Optimal. Leaf size=28 \[ \frac {c (a+b x)^3 \sqrt {c (a+b x)^2}}{4 b} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac {c (a+b x)^3 \sqrt {c (a+b x)^2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \left (c (a+b x)^2\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (c x^2\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\left (c \sqrt {c (a+b x)^2}\right ) \operatorname {Subst}\left (\int x^3 \, dx,x,a+b x\right )}{b (a+b x)}\\ &=\frac {c (a+b x)^3 \sqrt {c (a+b x)^2}}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.89 \[ \frac {(a+b x) \left (c (a+b x)^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 67, normalized size = 2.39 \[ \frac {{\left (b^{3} c x^{4} + 4 \, a b^{2} c x^{3} + 6 \, a^{2} b c x^{2} + 4 \, a^{3} c x\right )} \sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{4 \, {\left (b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 21, normalized size = 0.75 \[ \frac {{\left (b x + a\right )}^{4} c^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 51, normalized size = 1.82 \[ \frac {\left (b^{3} x^{3}+4 a \,b^{2} x^{2}+6 a^{2} b x +4 a^{3}\right ) \left (\left (b x +a \right )^{2} c \right )^{\frac {3}{2}} x}{4 \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 54, normalized size = 1.93 \[ \frac {1}{4} \, {\left (b^{2} c x^{2} + 2 \, a b c x + a^{2} c\right )}^{\frac {3}{2}} x + \frac {{\left (b^{2} c x^{2} + 2 \, a b c x + a^{2} c\right )}^{\frac {3}{2}} a}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 36, normalized size = 1.29 \[ \frac {\left (x\,b^2+a\,b\right )\,{\left (c\,a^2+2\,c\,a\,b\,x+c\,b^2\,x^2\right )}^{3/2}}{4\,b^2} \]
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 \[ \int \left (c \left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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