Optimal. Leaf size=68 \[ \frac {1}{2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}-\frac {c \sqrt {a+c x^4}}{2 x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {275, 277, 217, 206} \[ \frac {1}{2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )-\frac {c \sqrt {a+c x^4}}{2 x^2}-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+c x^2\right )^{3/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {\sqrt {a+c x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {a+c x^4}}{2 x^2}-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {a+c x^4}}{2 x^2}-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c^2 \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {a+c x^4}}\right )\\ &=-\frac {c \sqrt {a+c x^4}}{2 x^2}-\frac {\left (a+c x^4\right )^{3/2}}{6 x^6}+\frac {1}{2} c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.76 \[ -\frac {a \sqrt {a+c x^4} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {c x^4}{a}\right )}{6 x^6 \sqrt {\frac {c x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 116, normalized size = 1.71 \[ \left [\frac {3 \, c^{\frac {3}{2}} x^{6} \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) - 2 \, {\left (4 \, c x^{4} + a\right )} \sqrt {c x^{4} + a}}{12 \, x^{6}}, -\frac {3 \, \sqrt {-c} c x^{6} \arctan \left (\frac {\sqrt {-c} x^{2}}{\sqrt {c x^{4} + a}}\right ) + {\left (4 \, c x^{4} + a\right )} \sqrt {c x^{4} + a}}{6 \, x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 122, normalized size = 1.79 \[ -\frac {1}{4} \, c^{\frac {3}{2}} \log \left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{2} c^{\frac {3}{2}} + 2 \, a^{3} c^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 55, normalized size = 0.81 \[ \frac {c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2}-\frac {2 \sqrt {c \,x^{4}+a}\, c}{3 x^{2}}-\frac {\sqrt {c \,x^{4}+a}\, a}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 75, normalized size = 1.10 \[ -\frac {1}{4} \, c^{\frac {3}{2}} \log \left (-\frac {\sqrt {c} - \frac {\sqrt {c x^{4} + a}}{x^{2}}}{\sqrt {c} + \frac {\sqrt {c x^{4} + a}}{x^{2}}}\right ) - \frac {\sqrt {c x^{4} + a} c}{2 \, x^{2}} - \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x^4+a\right )}^{3/2}}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.48, size = 80, normalized size = 1.18 \[ - \frac {a \sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{6 x^{4}} - \frac {2 c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{3} - \frac {c^{\frac {3}{2}} \log {\left (\frac {a}{c x^{4}} \right )}}{4} + \frac {c^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{c x^{4}} + 1} + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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