Optimal. Leaf size=221 \[ -\frac {5 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{3/4} c^{9/4}}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}-\frac {5 x}{32 c^2 \left (a+c x^4\right )}-\frac {x^5}{8 c \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.14, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {288, 211, 1165, 628, 1162, 617, 204} \[ -\frac {5 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{3/4} c^{9/4}}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}-\frac {5 x}{32 c^2 \left (a+c x^4\right )}-\frac {x^5}{8 c \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 288
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+c x^4\right )^3} \, dx &=-\frac {x^5}{8 c \left (a+c x^4\right )^2}+\frac {5 \int \frac {x^4}{\left (a+c x^4\right )^2} \, dx}{8 c}\\ &=-\frac {x^5}{8 c \left (a+c x^4\right )^2}-\frac {5 x}{32 c^2 \left (a+c x^4\right )}+\frac {5 \int \frac {1}{a+c x^4} \, dx}{32 c^2}\\ &=-\frac {x^5}{8 c \left (a+c x^4\right )^2}-\frac {5 x}{32 c^2 \left (a+c x^4\right )}+\frac {5 \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{64 \sqrt {a} c^2}+\frac {5 \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{64 \sqrt {a} c^2}\\ &=-\frac {x^5}{8 c \left (a+c x^4\right )^2}-\frac {5 x}{32 c^2 \left (a+c x^4\right )}+\frac {5 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 \sqrt {a} c^{5/2}}+\frac {5 \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 \sqrt {a} c^{5/2}}-\frac {5 \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{3/4} c^{9/4}}-\frac {5 \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{3/4} c^{9/4}}\\ &=-\frac {x^5}{8 c \left (a+c x^4\right )^2}-\frac {5 x}{32 c^2 \left (a+c x^4\right )}-\frac {5 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}\\ &=-\frac {x^5}{8 c \left (a+c x^4\right )^2}-\frac {5 x}{32 c^2 \left (a+c x^4\right )}-\frac {5 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{3/4} c^{9/4}}-\frac {5 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{3/4} c^{9/4}}+\frac {5 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{3/4} c^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 201, normalized size = 0.91 \[ \frac {-\frac {5 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{a^{3/4}}+\frac {5 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{a^{3/4}}-\frac {10 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {10 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac {72 \sqrt [4]{c} x}{a+c x^4}+\frac {32 a \sqrt [4]{c} x}{\left (a+c x^4\right )^2}}{256 c^{9/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 245, normalized size = 1.11 \[ -\frac {36 \, c x^{5} - 20 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} \arctan \left (-a^{2} c^{7} x \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {3}{4}} + \sqrt {a^{2} c^{4} \sqrt {-\frac {1}{a^{3} c^{9}}} + x^{2}} a^{2} c^{7} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {3}{4}}\right ) - 5 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} \log \left (a c^{2} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} + x\right ) + 5 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} \log \left (-a c^{2} \left (-\frac {1}{a^{3} c^{9}}\right )^{\frac {1}{4}} + x\right ) + 20 \, a x}{128 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 204, normalized size = 0.92 \[ \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a c^{3}} + \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a c^{3}} + \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a c^{3}} - \frac {5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a c^{3}} - \frac {9 \, c x^{5} + 5 \, a x}{32 \, {\left (c x^{4} + a\right )}^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 163, normalized size = 0.74 \[ \frac {5 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{128 a \,c^{2}}+\frac {5 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{128 a \,c^{2}}+\frac {5 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{256 a \,c^{2}}+\frac {-\frac {9 x^{5}}{32 c}-\frac {5 a x}{32 c^{2}}}{\left (c \,x^{4}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 213, normalized size = 0.96 \[ -\frac {9 \, c x^{5} + 5 \, a x}{32 \, {\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} + \frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}}} + \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}}\right )}}{256 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 81, normalized size = 0.37 \[ -\frac {\frac {9\,x^5}{32\,c}+\frac {5\,a\,x}{32\,c^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}-\frac {5\,\mathrm {atan}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,c^{9/4}}-\frac {5\,\mathrm {atanh}\left (\frac {c^{1/4}\,x}{{\left (-a\right )}^{1/4}}\right )}{64\,{\left (-a\right )}^{3/4}\,c^{9/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 68, normalized size = 0.31 \[ \frac {- 5 a x - 9 c x^{5}}{32 a^{2} c^{2} + 64 a c^{3} x^{4} + 32 c^{4} x^{8}} + \operatorname {RootSum} {\left (268435456 t^{4} a^{3} c^{9} + 625, \left (t \mapsto t \log {\left (\frac {128 t a c^{2}}{5} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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