Optimal. Leaf size=140 \[ -\frac {a^3 c \sqrt {c \left (a+b x^2\right )^2}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a^2 b c \log (x) \sqrt {c \left (a+b x^2\right )^2}}{a+b x^2}+\frac {b^3 c x^4 \sqrt {c \left (a+b x^2\right )^2}}{4 \left (a+b x^2\right )}+\frac {3 a b^2 c x^2 \sqrt {c \left (a+b x^2\right )^2}}{2 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 184, normalized size of antiderivative = 1.31, 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 = {1989, 1112, 266, 43} \[ \frac {b^3 c x^4 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{4 \left (a+b x^2\right )}+\frac {3 a b^2 c x^2 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{2 \left (a+b x^2\right )}-\frac {a^3 c \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a^2 b c \log (x) \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{a+b x^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1112
Rule 1989
Rubi steps
\begin {align*} \int \frac {\left (c \left (a+b x^2\right )^2\right )^{3/2}}{x^3} \, dx &=\int \frac {\left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{x^3} \, dx\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \int \frac {\left (a b c+b^2 c x^2\right )^3}{x^3} \, dx}{b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \operatorname {Subst}\left (\int \frac {\left (a b c+b^2 c x\right )^3}{x^2} \, dx,x,x^2\right )}{2 b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \operatorname {Subst}\left (\int \left (3 a b^5 c^3+\frac {a^3 b^3 c^3}{x^2}+\frac {3 a^2 b^4 c^3}{x}+b^6 c^3 x\right ) \, dx,x,x^2\right )}{2 b^2 c \left (a b c+b^2 c x^2\right )}\\ &=-\frac {a^3 c \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a b^2 c x^2 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{2 \left (a+b x^2\right )}+\frac {b^3 c x^4 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{4 \left (a+b x^2\right )}+\frac {3 a^2 b c \sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \log (x)}{a+b x^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 0.46 \[ -\frac {\left (c \left (a+b x^2\right )^2\right )^{3/2} \left (2 a^3-12 a^2 b x^2 \log (x)-6 a b^2 x^4-b^3 x^6\right )}{4 x^2 \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 76, normalized size = 0.54 \[ \frac {{\left (b^{3} c x^{6} + 6 \, a b^{2} c x^{4} + 12 \, a^{2} b c x^{2} \log \relax (x) - 2 \, a^{3} c\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{4 \, {\left (b x^{4} + a x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 91, normalized size = 0.65 \[ \frac {1}{4} \, {\left (b^{3} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a b^{2} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a^{2} b \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {2 \, {\left (3 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{x^{2}}\right )} c^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 61, normalized size = 0.44 \[ \frac {\left (\left (b \,x^{2}+a \right )^{2} c \right )^{\frac {3}{2}} \left (b^{3} x^{6}+6 a \,b^{2} x^{4}+12 a^{2} b \,x^{2} \ln \relax (x )-2 a^{3}\right )}{4 \left (b \,x^{2}+a \right )^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 176, normalized size = 1.26 \[ \frac {3}{2} \, \left (-1\right )^{2 \, b^{2} c x^{2} + 2 \, a b c} a^{2} b c^{\frac {3}{2}} \log \left (2 \, b^{2} c x^{2} + 2 \, a b c\right ) - \frac {3}{2} \, \left (-1\right )^{2 \, a b c x^{2} + 2 \, a^{2} c} a^{2} b c^{\frac {3}{2}} \log \left (2 \, a b c + \frac {2 \, a^{2} c}{x^{2}}\right ) + \frac {3}{4} \, \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c} b^{2} c x^{2} + \frac {9}{4} \, \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c} a b c - \frac {{\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )}^{\frac {3}{2}}}{2 \, x^{2}} \]
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 \[ \int \frac {{\left (c\,{\left (b\,x^2+a\right )}^2\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (c \left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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