Optimal. Leaf size=162 \[ \frac{(b+2 c x) \sqrt{b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{64 c^3}-\frac{b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 c^2}+\frac{e \left (b x+c x^2\right )^{3/2} (d+e x)}{4 c} \]
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Rubi [A] time = 0.12639, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {742, 640, 612, 620, 206} \[ \frac{(b+2 c x) \sqrt{b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{64 c^3}-\frac{b^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}+\frac{5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 c^2}+\frac{e \left (b x+c x^2\right )^{3/2} (d+e x)}{4 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 640
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int (d+e x)^2 \sqrt{b x+c x^2} \, dx &=\frac{e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}+\frac{\int \left (\frac{1}{2} d (8 c d-3 b e)+\frac{5}{2} e (2 c d-b e) x\right ) \sqrt{b x+c x^2} \, dx}{4 c}\\ &=\frac{5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}+\frac{\left (c d (8 c d-3 b e)-\frac{5}{2} b e (2 c d-b e)\right ) \int \sqrt{b x+c x^2} \, dx}{8 c^2}\\ &=\frac{\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}+\frac{5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c^3}\\ &=\frac{\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}+\frac{5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c^3}\\ &=\frac{\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b+2 c x) \sqrt{b x+c x^2}}{64 c^3}+\frac{5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac{e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}-\frac{b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.298318, size = 164, normalized size = 1.01 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-2 b^2 c e (24 d+5 e x)+15 b^3 e^2+8 b c^2 \left (6 d^2+4 d e x+e^2 x^2\right )+16 c^3 x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )-\frac{3 b^{3/2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{192 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 287, normalized size = 1.8 \begin{align*}{\frac{{e}^{2}x}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{e}^{2}b}{24\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}{e}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{3}{e}^{2}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{e}^{2}{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{2\,de}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{bdex}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}de}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{de{b}^{3}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{d}^{2}x}{2}\sqrt{c{x}^{2}+bx}}+{\frac{{d}^{2}b}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}{d}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.66614, size = 757, normalized size = 4.67 \begin{align*} \left [\frac{3 \,{\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \sqrt{c} \log \left (2 \, c x + b - 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) + 2 \,{\left (48 \, c^{4} e^{2} x^{3} + 48 \, b c^{3} d^{2} - 48 \, b^{2} c^{2} d e + 15 \, b^{3} c e^{2} + 8 \,{\left (16 \, c^{4} d e + b c^{3} e^{2}\right )} x^{2} + 2 \,{\left (48 \, c^{4} d^{2} + 16 \, b c^{3} d e - 5 \, b^{2} c^{2} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{384 \, c^{4}}, \frac{3 \,{\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (48 \, c^{4} e^{2} x^{3} + 48 \, b c^{3} d^{2} - 48 \, b^{2} c^{2} d e + 15 \, b^{3} c e^{2} + 8 \,{\left (16 \, c^{4} d e + b c^{3} e^{2}\right )} x^{2} + 2 \,{\left (48 \, c^{4} d^{2} + 16 \, b c^{3} d e - 5 \, b^{2} c^{2} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{192 \, c^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x \left (b + c x\right )} \left (d + e x\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3201, size = 232, normalized size = 1.43 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, x e^{2} + \frac{16 \, c^{3} d e + b c^{2} e^{2}}{c^{3}}\right )} x + \frac{48 \, c^{3} d^{2} + 16 \, b c^{2} d e - 5 \, b^{2} c e^{2}}{c^{3}}\right )} x + \frac{3 \,{\left (16 \, b c^{2} d^{2} - 16 \, b^{2} c d e + 5 \, b^{3} e^{2}\right )}}{c^{3}}\right )} + \frac{{\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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