Optimal. Leaf size=99 \[ -\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c} \]
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Rubi [A] time = 0.106552, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{b^2 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{5/2}}+\frac{(b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{8 c^2}+\frac{e \left (b x+c x^2\right )^{3/2}}{3 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 11.3516, size = 87, normalized size = 0.88 \[ \frac{b^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} + \frac{e \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{8 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.167974, size = 110, normalized size = 1.11 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-3 b^2 e+2 b c (3 d+e x)+4 c^2 x (3 d+2 e x)\right )+\frac{3 b^2 (b e-2 c d) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{24 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.009, size = 157, normalized size = 1.6 \[{\frac{dx}{2}\sqrt{c{x}^{2}+bx}}+{\frac{bd}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}d}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e}{3\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{bex}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}e}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{e{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229839, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, c^{2} e x^{2} + 6 \, b c d - 3 \, b^{2} e + 2 \,{\left (6 \, c^{2} d + b c e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3 \,{\left (2 \, b^{2} c d - b^{3} e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{48 \, c^{\frac{5}{2}}}, \frac{{\left (8 \, c^{2} e x^{2} + 6 \, b c d - 3 \, b^{2} e + 2 \,{\left (6 \, c^{2} d + b c e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} - 3 \,{\left (2 \, b^{2} c d - b^{3} e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{24 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x \left (b + c x\right )} \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221164, size = 146, normalized size = 1.47 \[ \frac{1}{24} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \, x e + \frac{6 \, c^{2} d + b c e}{c^{2}}\right )} x + \frac{3 \,{\left (2 \, b c d - b^{2} e\right )}}{c^{2}}\right )} + \frac{{\left (2 \, b^{2} c d - b^{3} e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{16 \, c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(e*x + d),x, algorithm="giac")
[Out]