Optimal. Leaf size=25 \[ \frac {(a+b x) \sqrt {c (a+b x)^2}}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 25, 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} \[ \frac {(a+b x) \sqrt {c (a+b x)^2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \sqrt {c (a+b x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {c x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\sqrt {c (a+b x)^2} \operatorname {Subst}(\int x \, dx,x,a+b x)}{b (a+b x)}\\ &=\frac {(a+b x) \sqrt {c (a+b x)^2}}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.24 \[ \frac {c x (a+b x) (2 a+b x)}{2 \sqrt {c (a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 41, normalized size = 1.64 \[ \frac {\sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c} {\left (b x^{2} + 2 \, a x\right )}}{2 \, {\left (b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 36, normalized size = 1.44 \[ \frac {1}{2} \, {\left ({\left (b x^{2} + 2 \, a x\right )} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{2} \mathrm {sgn}\left (b x + a\right )}{b}\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.16 \[ \frac {\left (b x +2 a \right ) \sqrt {\left (b x +a \right )^{2} c}\, x}{2 b x +2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 54, normalized size = 2.16 \[ \frac {1}{2} \, \sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c} x + \frac {\sqrt {b^{2} c x^{2} + 2 \, a b c x + a^{2} c} a}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.47, size = 21, normalized size = 0.84 \[ \frac {\sqrt {c\,{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b} \]
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 \[ \int \sqrt {c \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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