Optimal. Leaf size=80 \[ \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 \sqrt {c}}+\frac {3}{8} b \sqrt {b x^2+c x^4}+\frac {\left (b x^2+c x^4\right )^{3/2}}{4 x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2018, 664, 620, 206} \begin {gather*} \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 \sqrt {c}}+\frac {3}{8} b \sqrt {b x^2+c x^4}+\frac {\left (b x^2+c x^4\right )^{3/2}}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 664
Rule 2018
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (b x+c x^2\right )^{3/2}}{x^2} \, dx,x,x^2\right )\\ &=\frac {\left (b x^2+c x^4\right )^{3/2}}{4 x^2}+\frac {1}{8} (3 b) \operatorname {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x} \, dx,x,x^2\right )\\ &=\frac {3}{8} b \sqrt {b x^2+c x^4}+\frac {\left (b x^2+c x^4\right )^{3/2}}{4 x^2}+\frac {1}{16} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {3}{8} b \sqrt {b x^2+c x^4}+\frac {\left (b x^2+c x^4\right )^{3/2}}{4 x^2}+\frac {1}{8} \left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {3}{8} b \sqrt {b x^2+c x^4}+\frac {\left (b x^2+c x^4\right )^{3/2}}{4 x^2}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 71, normalized size = 0.89 \begin {gather*} \frac {1}{8} \sqrt {x^2 \left (b+c x^2\right )} \left (\frac {3 b^{3/2} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {c} x \sqrt {\frac {c x^2}{b}+1}}+5 b+2 c x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 73, normalized size = 0.91 \begin {gather*} \frac {1}{8} \left (5 b+2 c x^2\right ) \sqrt {b x^2+c x^4}-\frac {3 b^2 \log \left (-2 \sqrt {c} \sqrt {b x^2+c x^4}+b+2 c x^2\right )}{16 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.58, size = 145, normalized size = 1.81 \begin {gather*} \left [\frac {3 \, b^{2} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c\right )}}{16 \, c}, -\frac {3 \, b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - \sqrt {c x^{4} + b x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c\right )}}{8 \, c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 68, normalized size = 0.85 \begin {gather*} -\frac {3 \, b^{2} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right ) \mathrm {sgn}\relax (x)}{8 \, \sqrt {c}} + \frac {3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{16 \, \sqrt {c}} + \frac {1}{8} \, {\left (2 \, c x^{2} \mathrm {sgn}\relax (x) + 5 \, b \mathrm {sgn}\relax (x)\right )} \sqrt {c x^{2} + b} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 84, normalized size = 1.05 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 b^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+3 \sqrt {c \,x^{2}+b}\, b \sqrt {c}\, x +2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}\, x \right )}{8 \left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}\, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.44, size = 70, normalized size = 0.88 \begin {gather*} \frac {3 \, b^{2} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{16 \, \sqrt {c}} + \frac {3}{8} \, \sqrt {c x^{4} + b x^{2}} b + \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{4 \, x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^3} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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