3.3.49 \(\int \frac {1}{(a-b x^2)^5} \, dx\) [249]

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

[Out]

1/8*x/a/(-b*x^2+a)^4+7/48*x/a^2/(-b*x^2+a)^3+35/192*x/a^3/(-b*x^2+a)^2+35/128*x/a^4/(-b*x^2+a)+35/128*arctanh(
x*b^(1/2)/a^(1/2))/a^(9/2)/b^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {205, 214} \begin {gather*} \frac {35 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{9/2} \sqrt {b}}+\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 {x}{8 a \left (a-b x^2\right )^4} \end {gather*}

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 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 214

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

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Mathematica [A]
time = 0.03, size = 79, normalized size = 0.79 \begin {gather*} \frac {\frac {\sqrt {a} x \left (279 a^3-511 a^2 b x^2+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}} \end {gather*}

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 = 0.05, size = 103, normalized size = 1.03

method	result	size
risch	\(\frac {-\frac {35 b^{3} x^{7}}{128 a^{4}}+\frac {385 b^{2} x^{5}}{384 a^{3}}-\frac {511 b \,x^{3}}{384 a^{2}}+\frac {93 x}{128 a}}{\left (-b \,x^{2}+a \right )^{4}}+\frac {35 \ln \left (b x +\sqrt {a b}\right )}{256 \sqrt {a b}\, a^{4}}-\frac {35 \ln \left (-b x +\sqrt {a b}\right )}{256 \sqrt {a b}\, a^{4}}\)	\(92\)
default	\(\frac {x}{8 a \left (-b \,x^{2}+a \right )^{4}}+\frac {\frac {7 x}{48 a \left (-b \,x^{2}+a \right )^{3}}+\frac {7 \left (\frac {5 x}{24 a \left (-b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {3 x}{8 a \left (-b \,x^{2}+a \right )}+\frac {3 \arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{6 a}\right )}{8 a}}{a}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 [A]
time = 0.48, size = 117, normalized size = 1.17 \begin {gather*} -\frac {105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x}{384 \, {\left (a^{4} b^{4} x^{8} - 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} - 4 \, a^{7} b x^{2} + a^{8}\right )}} - \frac {35 \, \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.08, size = 320, normalized size = 3.20 \begin {gather*} \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 {gather*}

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 = 0.23, size = 146, normalized size = 1.46 \begin {gather*} - \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 {gather*}

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 = 1.56, size = 71, normalized size = 0.71 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 4.79, size = 99, normalized size = 0.99 \begin {gather*} \frac {\frac {93\,x}{128\,a}-\frac {511\,b\,x^3}{384\,a^2}+\frac {385\,b^2\,x^5}{384\,a^3}-\frac {35\,b^3\,x^7}{128\,a^4}}{a^4-4\,a^3\,b\,x^2+6\,a^2\,b^2\,x^4-4\,a\,b^3\,x^6+b^4\,x^8}+\frac {35\,\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{128\,a^{9/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*x^2)^5,x)

[Out]

((93*x)/(128*a) - (511*b*x^3)/(384*a^2) + (385*b^2*x^5)/(384*a^3) - (35*b^3*x^7)/(128*a^4))/(a^4 + b^4*x^8 - 4
*a^3*b*x^2 - 4*a*b^3*x^6 + 6*a^2*b^2*x^4) + (35*atanh((b^(1/2)*x)/a^(1/2)))/(128*a^(9/2)*b^(1/2))

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