Optimal. Leaf size=40 \[ \frac{\left (b+2 c \sqrt{x}\right )^6}{192 c^4}-\frac{b \left (b+2 c \sqrt{x}\right )^5}{160 c^4} \]
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Rubi [A] time = 0.019456, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {28, 190, 43} \[ \frac{\left (b+2 c \sqrt{x}\right )^6}{192 c^4}-\frac{b \left (b+2 c \sqrt{x}\right )^5}{160 c^4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (\frac{b^2}{4 c}+b \sqrt{x}+c x\right )^2 \, dx &=\frac{\int \left (\frac{b}{2}+c \sqrt{x}\right )^4 \, dx}{c^2}\\ &=\frac{2 \operatorname{Subst}\left (\int x \left (\frac{b}{2}+c x\right )^4 \, dx,x,\sqrt{x}\right )}{c^2}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{b \left (\frac{b}{2}+c x\right )^4}{2 c}+\frac{\left (\frac{b}{2}+c x\right )^5}{c}\right ) \, dx,x,\sqrt{x}\right )}{c^2}\\ &=-\frac{b \left (b+2 c \sqrt{x}\right )^5}{160 c^4}+\frac{\left (b+2 c \sqrt{x}\right )^6}{192 c^4}\\ \end{align*}
Mathematica [A] time = 0.0257015, size = 29, normalized size = 0.72 \[ -\frac{\left (b-10 c \sqrt{x}\right ) \left (b+2 c \sqrt{x}\right )^5}{960 c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 52, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2}}+{\frac{b}{2\,c} \left ({\frac{8\,{c}^{2}}{5}{x}^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}}{3}{x}^{{\frac{3}{2}}}} \right ) }+{\frac{1}{3\,c} \left ({\frac{{b}^{2}}{4\,c}}+cx \right ) ^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0648, size = 73, normalized size = 1.82 \begin{align*} \frac{1}{3} \, c^{2} x^{3} + \frac{4}{5} \, b c x^{\frac{5}{2}} + \frac{1}{2} \, b^{2} x^{2} + \frac{b^{4} x}{16 \, c^{2}} + \frac{{\left (3 \, c x^{2} + 4 \, b x^{\frac{3}{2}}\right )} b^{2}}{12 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99539, size = 126, normalized size = 3.15 \begin{align*} \frac{80 \, c^{4} x^{3} + 180 \, b^{2} c^{2} x^{2} + 15 \, b^{4} x + 16 \,{\left (12 \, b c^{3} x^{2} + 5 \, b^{3} c x\right )} \sqrt{x}}{240 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.411806, size = 51, normalized size = 1.27 \begin{align*} \frac{b^{4} x}{16 c^{2}} + \frac{b^{3} x^{\frac{3}{2}}}{3 c} + \frac{3 b^{2} x^{2}}{4} + \frac{4 b c x^{\frac{5}{2}}}{5} + \frac{c^{2} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11856, size = 66, normalized size = 1.65 \begin{align*} \frac{80 \, c^{4} x^{3} + 192 \, b c^{3} x^{\frac{5}{2}} + 180 \, b^{2} c^{2} x^{2} + 80 \, b^{3} c x^{\frac{3}{2}} + 15 \, b^{4} x}{240 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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