Optimal. Leaf size=78 \[ -\frac {15 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{7/2}}-\frac {15}{16 a^3 x^2}+\frac {5}{16 a^2 x^2 \left (a+c x^4\right )}+\frac {1}{8 a x^2 \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {275, 290, 325, 205} \[ \frac {5}{16 a^2 x^2 \left (a+c x^4\right )}-\frac {15 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{7/2}}-\frac {15}{16 a^3 x^2}+\frac {1}{8 a x^2 \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 290
Rule 325
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+c x^4\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+c x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac {1}{8 a x^2 \left (a+c x^4\right )^2}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+c x^2\right )^2} \, dx,x,x^2\right )}{8 a}\\ &=\frac {1}{8 a x^2 \left (a+c x^4\right )^2}+\frac {5}{16 a^2 x^2 \left (a+c x^4\right )}+\frac {15 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+c x^2\right )} \, dx,x,x^2\right )}{16 a^2}\\ &=-\frac {15}{16 a^3 x^2}+\frac {1}{8 a x^2 \left (a+c x^4\right )^2}+\frac {5}{16 a^2 x^2 \left (a+c x^4\right )}-\frac {(15 c) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{16 a^3}\\ &=-\frac {15}{16 a^3 x^2}+\frac {1}{8 a x^2 \left (a+c x^4\right )^2}+\frac {5}{16 a^2 x^2 \left (a+c x^4\right )}-\frac {15 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 105, normalized size = 1.35 \[ \frac {-\frac {\sqrt {a} \left (8 a^2+25 a c x^4+15 c^2 x^8\right )}{x^2 \left (a+c x^4\right )^2}+15 \sqrt {c} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+15 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{16 a^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 218, normalized size = 2.79 \[ \left [-\frac {30 \, c^{2} x^{8} + 50 \, a c x^{4} - 15 \, {\left (c^{2} x^{10} + 2 \, a c x^{6} + a^{2} x^{2}\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{4} - 2 \, a x^{2} \sqrt {-\frac {c}{a}} - a}{c x^{4} + a}\right ) + 16 \, a^{2}}{32 \, {\left (a^{3} c^{2} x^{10} + 2 \, a^{4} c x^{6} + a^{5} x^{2}\right )}}, -\frac {15 \, c^{2} x^{8} + 25 \, a c x^{4} - 15 \, {\left (c^{2} x^{10} + 2 \, a c x^{6} + a^{2} x^{2}\right )} \sqrt {\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{a}}}{c x^{2}}\right ) + 8 \, a^{2}}{16 \, {\left (a^{3} c^{2} x^{10} + 2 \, a^{4} c x^{6} + a^{5} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 61, normalized size = 0.78 \[ -\frac {15 \, c \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3}} - \frac {7 \, c^{2} x^{6} + 9 \, a c x^{2}}{16 \, {\left (c x^{4} + a\right )}^{2} a^{3}} - \frac {1}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 70, normalized size = 0.90 \[ -\frac {7 c^{2} x^{6}}{16 \left (c \,x^{4}+a \right )^{2} a^{3}}-\frac {9 c \,x^{2}}{16 \left (c \,x^{4}+a \right )^{2} a^{2}}-\frac {15 c \arctan \left (\frac {c \,x^{2}}{\sqrt {a c}}\right )}{16 \sqrt {a c}\, a^{3}}-\frac {1}{2 a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 75, normalized size = 0.96 \[ -\frac {15 \, c^{2} x^{8} + 25 \, a c x^{4} + 8 \, a^{2}}{16 \, {\left (a^{3} c^{2} x^{10} + 2 \, a^{4} c x^{6} + a^{5} x^{2}\right )}} - \frac {15 \, c \arctan \left (\frac {c x^{2}}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 72, normalized size = 0.92 \[ -\frac {\frac {1}{2\,a}+\frac {25\,c\,x^4}{16\,a^2}+\frac {15\,c^2\,x^8}{16\,a^3}}{a^2\,x^2+2\,a\,c\,x^6+c^2\,x^{10}}-\frac {15\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x^2}{\sqrt {a}}\right )}{16\,a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 121, normalized size = 1.55 \[ \frac {15 \sqrt {- \frac {c}{a^{7}}} \log {\left (- \frac {a^{4} \sqrt {- \frac {c}{a^{7}}}}{c} + x^{2} \right )}}{32} - \frac {15 \sqrt {- \frac {c}{a^{7}}} \log {\left (\frac {a^{4} \sqrt {- \frac {c}{a^{7}}}}{c} + x^{2} \right )}}{32} + \frac {- 8 a^{2} - 25 a c x^{4} - 15 c^{2} x^{8}}{16 a^{5} x^{2} + 32 a^{4} c x^{6} + 16 a^{3} c^{2} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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