Optimal. Leaf size=91 \[ \frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{5/2}}-\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {\left (b x^2+c x^4\right )^{3/2}}{6 c} \]
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Rubi [A] time = 0.10, antiderivative size = 91, 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, 640, 612, 620, 206} \[ \frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{5/2}}-\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {\left (b x^2+c x^4\right )^{3/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rule 2018
Rubi steps
\begin {align*} \int x^3 \sqrt {b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \sqrt {b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {\left (b x^2+c x^4\right )^{3/2}}{6 c}-\frac {b \operatorname {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {\left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{32 c^2}\\ &=-\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {\left (b x^2+c x^4\right )^{3/2}}{6 c}+\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^2}\\ &=-\frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {\left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 103, normalized size = 1.13 \[ \frac {x \sqrt {b+c x^2} \left (3 b^3 \log \left (\sqrt {c} \sqrt {b+c x^2}+c x\right )+\sqrt {c} x \sqrt {b+c x^2} \left (-3 b^2+2 b c x^2+8 c^2 x^4\right )\right )}{48 c^{5/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 167, normalized size = 1.84 \[ \left [\frac {3 \, b^{3} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} x^{4} + 2 \, b c^{2} x^{2} - 3 \, b^{2} c\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, c^{3}}, -\frac {3 \, b^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - {\left (8 \, c^{3} x^{4} + 2 \, b c^{2} x^{2} - 3 \, b^{2} c\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 85, normalized size = 0.93 \[ \frac {1}{48} \, {\left (2 \, {\left (4 \, x^{2} \mathrm {sgn}\relax (x) + \frac {b \mathrm {sgn}\relax (x)}{c}\right )} x^{2} - \frac {3 \, b^{2} \mathrm {sgn}\relax (x)}{c^{2}}\right )} \sqrt {c x^{2} + b} x - \frac {b^{3} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right ) \mathrm {sgn}\relax (x)}{16 \, c^{\frac {5}{2}}} + \frac {b^{3} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\relax (x)}{32 \, c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 104, normalized size = 1.14 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (8 \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}} x^{3}+3 b^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+3 \sqrt {c \,x^{2}+b}\, b^{2} \sqrt {c}\, x -6 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b \sqrt {c}\, x \right )}{48 \sqrt {c \,x^{2}+b}\, c^{\frac {5}{2}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 97, normalized size = 1.07 \[ -\frac {\sqrt {c x^{4} + b x^{2}} b x^{2}}{8 \, c} + \frac {b^{3} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{32 \, c^{\frac {5}{2}}} - \frac {\sqrt {c x^{4} + b x^{2}} b^{2}}{16 \, c^{2}} + \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{6 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 77, normalized size = 0.85 \[ \frac {b^3\,\ln \left (\frac {2\,c\,x^2+b}{\sqrt {c}}+2\,\sqrt {c\,x^4+b\,x^2}\right )}{32\,c^{5/2}}+\frac {\sqrt {c\,x^4+b\,x^2}\,\left (-3\,b^2+2\,b\,c\,x^2+8\,c^2\,x^4\right )}{48\,c^2} \]
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 \[ \int x^{3} \sqrt {x^{2} \left (b + c x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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