3.3.99 \(\int \frac {1}{x^6 (a+b x^2)^2 (c+d x^2)} \, dx\) [299]

Optimal. Leaf size=250 \[ -\frac {7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac {7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}-\frac {7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3}{2 a^4 c^3 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac {b^{7/2} (7 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} (b c-a d)^2}-\frac {d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^2} \]

[Out]

1/10*(2*a*d-7*b*c)/a^2/c/(-a*d+b*c)/x^5+1/6*(-2*a^2*d^2-2*a*b*c*d+7*b^2*c^2)/a^3/c^2/(-a*d+b*c)/x^3+1/2*(2*a^3
*d^3+2*a^2*b*c*d^2+2*a*b^2*c^2*d-7*b^3*c^3)/a^4/c^3/(-a*d+b*c)/x+1/2*b/a/(-a*d+b*c)/x^5/(b*x^2+a)-1/2*b^(7/2)*
(-9*a*d+7*b*c)*arctan(x*b^(1/2)/a^(1/2))/a^(9/2)/(-a*d+b*c)^2-d^(9/2)*arctan(x*d^(1/2)/c^(1/2))/c^(7/2)/(-a*d+
b*c)^2

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Rubi [A]
time = 0.28, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {483, 597, 536, 211} \begin {gather*} -\frac {b^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (7 b c-9 a d)}{2 a^{9/2} (b c-a d)^2}-\frac {7 b c-2 a d}{10 a^2 c x^5 (b c-a d)}+\frac {-2 a^2 d^2-2 a b c d+7 b^2 c^2}{6 a^3 c^2 x^3 (b c-a d)}-\frac {-2 a^3 d^3-2 a^2 b c d^2-2 a b^2 c^2 d+7 b^3 c^3}{2 a^4 c^3 x (b c-a d)}-\frac {d^{9/2} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^2}+\frac {b}{2 a x^5 \left (a+b x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

-1/10*(7*b*c - 2*a*d)/(a^2*c*(b*c - a*d)*x^5) + (7*b^2*c^2 - 2*a*b*c*d - 2*a^2*d^2)/(6*a^3*c^2*(b*c - a*d)*x^3
) - (7*b^3*c^3 - 2*a*b^2*c^2*d - 2*a^2*b*c*d^2 - 2*a^3*d^3)/(2*a^4*c^3*(b*c - a*d)*x) + b/(2*a*(b*c - a*d)*x^5
*(a + b*x^2)) - (b^(7/2)*(7*b*c - 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2)*(b*c - a*d)^2) - (d^(9/2)*Arc
Tan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^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 483

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*(e*
x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a*d)
*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n
*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ
[p, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac {\int \frac {-7 b c+2 a d-7 b d x^2}{x^6 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=-\frac {7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac {b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}+\frac {\int \frac {-5 \left (7 b^2 c^2-2 a b c d-2 a^2 d^2\right )-5 b d (7 b c-2 a d) x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{10 a^2 c (b c-a d)}\\ &=-\frac {7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac {7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}+\frac {b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac {\int \frac {-15 \left (7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3\right )-15 b d \left (7 b^2 c^2-2 a b c d-2 a^2 d^2\right ) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{30 a^3 c^2 (b c-a d)}\\ &=-\frac {7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac {7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}-\frac {7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3}{2 a^4 c^3 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}+\frac {\int \frac {-15 \left (7 b^4 c^4-2 a b^3 c^3 d-2 a^2 b^2 c^2 d^2-2 a^3 b c d^3-2 a^4 d^4\right )-15 b d \left (7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{30 a^4 c^3 (b c-a d)}\\ &=-\frac {7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac {7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}-\frac {7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3}{2 a^4 c^3 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac {d^5 \int \frac {1}{c+d x^2} \, dx}{c^3 (b c-a d)^2}-\frac {\left (b^4 (7 b c-9 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^4 (b c-a d)^2}\\ &=-\frac {7 b c-2 a d}{10 a^2 c (b c-a d) x^5}+\frac {7 b^2 c^2-2 a b c d-2 a^2 d^2}{6 a^3 c^2 (b c-a d) x^3}-\frac {7 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3}{2 a^4 c^3 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^5 \left (a+b x^2\right )}-\frac {b^{7/2} (7 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} (b c-a d)^2}-\frac {d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^2}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 179, normalized size = 0.72 \begin {gather*} -\frac {1}{5 a^2 c x^5}+\frac {2 b c+a d}{3 a^3 c^2 x^3}+\frac {-3 b^2 c^2-2 a b c d-a^2 d^2}{a^4 c^3 x}+\frac {b^4 x}{2 a^4 (-b c+a d) \left (a+b x^2\right )}+\frac {b^{7/2} (-7 b c+9 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} (-b c+a d)^2}-\frac {d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x]

[Out]

-1/5*1/(a^2*c*x^5) + (2*b*c + a*d)/(3*a^3*c^2*x^3) + (-3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)/(a^4*c^3*x) + (b^4*x)/
(2*a^4*(-(b*c) + a*d)*(a + b*x^2)) + (b^(7/2)*(-7*b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(9/2)*(-(b*c)
+ a*d)^2) - (d^(9/2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(c^(7/2)*(b*c - a*d)^2)

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Maple [A]
time = 0.16, size = 161, normalized size = 0.64

method	result	size
default	\(\frac {b^{4} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (9 a d -7 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4} \left (a d -b c \right )^{2}}-\frac {d^{5} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{c^{3} \left (a d -b c \right )^{2} \sqrt {c d}}-\frac {1}{5 a^{2} c \,x^{5}}-\frac {-a d -2 b c}{3 a^{3} c^{2} x^{3}}-\frac {a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{a^{4} c^{3} x}\)	\(161\)
risch	\(\frac {-\frac {b \left (2 a^{3} d^{3}+2 a^{2} b c \,d^{2}+2 a \,b^{2} c^{2} d -7 b^{3} c^{3}\right ) x^{6}}{2 a^{4} c^{3} \left (a d -b c \right )}-\frac {\left (3 a^{2} d^{2}+5 a b c d +7 b^{2} c^{2}\right ) x^{4}}{3 a^{3} c^{3}}+\frac {\left (5 a d +7 b c \right ) x^{2}}{15 a^{2} c^{2}}-\frac {1}{5 a c}}{x^{5} \left (b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (d^{4} a^{13}-4 c \,d^{3} a^{12} b +6 c^{2} d^{2} a^{11} b^{2}-4 c^{3} d \,a^{10} b^{3}+a^{9} b^{4} c^{4}\right ) \textit {\_Z}^{2}+81 a^{2} b^{7} d^{2}-126 a \,b^{8} c d +49 b^{9} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a^{17} c^{7} d^{8}-22 a^{16} b \,c^{8} d^{7}+72 a^{15} b^{2} c^{9} d^{6}-138 a^{14} b^{3} c^{10} d^{5}+170 a^{13} b^{4} c^{11} d^{4}-138 a^{12} b^{5} c^{12} d^{3}+72 a^{11} b^{6} c^{13} d^{2}-22 a^{10} b^{7} c^{14} d +3 a^{9} b^{8} c^{15}\right ) \textit {\_R}^{4}+\left (8 a^{13} d^{13}-32 a^{12} b c \,d^{12}+52 a^{11} b^{2} c^{2} d^{11}-40 a^{10} b^{3} c^{3} d^{10}+12 a^{9} b^{4} c^{4} d^{9}+243 a^{6} b^{7} c^{7} d^{6}-1188 a^{5} b^{8} c^{8} d^{5}+2460 a^{4} b^{9} c^{9} d^{4}-2776 a^{3} b^{10} c^{10} d^{3}+1807 a^{2} b^{11} c^{11} d^{2}-644 a \,b^{12} c^{12} d +98 b^{13} c^{13}\right ) \textit {\_R}^{2}+648 a^{2} b^{7} d^{11}-1008 a \,b^{8} c \,d^{10}+392 b^{9} c^{2} d^{9}\right ) x +\left (2 a^{15} c^{4} d^{10}-8 a^{14} b \,c^{5} d^{9}+12 a^{13} b^{2} c^{6} d^{8}-8 a^{12} b^{3} c^{7} d^{7}+2 a^{11} b^{4} c^{8} d^{6}-9 a^{10} b^{5} c^{9} d^{5}+43 a^{9} b^{6} c^{10} d^{4}-82 a^{8} b^{7} c^{11} d^{3}+78 a^{7} b^{8} c^{12} d^{2}-37 a^{6} b^{9} c^{13} d +7 a^{5} b^{10} c^{14}\right ) \textit {\_R}^{3}+\left (-36 a^{6} b^{5} c^{2} d^{10}+28 a^{5} b^{6} c^{3} d^{9}+162 a^{4} b^{7} c^{4} d^{8}-252 a^{3} b^{8} c^{5} d^{7}+98 a^{2} b^{9} c^{6} d^{6}\right ) \textit {\_R} \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (d^{4} c^{7} a^{4}-4 a^{3} b \,c^{8} d^{3}+6 a^{2} b^{2} c^{9} d^{2}-4 a \,b^{3} c^{10} d +b^{4} c^{11}\right ) \textit {\_Z}^{2}+d^{9}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (12 a^{17} c^{7} d^{8}-88 a^{16} b \,c^{8} d^{7}+288 a^{15} b^{2} c^{9} d^{6}-552 a^{14} b^{3} c^{10} d^{5}+680 a^{13} b^{4} c^{11} d^{4}-552 a^{12} b^{5} c^{12} d^{3}+288 a^{11} b^{6} c^{13} d^{2}-88 a^{10} b^{7} c^{14} d +12 a^{9} b^{8} c^{15}\right ) \textit {\_R}^{4}+\left (8 a^{13} d^{13}-32 a^{12} b c \,d^{12}+52 a^{11} b^{2} c^{2} d^{11}-40 a^{10} b^{3} c^{3} d^{10}+12 a^{9} b^{4} c^{4} d^{9}+243 a^{6} b^{7} c^{7} d^{6}-1188 a^{5} b^{8} c^{8} d^{5}+2460 a^{4} b^{9} c^{9} d^{4}-2776 a^{3} b^{10} c^{10} d^{3}+1807 a^{2} b^{11} c^{11} d^{2}-644 a \,b^{12} c^{12} d +98 b^{13} c^{13}\right ) \textit {\_R}^{2}+162 a^{2} b^{7} d^{11}-252 a \,b^{8} c \,d^{10}+98 b^{9} c^{2} d^{9}\right ) x +\left (4 a^{15} c^{4} d^{10}-16 a^{14} b \,c^{5} d^{9}+24 a^{13} b^{2} c^{6} d^{8}-16 a^{12} b^{3} c^{7} d^{7}+4 a^{11} b^{4} c^{8} d^{6}-18 a^{10} b^{5} c^{9} d^{5}+86 a^{9} b^{6} c^{10} d^{4}-164 a^{8} b^{7} c^{11} d^{3}+156 a^{7} b^{8} c^{12} d^{2}-74 a^{6} b^{9} c^{13} d +14 a^{5} b^{10} c^{14}\right ) \textit {\_R}^{3}+\left (-18 a^{6} b^{5} c^{2} d^{10}+14 a^{5} b^{6} c^{3} d^{9}+81 a^{4} b^{7} c^{4} d^{8}-126 a^{3} b^{8} c^{5} d^{7}+49 a^{2} b^{9} c^{6} d^{6}\right ) \textit {\_R} \right )\right )}{2}\)	\(1367\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^4/a^4/(a*d-b*c)^2*((1/2*a*d-1/2*b*c)*x/(b*x^2+a)+1/2*(9*a*d-7*b*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/c^
3*d^5/(a*d-b*c)^2/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2))-1/5/a^2/c/x^5-1/3*(-a*d-2*b*c)/a^3/c^2/x^3-(a^2*d^2+2*a*
b*c*d+3*b^2*c^2)/a^4/c^3/x

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Maxima [A]
time = 0.53, size = 303, normalized size = 1.21 \begin {gather*} -\frac {d^{5} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt {c d}} - \frac {{\left (7 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )} \sqrt {a b}} - \frac {6 \, a^{3} b c^{3} - 6 \, a^{4} c^{2} d + 15 \, {\left (7 \, b^{4} c^{3} - 2 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{6} + 10 \, {\left (7 \, a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d - 2 \, a^{3} b c d^{2} - 3 \, a^{4} d^{3}\right )} x^{4} - 2 \, {\left (7 \, a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d - 5 \, a^{4} c d^{2}\right )} x^{2}}{30 \, {\left ({\left (a^{4} b^{2} c^{4} - a^{5} b c^{3} d\right )} x^{7} + {\left (a^{5} b c^{4} - a^{6} c^{3} d\right )} x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^5*arctan(d*x/sqrt(c*d))/((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt(c*d)) - 1/2*(7*b^5*c - 9*a*b^4*d)*arcta
n(b*x/sqrt(a*b))/((a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*sqrt(a*b)) - 1/30*(6*a^3*b*c^3 - 6*a^4*c^2*d + 15*(7*b
^4*c^3 - 2*a*b^3*c^2*d - 2*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*x^6 + 10*(7*a*b^3*c^3 - 2*a^2*b^2*c^2*d - 2*a^3*b*c*d^
2 - 3*a^4*d^3)*x^4 - 2*(7*a^2*b^2*c^3 - 2*a^3*b*c^2*d - 5*a^4*c*d^2)*x^2)/((a^4*b^2*c^4 - a^5*b*c^3*d)*x^7 + (
a^5*b*c^4 - a^6*c^3*d)*x^5)

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Fricas [A]
time = 4.56, size = 1489, normalized size = 5.96 \begin {gather*} \left [-\frac {12 \, a^{3} b^{2} c^{4} - 24 \, a^{4} b c^{3} d + 12 \, a^{5} c^{2} d^{2} + 30 \, {\left (7 \, b^{5} c^{4} - 9 \, a b^{4} c^{3} d + 2 \, a^{4} b d^{4}\right )} x^{6} + 20 \, {\left (7 \, a b^{4} c^{4} - 9 \, a^{2} b^{3} c^{3} d - a^{4} b c d^{3} + 3 \, a^{5} d^{4}\right )} x^{4} - 4 \, {\left (7 \, a^{2} b^{3} c^{4} - 9 \, a^{3} b^{2} c^{3} d - 3 \, a^{4} b c^{2} d^{2} + 5 \, a^{5} c d^{3}\right )} x^{2} + 15 \, {\left ({\left (7 \, b^{5} c^{4} - 9 \, a b^{4} c^{3} d\right )} x^{7} + {\left (7 \, a b^{4} c^{4} - 9 \, a^{2} b^{3} c^{3} d\right )} 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 ) - 30 \, {\left (a^{4} b d^{4} x^{7} + a^{5} d^{4} x^{5}\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{60 \, {\left ({\left (a^{4} b^{3} c^{5} - 2 \, a^{5} b^{2} c^{4} d + a^{6} b c^{3} d^{2}\right )} x^{7} + {\left (a^{5} b^{2} c^{5} - 2 \, a^{6} b c^{4} d + a^{7} c^{3} d^{2}\right )} x^{5}\right )}}, -\frac {12 \, a^{3} b^{2} c^{4} - 24 \, a^{4} b c^{3} d + 12 \, a^{5} c^{2} d^{2} + 30 \, {\left (7 \, b^{5} c^{4} - 9 \, a b^{4} c^{3} d + 2 \, a^{4} b d^{4}\right )} x^{6} + 20 \, {\left (7 \, a b^{4} c^{4} - 9 \, a^{2} b^{3} c^{3} d - a^{4} b c d^{3} + 3 \, a^{5} d^{4}\right )} x^{4} - 4 \, {\left (7 \, a^{2} b^{3} c^{4} - 9 \, a^{3} b^{2} c^{3} d - 3 \, a^{4} b c^{2} d^{2} + 5 \, a^{5} c d^{3}\right )} x^{2} + 60 \, {\left (a^{4} b d^{4} x^{7} + a^{5} d^{4} x^{5}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + 15 \, {\left ({\left (7 \, b^{5} c^{4} - 9 \, a b^{4} c^{3} d\right )} x^{7} + {\left (7 \, a b^{4} c^{4} - 9 \, a^{2} b^{3} c^{3} d\right )} 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 )}{60 \, {\left ({\left (a^{4} b^{3} c^{5} - 2 \, a^{5} b^{2} c^{4} d + a^{6} b c^{3} d^{2}\right )} x^{7} + {\left (a^{5} b^{2} c^{5} - 2 \, a^{6} b c^{4} d + a^{7} c^{3} d^{2}\right )} x^{5}\right )}}, -\frac {6 \, a^{3} b^{2} c^{4} - 12 \, a^{4} b c^{3} d + 6 \, a^{5} c^{2} d^{2} + 15 \, {\left (7 \, b^{5} c^{4} - 9 \, a b^{4} c^{3} d + 2 \, a^{4} b d^{4}\right )} x^{6} + 10 \, {\left (7 \, a b^{4} c^{4} - 9 \, a^{2} b^{3} c^{3} d - a^{4} b c d^{3} + 3 \, a^{5} d^{4}\right )} x^{4} - 2 \, {\left (7 \, a^{2} b^{3} c^{4} - 9 \, a^{3} b^{2} c^{3} d - 3 \, a^{4} b c^{2} d^{2} + 5 \, a^{5} c d^{3}\right )} x^{2} + 15 \, {\left ({\left (7 \, b^{5} c^{4} - 9 \, a b^{4} c^{3} d\right )} x^{7} + {\left (7 \, a b^{4} c^{4} - 9 \, a^{2} b^{3} c^{3} d\right )} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - 15 \, {\left (a^{4} b d^{4} x^{7} + a^{5} d^{4} x^{5}\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{30 \, {\left ({\left (a^{4} b^{3} c^{5} - 2 \, a^{5} b^{2} c^{4} d + a^{6} b c^{3} d^{2}\right )} x^{7} + {\left (a^{5} b^{2} c^{5} - 2 \, a^{6} b c^{4} d + a^{7} c^{3} d^{2}\right )} x^{5}\right )}}, -\frac {6 \, a^{3} b^{2} c^{4} - 12 \, a^{4} b c^{3} d + 6 \, a^{5} c^{2} d^{2} + 15 \, {\left (7 \, b^{5} c^{4} - 9 \, a b^{4} c^{3} d + 2 \, a^{4} b d^{4}\right )} x^{6} + 10 \, {\left (7 \, a b^{4} c^{4} - 9 \, a^{2} b^{3} c^{3} d - a^{4} b c d^{3} + 3 \, a^{5} d^{4}\right )} x^{4} - 2 \, {\left (7 \, a^{2} b^{3} c^{4} - 9 \, a^{3} b^{2} c^{3} d - 3 \, a^{4} b c^{2} d^{2} + 5 \, a^{5} c d^{3}\right )} x^{2} + 15 \, {\left ({\left (7 \, b^{5} c^{4} - 9 \, a b^{4} c^{3} d\right )} x^{7} + {\left (7 \, a b^{4} c^{4} - 9 \, a^{2} b^{3} c^{3} d\right )} x^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 30 \, {\left (a^{4} b d^{4} x^{7} + a^{5} d^{4} x^{5}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right )}{30 \, {\left ({\left (a^{4} b^{3} c^{5} - 2 \, a^{5} b^{2} c^{4} d + a^{6} b c^{3} d^{2}\right )} x^{7} + {\left (a^{5} b^{2} c^{5} - 2 \, a^{6} b c^{4} d + a^{7} c^{3} d^{2}\right )} 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)^2/(d*x^2+c),x, algorithm="fricas")

[Out]

[-1/60*(12*a^3*b^2*c^4 - 24*a^4*b*c^3*d + 12*a^5*c^2*d^2 + 30*(7*b^5*c^4 - 9*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 +
20*(7*a*b^4*c^4 - 9*a^2*b^3*c^3*d - a^4*b*c*d^3 + 3*a^5*d^4)*x^4 - 4*(7*a^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a^4*
b*c^2*d^2 + 5*a^5*c*d^3)*x^2 + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a*b^4*c^4 - 9*a^2*b^3*c^3*d)*x^5)*sqrt
(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 30*(a^4*b*d^4*x^7 + a^5*d^4*x^5)*sqrt(-d/c)*log((d*x^
2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^2*c^5 -
2*a^6*b*c^4*d + a^7*c^3*d^2)*x^5), -1/60*(12*a^3*b^2*c^4 - 24*a^4*b*c^3*d + 12*a^5*c^2*d^2 + 30*(7*b^5*c^4 - 9
*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 + 20*(7*a*b^4*c^4 - 9*a^2*b^3*c^3*d - a^4*b*c*d^3 + 3*a^5*d^4)*x^4 - 4*(7*a^2*
b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a^4*b*c^2*d^2 + 5*a^5*c*d^3)*x^2 + 60*(a^4*b*d^4*x^7 + a^5*d^4*x^5)*sqrt(d/c)*ar
ctan(x*sqrt(d/c)) + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a*b^4*c^4 - 9*a^2*b^3*c^3*d)*x^5)*sqrt(-b/a)*log(
(b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/((a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^2*c
^5 - 2*a^6*b*c^4*d + a^7*c^3*d^2)*x^5), -1/30*(6*a^3*b^2*c^4 - 12*a^4*b*c^3*d + 6*a^5*c^2*d^2 + 15*(7*b^5*c^4
- 9*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 + 10*(7*a*b^4*c^4 - 9*a^2*b^3*c^3*d - a^4*b*c*d^3 + 3*a^5*d^4)*x^4 - 2*(7*a
^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a^4*b*c^2*d^2 + 5*a^5*c*d^3)*x^2 + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (7*a
*b^4*c^4 - 9*a^2*b^3*c^3*d)*x^5)*sqrt(b/a)*arctan(x*sqrt(b/a)) - 15*(a^4*b*d^4*x^7 + a^5*d^4*x^5)*sqrt(-d/c)*l
og((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)))/((a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^
2*c^5 - 2*a^6*b*c^4*d + a^7*c^3*d^2)*x^5), -1/30*(6*a^3*b^2*c^4 - 12*a^4*b*c^3*d + 6*a^5*c^2*d^2 + 15*(7*b^5*c
^4 - 9*a*b^4*c^3*d + 2*a^4*b*d^4)*x^6 + 10*(7*a*b^4*c^4 - 9*a^2*b^3*c^3*d - a^4*b*c*d^3 + 3*a^5*d^4)*x^4 - 2*(
7*a^2*b^3*c^4 - 9*a^3*b^2*c^3*d - 3*a^4*b*c^2*d^2 + 5*a^5*c*d^3)*x^2 + 15*((7*b^5*c^4 - 9*a*b^4*c^3*d)*x^7 + (
7*a*b^4*c^4 - 9*a^2*b^3*c^3*d)*x^5)*sqrt(b/a)*arctan(x*sqrt(b/a)) + 30*(a^4*b*d^4*x^7 + a^5*d^4*x^5)*sqrt(d/c)
*arctan(x*sqrt(d/c)))/((a^4*b^3*c^5 - 2*a^5*b^2*c^4*d + a^6*b*c^3*d^2)*x^7 + (a^5*b^2*c^5 - 2*a^6*b*c^4*d + a^
7*c^3*d^2)*x^5)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 1.00, size = 207, normalized size = 0.83 \begin {gather*} -\frac {d^{5} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )} \sqrt {c d}} - \frac {b^{4} x}{2 \, {\left (a^{4} b c - a^{5} d\right )} {\left (b x^{2} + a\right )}} - \frac {{\left (7 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}\right )} \sqrt {a b}} - \frac {45 \, b^{2} c^{2} x^{4} + 30 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} - 10 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, a^{4} c^{3} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^5*arctan(d*x/sqrt(c*d))/((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt(c*d)) - 1/2*b^4*x/((a^4*b*c - a^5*d)*(b
*x^2 + a)) - 1/2*(7*b^5*c - 9*a*b^4*d)*arctan(b*x/sqrt(a*b))/((a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*sqrt(a*b))
- 1/15*(45*b^2*c^2*x^4 + 30*a*b*c*d*x^4 + 15*a^2*d^2*x^4 - 10*a*b*c^2*x^2 - 5*a^2*c*d*x^2 + 3*a^2*c^2)/(a^4*c
^3*x^5)

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Mupad [B]
time = 0.66, size = 2737, normalized size = 10.95 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(a + b*x^2)^2*(c + d*x^2)),x)

[Out]

(atan((a^9*d*x*(-c^7*d^9)^(3/2)*4i + b^9*c^16*d*x*(-c^7*d^9)^(1/2)*49i + a^2*b^7*c^14*d^3*x*(-c^7*d^9)^(1/2)*8
1i - a*b^8*c^15*d^2*x*(-c^7*d^9)^(1/2)*126i)/(4*a^9*c^11*d^14 - 49*b^9*c^20*d^5 + 126*a*b^8*c^19*d^6 - 81*a^2*
b^7*c^18*d^7))*(-c^7*d^9)^(1/2)*1i)/(b^2*c^9 + a^2*c^7*d^2 - 2*a*b*c^8*d) - (1/(5*a*c) - (x^2*(5*a*d + 7*b*c))
/(15*a^2*c^2) + (x^4*(3*a^2*d^2 + 7*b^2*c^2 + 5*a*b*c*d))/(3*a^3*c^3) + (x^6*(2*a^3*b*d^3 - 7*b^4*c^3 + 2*a^2*
b^2*c*d^2 + 2*a*b^3*c^2*d))/(2*a^4*c^3*(a*d - b*c)))/(a*x^5 + b*x^7) - (atan((((x*(784*a^12*b^14*c^20*d^3 - 43
68*a^13*b^13*c^19*d^4 + 9696*a^14*b^12*c^18*d^5 - 10720*a^15*b^11*c^17*d^6 + 5904*a^16*b^10*c^16*d^7 - 1296*a^
17*b^9*c^15*d^8 + 64*a^20*b^6*c^12*d^11 - 192*a^21*b^5*c^11*d^12 + 192*a^22*b^4*c^10*d^13 - 64*a^23*b^3*c^9*d^
14) + ((9*a*d - 7*b*c)*(-a^9*b^7)^(1/2)*(2816*a^17*b^11*c^21*d^3 - 448*a^16*b^12*c^22*d^2 - 7360*a^18*b^10*c^2
0*d^4 + 10240*a^19*b^9*c^19*d^5 - 8000*a^20*b^8*c^18*d^6 + 3200*a^21*b^7*c^17*d^7 + 64*a^22*b^6*c^16*d^8 - 128
0*a^23*b^5*c^15*d^9 + 1280*a^24*b^4*c^14*d^10 - 640*a^25*b^3*c^13*d^11 + 128*a^26*b^2*c^12*d^12 + (x*(9*a*d -
7*b*c)*(-a^9*b^7)^(1/2)*(256*a^20*b^10*c^23*d^2 - 1536*a^21*b^9*c^22*d^3 + 3584*a^22*b^8*c^21*d^4 - 3584*a^23*
b^7*c^20*d^5 + 3584*a^25*b^5*c^18*d^7 - 3584*a^26*b^4*c^17*d^8 + 1536*a^27*b^3*c^16*d^9 - 256*a^28*b^2*c^15*d^
10))/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d))))/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d)))*(9*a*d - 7*b*c
)*(-a^9*b^7)^(1/2)*1i)/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d)) + ((x*(784*a^12*b^14*c^20*d^3 - 4368*a^13*b
^13*c^19*d^4 + 9696*a^14*b^12*c^18*d^5 - 10720*a^15*b^11*c^17*d^6 + 5904*a^16*b^10*c^16*d^7 - 1296*a^17*b^9*c^
15*d^8 + 64*a^20*b^6*c^12*d^11 - 192*a^21*b^5*c^11*d^12 + 192*a^22*b^4*c^10*d^13 - 64*a^23*b^3*c^9*d^14) + ((9
*a*d - 7*b*c)*(-a^9*b^7)^(1/2)*(448*a^16*b^12*c^22*d^2 - 2816*a^17*b^11*c^21*d^3 + 7360*a^18*b^10*c^20*d^4 - 1
0240*a^19*b^9*c^19*d^5 + 8000*a^20*b^8*c^18*d^6 - 3200*a^21*b^7*c^17*d^7 - 64*a^22*b^6*c^16*d^8 + 1280*a^23*b^
5*c^15*d^9 - 1280*a^24*b^4*c^14*d^10 + 640*a^25*b^3*c^13*d^11 - 128*a^26*b^2*c^12*d^12 + (x*(9*a*d - 7*b*c)*(-
a^9*b^7)^(1/2)*(256*a^20*b^10*c^23*d^2 - 1536*a^21*b^9*c^22*d^3 + 3584*a^22*b^8*c^21*d^4 - 3584*a^23*b^7*c^20*
d^5 + 3584*a^25*b^5*c^18*d^7 - 3584*a^26*b^4*c^17*d^8 + 1536*a^27*b^3*c^16*d^9 - 256*a^28*b^2*c^15*d^10))/(4*(
a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d))))/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d)))*(9*a*d - 7*b*c)*(-a^9*b
^7)^(1/2)*1i)/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d)))/(((x*(784*a^12*b^14*c^20*d^3 - 4368*a^13*b^13*c^19*
d^4 + 9696*a^14*b^12*c^18*d^5 - 10720*a^15*b^11*c^17*d^6 + 5904*a^16*b^10*c^16*d^7 - 1296*a^17*b^9*c^15*d^8 +
64*a^20*b^6*c^12*d^11 - 192*a^21*b^5*c^11*d^12 + 192*a^22*b^4*c^10*d^13 - 64*a^23*b^3*c^9*d^14) + ((9*a*d - 7*
b*c)*(-a^9*b^7)^(1/2)*(448*a^16*b^12*c^22*d^2 - 2816*a^17*b^11*c^21*d^3 + 7360*a^18*b^10*c^20*d^4 - 10240*a^19
*b^9*c^19*d^5 + 8000*a^20*b^8*c^18*d^6 - 3200*a^21*b^7*c^17*d^7 - 64*a^22*b^6*c^16*d^8 + 1280*a^23*b^5*c^15*d^
9 - 1280*a^24*b^4*c^14*d^10 + 640*a^25*b^3*c^13*d^11 - 128*a^26*b^2*c^12*d^12 + (x*(9*a*d - 7*b*c)*(-a^9*b^7)^
(1/2)*(256*a^20*b^10*c^23*d^2 - 1536*a^21*b^9*c^22*d^3 + 3584*a^22*b^8*c^21*d^4 - 3584*a^23*b^7*c^20*d^5 + 358
4*a^25*b^5*c^18*d^7 - 3584*a^26*b^4*c^17*d^8 + 1536*a^27*b^3*c^16*d^9 - 256*a^28*b^2*c^15*d^10))/(4*(a^11*d^2
+ a^9*b^2*c^2 - 2*a^10*b*c*d))))/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d)))*(9*a*d - 7*b*c)*(-a^9*b^7)^(1/2)
)/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d)) - ((x*(784*a^12*b^14*c^20*d^3 - 4368*a^13*b^13*c^19*d^4 + 9696*a
^14*b^12*c^18*d^5 - 10720*a^15*b^11*c^17*d^6 + 5904*a^16*b^10*c^16*d^7 - 1296*a^17*b^9*c^15*d^8 + 64*a^20*b^6*
c^12*d^11 - 192*a^21*b^5*c^11*d^12 + 192*a^22*b^4*c^10*d^13 - 64*a^23*b^3*c^9*d^14) + ((9*a*d - 7*b*c)*(-a^9*b
^7)^(1/2)*(2816*a^17*b^11*c^21*d^3 - 448*a^16*b^12*c^22*d^2 - 7360*a^18*b^10*c^20*d^4 + 10240*a^19*b^9*c^19*d^
5 - 8000*a^20*b^8*c^18*d^6 + 3200*a^21*b^7*c^17*d^7 + 64*a^22*b^6*c^16*d^8 - 1280*a^23*b^5*c^15*d^9 + 1280*a^2
4*b^4*c^14*d^10 - 640*a^25*b^3*c^13*d^11 + 128*a^26*b^2*c^12*d^12 + (x*(9*a*d - 7*b*c)*(-a^9*b^7)^(1/2)*(256*a
^20*b^10*c^23*d^2 - 1536*a^21*b^9*c^22*d^3 + 3584*a^22*b^8*c^21*d^4 - 3584*a^23*b^7*c^20*d^5 + 3584*a^25*b^5*c
^18*d^7 - 3584*a^26*b^4*c^17*d^8 + 1536*a^27*b^3*c^16*d^9 - 256*a^28*b^2*c^15*d^10))/(4*(a^11*d^2 + a^9*b^2*c^
2 - 2*a^10*b*c*d))))/(4*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d)))*(9*a*d - 7*b*c)*(-a^9*b^7)^(1/2))/(4*(a^11*d
^2 + a^9*b^2*c^2 - 2*a^10*b*c*d)) + 784*a^12*b^12*c^15*d^7 - 2800*a^13*b^11*c^14*d^8 + 3312*a^14*b^10*c^13*d^9
- 1296*a^15*b^9*c^12*d^10 + 224*a^16*b^8*c^11*d^11 - 512*a^17*b^7*c^10*d^12 + 288*a^18*b^6*c^9*d^13))*(9*a*d
- 7*b*c)*(-a^9*b^7)^(1/2)*1i)/(2*(a^11*d^2 + a^9*b^2*c^2 - 2*a^10*b*c*d))

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