Optimal. Leaf size=88 \[ \frac {b \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c}+\frac {1}{6} \left (b x^2+c x^4\right )^{3/2}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{3/2}} \]
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Rubi [A]
time = 0.06, 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 = {2043, 678, 626,
634, 212} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 634
Rule 678
Rule 2043
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x} \, dx &=\frac {1}{2} \text {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 \text {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 \text {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 \text {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 = 0.09, size = 102, normalized size = 1.16 \begin {gather*} \frac {x \sqrt {b+c x^2} \left (\sqrt {c} x \sqrt {b+c x^2} \left (3 b^2+14 b c x^2+8 c^2 x^4\right )+3 b^3 \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )\right )}{48 c^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 102, normalized size = 1.16
method | result | size |
risch | \(\frac {\left (8 c^{2} x^{4}+14 b c \,x^{2}+3 b^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{48 c}-\frac {b^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{16 c^{\frac {3}{2}} x \sqrt {c \,x^{2}+b}}\) | \(90\) |
default | \(\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (8 x \left (c \,x^{2}+b \right )^{\frac {5}{2}} \sqrt {c}-2 \sqrt {c}\, \left (c \,x^{2}+b \right )^{\frac {3}{2}} b x -3 \sqrt {c}\, \sqrt {c \,x^{2}+b}\, b^{2} x -3 \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+b}\right ) b^{3}\right )}{48 x^{3} \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}}}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 91, normalized size = 1.03 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 166, normalized size = 1.89 \begin {gather*} \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 ] \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}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.71, size = 84, normalized size = 0.95 \begin {gather*} \frac {b^{3} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right ) \mathrm {sgn}\left (x\right )}{16 \, c^{\frac {3}{2}}} - \frac {b^{3} \log \left ({\left | b \right |}\right ) \mathrm {sgn}\left (x\right )}{32 \, c^{\frac {3}{2}}} + \frac {1}{48} \, {\left (2 \, {\left (4 \, c x^{2} \mathrm {sgn}\left (x\right ) + 7 \, b \mathrm {sgn}\left (x\right )\right )} x^{2} + \frac {3 \, b^{2} \mathrm {sgn}\left (x\right )}{c}\right )} \sqrt {c x^{2} + b} x \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} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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