Optimal. Leaf size=32 \[ \frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^2}}{8 b} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1591, 15, 30} \[ \frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 1591
Rubi steps
\begin {align*} \int x \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (c x^2\right )^{3/2} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac {\left (c \sqrt {c \left (a+b x^2\right )^2}\right ) \operatorname {Subst}\left (\int x^3 \, dx,x,a+b x^2\right )}{2 b \left (a+b x^2\right )}\\ &=\frac {c \left (a+b x^2\right )^3 \sqrt {c \left (a+b x^2\right )^2}}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.91 \[ \frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^2\right )^{3/2}}{8 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 73, normalized size = 2.28 \[ \frac {{\left (b^{3} c x^{8} + 4 \, a b^{2} c x^{6} + 6 \, a^{2} b c x^{4} + 4 \, a^{3} c x^{2}\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{8 \, {\left (b x^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 25, normalized size = 0.78 \[ \frac {{\left (b x^{2} + a\right )}^{4} c^{\frac {3}{2}} \mathrm {sgn}\left (b x^{2} + a\right )}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 59, normalized size = 1.84 \[ \frac {\left (b^{3} x^{6}+4 a \,b^{2} x^{4}+6 a^{2} b \,x^{2}+4 a^{3}\right ) \left (\left (b \,x^{2}+a \right )^{2} c \right )^{\frac {3}{2}} x^{2}}{8 \left (b \,x^{2}+a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.93, size = 60, normalized size = 1.88 \[ \frac {1}{8} \, {\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )}^{\frac {3}{2}} x^{2} + \frac {{\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.84, size = 40, normalized size = 1.25 \[ \frac {\left (b^2\,x^2+a\,b\right )\,{\left (c\,a^2+2\,c\,a\,b\,x^2+c\,b^2\,x^4\right )}^{3/2}}{8\,b^2} \]
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 \[ \int x \left (c \left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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