Optimal. Leaf size=90 \[ -\frac {d (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{5/2}}+\frac {x^2 \sqrt {c+\frac {d}{x^2}} (4 b c-3 a d)}{8 c^2}+\frac {a x^4 \sqrt {c+\frac {d}{x^2}}}{4 c} \]
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Rubi [A] time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {446, 78, 51, 63, 208} \[ \frac {x^2 \sqrt {c+\frac {d}{x^2}} (4 b c-3 a d)}{8 c^2}-\frac {d (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{5/2}}+\frac {a x^4 \sqrt {c+\frac {d}{x^2}}}{4 c} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) x^3}{\sqrt {c+\frac {d}{x^2}}} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {a+b x}{x^3 \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {a \sqrt {c+\frac {d}{x^2}} x^4}{4 c}-\frac {\left (2 b c-\frac {3 a d}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )}{4 c}\\ &=\frac {(4 b c-3 a d) \sqrt {c+\frac {d}{x^2}} x^2}{8 c^2}+\frac {a \sqrt {c+\frac {d}{x^2}} x^4}{4 c}+\frac {(d (4 b c-3 a d)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,\frac {1}{x^2}\right )}{16 c^2}\\ &=\frac {(4 b c-3 a d) \sqrt {c+\frac {d}{x^2}} x^2}{8 c^2}+\frac {a \sqrt {c+\frac {d}{x^2}} x^4}{4 c}+\frac {(4 b c-3 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+\frac {d}{x^2}}\right )}{8 c^2}\\ &=\frac {(4 b c-3 a d) \sqrt {c+\frac {d}{x^2}} x^2}{8 c^2}+\frac {a \sqrt {c+\frac {d}{x^2}} x^4}{4 c}-\frac {d (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 95, normalized size = 1.06 \[ \frac {\sqrt {c} x \left (c x^2+d\right ) \left (2 a c x^2-3 a d+4 b c\right )+d \sqrt {c x^2+d} (3 a d-4 b c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+d}}\right )}{8 c^{5/2} x \sqrt {c+\frac {d}{x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 192, normalized size = 2.13 \[ \left [-\frac {{\left (4 \, b c d - 3 \, a d^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left (2 \, a c^{2} x^{4} + {\left (4 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, c^{3}}, \frac {{\left (4 \, b c d - 3 \, a d^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (2 \, a c^{2} x^{4} + {\left (4 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{8 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 113, normalized size = 1.26 \[ \frac {1}{8} \, \sqrt {c x^{4} + d x^{2}} {\left (\frac {2 \, a x^{2}}{c} + \frac {4 \, b c - 3 \, a d}{c^{2}}\right )} + \frac {{\left (4 \, b c d - 3 \, a d^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )} \sqrt {c} - d \right |}\right )}{16 \, c^{\frac {5}{2}}} - \frac {4 \, b c d \log \left ({\left | d \right |}\right ) - 3 \, a d^{2} \log \left ({\left | d \right |}\right )}{16 \, c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 129, normalized size = 1.43 \[ \frac {\sqrt {c \,x^{2}+d}\, \left (2 \sqrt {c \,x^{2}+d}\, a \,c^{\frac {5}{2}} x^{3}+3 a c \,d^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )-4 b \,c^{2} d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )-3 \sqrt {c \,x^{2}+d}\, a \,c^{\frac {3}{2}} d x +4 \sqrt {c \,x^{2}+d}\, b \,c^{\frac {5}{2}} x \right )}{8 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, c^{\frac {7}{2}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.17, size = 178, normalized size = 1.98 \[ \frac {1}{4} \, b {\left (\frac {2 \, \sqrt {c + \frac {d}{x^{2}}} d}{{\left (c + \frac {d}{x^{2}}\right )} c - c^{2}} + \frac {d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {1}{16} \, a {\left (\frac {3 \, d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} - 5 \, \sqrt {c + \frac {d}{x^{2}}} c d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} c^{2} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} c^{3} + c^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.35, size = 99, normalized size = 1.10 \[ \frac {5\,a\,x^4\,\sqrt {c+\frac {d}{x^2}}}{8\,c}-\frac {3\,a\,x^4\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{8\,c^2}+\frac {b\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2\,c}-\frac {b\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2\,c^{3/2}}+\frac {3\,a\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8\,c^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 71.06, size = 150, normalized size = 1.67 \[ \frac {a x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a \sqrt {d} x^{3}}{8 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {3 a d^{\frac {3}{2}} x}{8 c^{2} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 a d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 c^{\frac {5}{2}}} + \frac {b \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2 c} - \frac {b d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2 c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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