Optimal. Leaf size=88 \[ -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{3/2}}+\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {1}{6} \left (b x^2+c x^4\right )^{3/2} \]
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Rubi [A] time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2018, 664, 612, 620, 206} \[ -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{3/2}}+\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {1}{6} \left (b x^2+c x^4\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 664
Rule 2018
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{6} \left (b x^2+c x^4\right )^{3/2}+\frac {1}{4} b \operatorname {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {1}{6} \left (b x^2+c x^4\right )^{3/2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{32 c}\\ &=\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {1}{6} \left (b x^2+c x^4\right )^{3/2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c}\\ &=\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {1}{6} \left (b x^2+c x^4\right )^{3/2}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 104, normalized size = 1.18 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {c} x \sqrt {\frac {c x^2}{b}+1} \left (3 b^2+14 b c x^2+8 c^2 x^4\right )-3 b^{5/2} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )\right )}{48 c^{3/2} x \sqrt {\frac {c x^2}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 166, normalized size = 1.89 \[ \left [\frac {3 \, b^{3} \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, c^{2}}, \frac {3 \, b^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 84, normalized size = 0.95 \[ \frac {b^{3} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right ) \mathrm {sgn}\relax (x)}{16 \, c^{\frac {3}{2}}} - \frac {b^{3} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{32 \, c^{\frac {3}{2}}} + \frac {1}{48} \, {\left (2 \, {\left (4 \, c x^{2} \mathrm {sgn}\relax (x) + 7 \, b \mathrm {sgn}\relax (x)\right )} x^{2} + \frac {3 \, b^{2} \mathrm {sgn}\relax (x)}{c}\right )} \sqrt {c x^{2} + b} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 102, normalized size = 1.16 \[ \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (-3 b^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )-3 \sqrt {c \,x^{2}+b}\, b^{2} \sqrt {c}\, x -2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b \sqrt {c}\, x +8 \left (c \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {c}\, x \right )}{48 \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 91, normalized size = 1.03 \[ \frac {1}{8} \, \sqrt {c x^{4} + b x^{2}} b x^{2} - \frac {b^{3} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{32 \, c^{\frac {3}{2}}} + \frac {1}{6} \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} + \frac {\sqrt {c x^{4} + b x^{2}} b^{2}}{16 \, c} \]
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 \[ \int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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