35 x}{128 a^4 \left (a-b x^2\right )}+\frac{35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac{7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac{35 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b}}+\frac{x}{8 a \left (a-b x^2\right )^4} \]

[Out]

x/(8*a*(a - b*x^2)^4) + (7*x)/(48*a^2*(a - b*x^2)^3) + (35*x)/(192*a^3*(a - b*x^2)^2) + (35*x)/(128*a^4*(a - b
*x^2)) + (35*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(9/2)*Sqrt[b])

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Rubi [A]  time = 0.0336952, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {199, 208} \[ \frac{35 x}{128 a^4 \left (a-b x^2\right )}+\frac{35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac{7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac{35 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b}}+\frac{x}{8 a \left (a-b x^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(-5),x]

[Out]

x/(8*a*(a - b*x^2)^4) + (7*x)/(48*a^2*(a - b*x^2)^3) + (35*x)/(192*a^3*(a - b*x^2)^2) + (35*x)/(128*a^4*(a - b
*x^2)) + (35*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(9/2)*Sqrt[b])

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{1}{\left (a-b x^2\right )^5} \, dx &=\frac{x}{8 a \left (a-b x^2\right )^4}+\frac{7 \int \frac{1}{\left (a-b x^2\right )^4} \, dx}{8 a}\\ &=\frac{x}{8 a \left (a-b x^2\right )^4}+\frac{7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac{35 \int \frac{1}{\left (a-b x^2\right )^3} \, dx}{48 a^2}\\ &=\frac{x}{8 a \left (a-b x^2\right )^4}+\frac{7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac{35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac{35 \int \frac{1}{\left (a-b x^2\right )^2} \, dx}{64 a^3}\\ &=\frac{x}{8 a \left (a-b x^2\right )^4}+\frac{7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac{35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac{35 x}{128 a^4 \left (a-b x^2\right )}+\frac{35 \int \frac{1}{a-b x^2} \, dx}{128 a^4}\\ &=\frac{x}{8 a \left (a-b x^2\right )^4}+\frac{7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac{35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac{35 x}{128 a^4 \left (a-b x^2\right )}+\frac{35 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{9/2} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0438093, size = 79, normalized size = 0.79 \[ \frac{\frac{\sqrt{a} x \left (-511 a^2 b x^2+279 a^3+385 a b^2 x^4-105 b^3 x^6\right )}{\left (a-b x^2\right )^4}+\frac{105 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}}{384 a^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(-5),x]

[Out]

((Sqrt[a]*x*(279*a^3 - 511*a^2*b*x^2 + 385*a*b^2*x^4 - 105*b^3*x^6))/(a - b*x^2)^4 + (105*ArcTanh[(Sqrt[b]*x)/
Sqrt[a]])/Sqrt[b])/(384*a^(9/2))

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Maple [A]  time = 0.004, size = 107, normalized size = 1.1 \begin{align*}{\frac{x}{8\,a \left ( b{x}^{2}-a \right ) ^{4}}}+{\frac{7}{8\,a} \left ( -{\frac{x}{6\,a \left ( b{x}^{2}-a \right ) ^{3}}}-{\frac{5}{6\,a} \left ( -{\frac{x}{4\,a \left ( b{x}^{2}-a \right ) ^{2}}}-{\frac{3}{4\,a} \left ( -{\frac{x}{2\,a \left ( b{x}^{2}-a \right ) }}+{\frac{1}{2\,a}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x/a/(b*x^2-a)^4+7/8/a*(-1/6*x/a/(b*x^2-a)^3-5/6/a*(-1/4*x/a/(b*x^2-a)^2-3/4/a*(-1/2*x/a/(b*x^2-a)+1/2/a/(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(1/(-b*x^2+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26721, size = 694, normalized size = 6.94 \begin{align*} \left [-\frac{210 \, a b^{4} x^{7} - 770 \, a^{2} b^{3} x^{5} + 1022 \, a^{3} b^{2} x^{3} - 558 \, a^{4} b x - 105 \,{\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^{5} b^{5} x^{8} - 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} - 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}, -\frac{105 \, a b^{4} x^{7} - 385 \, a^{2} b^{3} x^{5} + 511 \, a^{3} b^{2} x^{3} - 279 \, a^{4} b x + 105 \,{\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^{5} b^{5} x^{8} - 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} - 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+a)^5,x, algorithm="fricas")

[Out]

[-1/768*(210*a*b^4*x^7 - 770*a^2*b^3*x^5 + 1022*a^3*b^2*x^3 - 558*a^4*b*x - 105*(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^5*b^5*x^8 - 4*a^6*b^4
*x^6 + 6*a^7*b^3*x^4 - 4*a^8*b^2*x^2 + a^9*b), -1/384*(105*a*b^4*x^7 - 385*a^2*b^3*x^5 + 511*a^3*b^2*x^3 - 279
*a^4*b*x + 105*(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^5*b^5*x^8 - 4*a^6*b^4*x^6 + 6*a^7*b^3*x^4 - 4*a^8*b^2*x^2 + a^9*b)]

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Sympy [A]  time = 0.981893, size = 146, normalized size = 1.46 \begin{align*} - \frac{35 \sqrt{\frac{1}{a^{9} b}} \log{\left (- a^{5} \sqrt{\frac{1}{a^{9} b}} + x \right )}}{256} + \frac{35 \sqrt{\frac{1}{a^{9} b}} \log{\left (a^{5} \sqrt{\frac{1}{a^{9} b}} + x \right )}}{256} - \frac{- 279 a^{3} x + 511 a^{2} b x^{3} - 385 a b^{2} x^{5} + 105 b^{3} x^{7}}{384 a^{8} - 1536 a^{7} b x^{2} + 2304 a^{6} b^{2} x^{4} - 1536 a^{5} b^{3} x^{6} + 384 a^{4} b^{4} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x**2+a)**5,x)

[Out]

-35*sqrt(1/(a**9*b))*log(-a**5*sqrt(1/(a**9*b)) + x)/256 + 35*sqrt(1/(a**9*b))*log(a**5*sqrt(1/(a**9*b)) + x)/
256 - (-279*a**3*x + 511*a**2*b*x**3 - 385*a*b**2*x**5 + 105*b**3*x**7)/(384*a**8 - 1536*a**7*b*x**2 + 2304*a*
*6*b**2*x**4 - 1536*a**5*b**3*x**6 + 384*a**4*b**4*x**8)

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Giac [A]  time = 2.52578, size = 96, normalized size = 0.96 \begin{align*} -\frac{35 \, \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{128 \, \sqrt{-a b} a^{4}} - \frac{105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x}{384 \,{\left (b x^{2} - a\right )}^{4} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x^2+a)^5,x, algorithm="giac")

[Out]

-35/128*arctan(b*x/sqrt(-a*b))/(sqrt(-a*b)*a^4) - 1/384*(105*b^3*x^7 - 385*a*b^2*x^5 + 511*a^2*b*x^3 - 279*a^3
*x)/((b*x^2 - a)^4*a^4)