3.2.90 \(\int \frac {1}{x^6 (a+b x^2)^3} \, dx\) [190]

Optimal. Leaf size=100 \[ -\frac {63}{40 a^3 x^5}+\frac {21 b}{8 a^4 x^3}-\frac {63 b^2}{8 a^5 x}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}+\frac {9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac {63 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2}} \]

[Out]

-63/40/a^3/x^5+21/8*b/a^4/x^3-63/8*b^2/a^5/x+1/4/a/x^5/(b*x^2+a)^2+9/8/a^2/x^5/(b*x^2+a)-63/8*b^(5/2)*arctan(x
*b^(1/2)/a^(1/2))/a^(11/2)

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Rubi [A]
time = 0.03, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {296, 331, 211} \begin {gather*} -\frac {63 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2}}-\frac {63 b^2}{8 a^5 x}+\frac {21 b}{8 a^4 x^3}-\frac {63}{40 a^3 x^5}+\frac {9}{8 a^2 x^5 \left (a+b x^2\right )}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^2)^3),x]

[Out]

-63/(40*a^3*x^5) + (21*b)/(8*a^4*x^3) - (63*b^2)/(8*a^5*x) + 1/(4*a*x^5*(a + b*x^2)^2) + 9/(8*a^2*x^5*(a + b*x
^2)) - (63*b^(5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2))

Rule 211

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

Rule 296

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

Rule 331

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

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (a+b x^2\right )^3} \, dx &=\frac {1}{4 a x^5 \left (a+b x^2\right )^2}+\frac {9 \int \frac {1}{x^6 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac {1}{4 a x^5 \left (a+b x^2\right )^2}+\frac {9}{8 a^2 x^5 \left (a+b x^2\right )}+\frac {63 \int \frac {1}{x^6 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac {63}{40 a^3 x^5}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}+\frac {9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac {(63 b) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{8 a^3}\\ &=-\frac {63}{40 a^3 x^5}+\frac {21 b}{8 a^4 x^3}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}+\frac {9}{8 a^2 x^5 \left (a+b x^2\right )}+\frac {\left (63 b^2\right ) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{8 a^4}\\ &=-\frac {63}{40 a^3 x^5}+\frac {21 b}{8 a^4 x^3}-\frac {63 b^2}{8 a^5 x}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}+\frac {9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac {\left (63 b^3\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^5}\\ &=-\frac {63}{40 a^3 x^5}+\frac {21 b}{8 a^4 x^3}-\frac {63 b^2}{8 a^5 x}+\frac {1}{4 a x^5 \left (a+b x^2\right )^2}+\frac {9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac {63 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 90, normalized size = 0.90 \begin {gather*} -\frac {8 a^4-24 a^3 b x^2+168 a^2 b^2 x^4+525 a b^3 x^6+315 b^4 x^8}{40 a^5 x^5 \left (a+b x^2\right )^2}-\frac {63 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^2)^3),x]

[Out]

-1/40*(8*a^4 - 24*a^3*b*x^2 + 168*a^2*b^2*x^4 + 525*a*b^3*x^6 + 315*b^4*x^8)/(a^5*x^5*(a + b*x^2)^2) - (63*b^(
5/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(11/2))

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Maple [A]
time = 0.06, size = 75, normalized size = 0.75

method	result	size
default	\(-\frac {b^{3} \left (\frac {\frac {15}{8} b \,x^{3}+\frac {17}{8} a x}{\left (b \,x^{2}+a \right )^{2}}+\frac {63 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{5}}-\frac {1}{5 a^{3} x^{5}}+\frac {b}{a^{4} x^{3}}-\frac {6 b^{2}}{a^{5} x}\)	\(75\)
risch	\(\frac {-\frac {63 b^{4} x^{8}}{8 a^{5}}-\frac {105 b^{3} x^{6}}{8 a^{4}}-\frac {21 b^{2} x^{4}}{5 a^{3}}+\frac {3 b \,x^{2}}{5 a^{2}}-\frac {1}{5 a}}{x^{5} \left (b \,x^{2}+a \right )^{2}}+\frac {63 \sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right )}{16 a^{6}}-\frac {63 \sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right )}{16 a^{6}}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/a^5*b^3*((15/8*b*x^3+17/8*a*x)/(b*x^2+a)^2+63/8/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/5/a^3/x^5+b/a^4/x^3-
6*b^2/a^5/x

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Maxima [A]
time = 0.50, size = 97, normalized size = 0.97 \begin {gather*} -\frac {315 \, b^{4} x^{8} + 525 \, a b^{3} x^{6} + 168 \, a^{2} b^{2} x^{4} - 24 \, a^{3} b x^{2} + 8 \, a^{4}}{40 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} - \frac {63 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/40*(315*b^4*x^8 + 525*a*b^3*x^6 + 168*a^2*b^2*x^4 - 24*a^3*b*x^2 + 8*a^4)/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*
x^5) - 63/8*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5)

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Fricas [A]
time = 1.29, size = 264, normalized size = 2.64 \begin {gather*} \left [-\frac {630 \, b^{4} x^{8} + 1050 \, a b^{3} x^{6} + 336 \, a^{2} b^{2} x^{4} - 48 \, a^{3} b x^{2} + 16 \, a^{4} - 315 \, {\left (b^{4} x^{9} + 2 \, a b^{3} x^{7} + a^{2} b^{2} x^{5}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{80 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}, -\frac {315 \, b^{4} x^{8} + 525 \, a b^{3} x^{6} + 168 \, a^{2} b^{2} x^{4} - 24 \, a^{3} b x^{2} + 8 \, a^{4} + 315 \, {\left (b^{4} x^{9} + 2 \, a b^{3} x^{7} + a^{2} b^{2} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{40 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/80*(630*b^4*x^8 + 1050*a*b^3*x^6 + 336*a^2*b^2*x^4 - 48*a^3*b*x^2 + 16*a^4 - 315*(b^4*x^9 + 2*a*b^3*x^7 +
a^2*b^2*x^5)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5)
, -1/40*(315*b^4*x^8 + 525*a*b^3*x^6 + 168*a^2*b^2*x^4 - 24*a^3*b*x^2 + 8*a^4 + 315*(b^4*x^9 + 2*a*b^3*x^7 + a
^2*b^2*x^5)*sqrt(b/a)*arctan(x*sqrt(b/a)))/(a^5*b^2*x^9 + 2*a^6*b*x^7 + a^7*x^5)]

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Sympy [A]
time = 0.23, size = 150, normalized size = 1.50 \begin {gather*} \frac {63 \sqrt {- \frac {b^{5}}{a^{11}}} \log {\left (- \frac {a^{6} \sqrt {- \frac {b^{5}}{a^{11}}}}{b^{3}} + x \right )}}{16} - \frac {63 \sqrt {- \frac {b^{5}}{a^{11}}} \log {\left (\frac {a^{6} \sqrt {- \frac {b^{5}}{a^{11}}}}{b^{3}} + x \right )}}{16} + \frac {- 8 a^{4} + 24 a^{3} b x^{2} - 168 a^{2} b^{2} x^{4} - 525 a b^{3} x^{6} - 315 b^{4} x^{8}}{40 a^{7} x^{5} + 80 a^{6} b x^{7} + 40 a^{5} b^{2} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(b*x**2+a)**3,x)

[Out]

63*sqrt(-b**5/a**11)*log(-a**6*sqrt(-b**5/a**11)/b**3 + x)/16 - 63*sqrt(-b**5/a**11)*log(a**6*sqrt(-b**5/a**11
)/b**3 + x)/16 + (-8*a**4 + 24*a**3*b*x**2 - 168*a**2*b**2*x**4 - 525*a*b**3*x**6 - 315*b**4*x**8)/(40*a**7*x*
*5 + 80*a**6*b*x**7 + 40*a**5*b**2*x**9)

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Giac [A]
time = 1.78, size = 80, normalized size = 0.80 \begin {gather*} -\frac {63 \, b^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{5}} - \frac {15 \, b^{4} x^{3} + 17 \, a b^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{5}} - \frac {30 \, b^{2} x^{4} - 5 \, a b x^{2} + a^{2}}{5 \, a^{5} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-63/8*b^3*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5) - 1/8*(15*b^4*x^3 + 17*a*b^3*x)/((b*x^2 + a)^2*a^5) - 1/5*(30*
b^2*x^4 - 5*a*b*x^2 + a^2)/(a^5*x^5)

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Mupad [B]
time = 5.02, size = 92, normalized size = 0.92 \begin {gather*} -\frac {\frac {1}{5\,a}-\frac {3\,b\,x^2}{5\,a^2}+\frac {21\,b^2\,x^4}{5\,a^3}+\frac {105\,b^3\,x^6}{8\,a^4}+\frac {63\,b^4\,x^8}{8\,a^5}}{a^2\,x^5+2\,a\,b\,x^7+b^2\,x^9}-\frac {63\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,a^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(a + b*x^2)^3),x)

[Out]

- (1/(5*a) - (3*b*x^2)/(5*a^2) + (21*b^2*x^4)/(5*a^3) + (105*b^3*x^6)/(8*a^4) + (63*b^4*x^8)/(8*a^5))/(a^2*x^5
+ b^2*x^9 + 2*a*b*x^7) - (63*b^(5/2)*atan((b^(1/2)*x)/a^(1/2)))/(8*a^(11/2))

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