Optimal. Leaf size=139 \[ \frac {a^3 c \log (x) \sqrt {c \left (a+b x^2\right )^2}}{a+b x^2}+\frac {3 a^2 b c x^2 \sqrt {c \left (a+b x^2\right )^2}}{2 \left (a+b x^2\right )}+\frac {b^3 c x^6 \sqrt {c \left (a+b x^2\right )^2}}{6 \left (a+b x^2\right )}+\frac {3 a b^2 c x^4 \sqrt {c \left (a+b x^2\right )^2}}{4 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 183, normalized size of antiderivative = 1.32, 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} \begin {gather*} \frac {b^3 c x^6 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{6 \left (a+b x^2\right )}+\frac {3 a b^2 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 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 \log (x) \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{a+b x^2} \end {gather*}
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} \, dx &=\int \frac {\left (a^2 c+2 a b c x^2+b^2 c x^4\right )^{3/2}}{x} \, 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} \, 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} \, 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^2 b^4 c^3+\frac {a^3 b^3 c^3}{x}+3 a b^5 c^3 x+b^6 c^3 x^2\right ) \, dx,x,x^2\right )}{2 b^2 c \left (a b c+b^2 c x^2\right )}\\ &=\frac {3 a^2 b 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 {3 a b^2 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 {b^3 c x^6 \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{6 \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} \log (x)}{a+b x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.45 \begin {gather*} \frac {\left (c \left (a+b x^2\right )^2\right )^{3/2} \left (12 a^3 \log (x)+b x^2 \left (18 a^2+9 a b x^2+2 b^2 x^4\right )\right )}{12 \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.57, size = 263, normalized size = 1.89 \begin {gather*} \frac {1}{24} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4} \left (11 a^2 c+7 a b c x^2+2 b^2 c x^4\right )+\frac {1}{24} \left (-18 a^2 c x^2 \sqrt {b^2 c}-9 a b c x^4 \sqrt {b^2 c}-2 b^2 c x^6 \sqrt {b^2 c}\right )+\frac {1}{2} a^3 c^{3/2} \tanh ^{-1}\left (\frac {x^2 \sqrt {b^2 c}}{a \sqrt {c}}-\frac {\sqrt {a^2 c+2 a b c x^2+b^2 c x^4}}{a \sqrt {c}}\right )-\frac {a^3 c \sqrt {b^2 c} \log \left (x^2 \left (a b c+b^2 c x^2\right )-x^2 \sqrt {b^2 c} \sqrt {a^2 c+2 a b c x^2+b^2 c x^4}\right )}{4 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 73, normalized size = 0.53 \begin {gather*} \frac {{\left (2 \, b^{3} c x^{6} + 9 \, a b^{2} c x^{4} + 18 \, a^{2} b c x^{2} + 12 \, a^{3} c \log \relax (x)\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{12 \, {\left (b x^{2} + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 73, normalized size = 0.53 \begin {gather*} \frac {1}{12} \, {\left (2 \, b^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + 9 \, a b^{2} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 18 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + 6 \, a^{3} \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right )\right )} c^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 59, normalized size = 0.42 \begin {gather*} \frac {\left (\left (b \,x^{2}+a \right )^{2} c \right )^{\frac {3}{2}} \left (2 b^{3} x^{6}+9 a \,b^{2} x^{4}+18 a^{2} b \,x^{2}+12 a^{3} \ln \relax (x )\right )}{12 \left (b \,x^{2}+a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 171, normalized size = 1.23 \begin {gather*} \frac {1}{2} \, \left (-1\right )^{2 \, b^{2} c x^{2} + 2 \, a b c} a^{3} c^{\frac {3}{2}} \log \left (2 \, b^{2} c x^{2} + 2 \, a b c\right ) - \frac {1}{2} \, \left (-1\right )^{2 \, a b c x^{2} + 2 \, a^{2} c} a^{3} c^{\frac {3}{2}} \log \left (2 \, a b c + \frac {2 \, a^{2} c}{x^{2}}\right ) + \frac {1}{4} \, \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c} a b c x^{2} + \frac {3}{4} \, \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c} a^{2} c + \frac {1}{6} \, {\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )}^{\frac {3}{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\,{\left (b\,x^2+a\right )}^2\right )}^{3/2}}{x} \,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 (c \left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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