Optimal. Leaf size=41 \[ a^2 x+\frac {4 a b (c+d x)^{3/2}}{3 d}+\frac {b^2 (c+d x)^2}{2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {247, 190, 43} \[ a^2 x+\frac {4 a b (c+d x)^{3/2}}{3 d}+\frac {b^2 (c+d x)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rule 247
Rubi steps
\begin {align*} \int \left (a+b \sqrt {c+d x}\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sqrt {x}\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int x (a+b x)^2 \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=a^2 x+\frac {4 a b (c+d x)^{3/2}}{3 d}+\frac {b^2 (c+d x)^2}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 0.98 \[ \frac {6 a^2 d x+8 a b (c+d x)^{3/2}+3 b^2 (c+d x)^2}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 49, normalized size = 1.20 \[ \frac {3 \, b^{2} d^{2} x^{2} + 6 \, {\left (b^{2} c + a^{2}\right )} d x + 8 \, {\left (a b d x + a b c\right )} \sqrt {d x + c}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 82, normalized size = 2.00 \[ \frac {6 \, {\left (d x + c\right )} b^{2} c + 24 \, \sqrt {d x + c} a b c + 6 \, {\left (d x + c\right )} a^{2} + 8 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a b + 3 \, {\left ({\left (d x + c\right )}^{2} - 2 \, {\left (d x + c\right )} c\right )} b^{2}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.85 \[ a^{2} x +\left (\frac {1}{2} d \,x^{2}+c x \right ) b^{2}+\frac {4 \left (d x +c \right )^{\frac {3}{2}} a b}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 35, normalized size = 0.85 \[ \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} b^{2} + a^{2} x + \frac {4 \, {\left (d x + c\right )}^{\frac {3}{2}} a b}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 36, normalized size = 0.88 \[ \frac {3\,b^2\,{\left (c+d\,x\right )}^2+8\,a\,b\,{\left (c+d\,x\right )}^{3/2}+6\,a^2\,d\,x}{6\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 68, normalized size = 1.66 \[ \begin {cases} a^{2} x + \frac {4 a b c \sqrt {c + d x}}{3 d} + \frac {4 a b x \sqrt {c + d x}}{3} + b^{2} c x + \frac {b^{2} d x^{2}}{2} & \text {for}\: d \neq 0 \\x \left (a + b \sqrt {c}\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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