Optimal. Leaf size=57 \[ \frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{4 \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ \frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{4 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)^2}{\sqrt {c x^2}} \, dx &=\frac {x \int x (a+b x)^2 \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{4 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.61 \[ \frac {x^3 \left (6 a^2+8 a b x+3 b^2 x^2\right )}{12 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 34, normalized size = 0.60 \[ \frac {{\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} x\right )} \sqrt {c x^{2}}}{12 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 38, normalized size = 0.67 \[ \frac {1}{12} \, \sqrt {c x^{2}} {\left ({\left (\frac {3 \, b^{2} x}{c} + \frac {8 \, a b}{c}\right )} x + \frac {6 \, a^{2}}{c}\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.56 \[ \frac {\left (3 b^{2} x^{2}+8 a b x +6 a^{2}\right ) x^{3}}{12 \sqrt {c \,x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 47, normalized size = 0.82 \[ \frac {\sqrt {c x^{2}} b^{2} x^{3}}{4 \, c} + \frac {2 \, \sqrt {c x^{2}} a b x^{2}}{3 \, c} + \frac {a^{2} x^{2}}{2 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2\,{\left (a+b\,x\right )}^2}{\sqrt {c\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 61, normalized size = 1.07 \[ \frac {a^{2} x^{3}}{2 \sqrt {c} \sqrt {x^{2}}} + \frac {2 a b x^{4}}{3 \sqrt {c} \sqrt {x^{2}}} + \frac {b^{2} x^{5}}{4 \sqrt {c} \sqrt {x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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