Optimal. Leaf size=80 \[ \frac {8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac {4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x} \]
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Rubi [A] time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2016, 2002, 2014} \begin {gather*} \frac {8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac {4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x^2 \left (b x^2+c x^4\right )^{3/2} \, dx &=\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x}-\frac {(4 b) \int \left (b x^2+c x^4\right )^{3/2} \, dx}{9 c}\\ &=-\frac {4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x}+\frac {\left (8 b^2\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{63 c^2}\\ &=\frac {8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac {4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac {\left (b x^2+c x^4\right )^{5/2}}{9 c x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.66 \begin {gather*} \frac {x \left (b+c x^2\right )^3 \left (8 b^2-20 b c x^2+35 c^2 x^4\right )}{315 c^3 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 68, normalized size = 0.85 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (8 b^4-4 b^3 c x^2+3 b^2 c^2 x^4+50 b c^3 x^6+35 c^4 x^8\right )}{315 c^3 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 64, normalized size = 0.80 \begin {gather*} \frac {{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt {c x^{4} + b x^{2}}}{315 \, c^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 60, normalized size = 0.75 \begin {gather*} -\frac {8 \, b^{\frac {9}{2}} \mathrm {sgn}\relax (x)}{315 \, c^{3}} + \frac {35 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} \mathrm {sgn}\relax (x) - 90 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} b \mathrm {sgn}\relax (x) + 63 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} b^{2} \mathrm {sgn}\relax (x)}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.62 \begin {gather*} \frac {\left (c \,x^{2}+b \right ) \left (35 c^{2} x^{4}-20 b c \,x^{2}+8 b^{2}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{315 c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 57, normalized size = 0.71 \begin {gather*} \frac {{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt {c x^{2} + b}}{315 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.19, size = 51, normalized size = 0.64 \begin {gather*} \frac {{\left (c\,x^2+b\right )}^2\,\sqrt {c\,x^4+b\,x^2}\,\left (8\,b^2-20\,b\,c\,x^2+35\,c^2\,x^4\right )}{315\,c^3\,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 x^{2} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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