Optimal. Leaf size=137 \[ \frac{3 b^4 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{7/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{128 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{16 c^2}+\frac{e \left (b x+c x^2\right )^{5/2}}{5 c} \]
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Rubi [A] time = 0.149719, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 b^4 (2 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{128 c^{7/2}}-\frac{3 b^2 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e)}{128 c^3}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{16 c^2}+\frac{e \left (b x+c x^2\right )^{5/2}}{5 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 16.3061, size = 126, normalized size = 0.92 \[ - \frac{3 b^{4} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{128 c^{\frac{7}{2}}} + \frac{3 b^{2} \left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{128 c^{3}} + \frac{e \left (b x + c x^{2}\right )^{\frac{5}{2}}}{5 c} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{16 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.242282, size = 148, normalized size = 1.08 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (15 b^4 e-10 b^3 c (3 d+e x)+4 b^2 c^2 x (5 d+2 e x)+16 b c^3 x^2 (15 d+11 e x)+32 c^4 x^3 (5 d+4 e x)\right )-\frac{15 b^4 (b e-2 c d) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{640 c^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.009, size = 239, normalized size = 1.7 \[{\frac{dx}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{bd}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}dx}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,d{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,d{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{e}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{bex}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}e}{16\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,e{b}^{3}x}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,e{b}^{4}}{128\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,e{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+b*x)^(3/2),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((c*x^2 + b*x)^(3/2)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233382, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (128 \, c^{4} e x^{4} - 30 \, b^{3} c d + 15 \, b^{4} e + 16 \,{\left (10 \, c^{4} d + 11 \, b c^{3} e\right )} x^{3} + 8 \,{\left (30 \, b c^{3} d + b^{2} c^{2} e\right )} x^{2} + 10 \,{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 15 \,{\left (2 \, b^{4} c d - b^{5} e\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{1280 \, c^{\frac{7}{2}}}, \frac{{\left (128 \, c^{4} e x^{4} - 30 \, b^{3} c d + 15 \, b^{4} e + 16 \,{\left (10 \, c^{4} d + 11 \, b c^{3} e\right )} x^{3} + 8 \,{\left (30 \, b c^{3} d + b^{2} c^{2} e\right )} x^{2} + 10 \,{\left (2 \, b^{2} c^{2} d - b^{3} c e\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 15 \,{\left (2 \, b^{4} c d - b^{5} e\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{640 \, \sqrt{-c} c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(e*x + d),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221651, size = 231, normalized size = 1.69 \[ \frac{1}{640} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x e + \frac{10 \, c^{5} d + 11 \, b c^{4} e}{c^{4}}\right )} x + \frac{30 \, b c^{4} d + b^{2} c^{3} e}{c^{4}}\right )} x + \frac{5 \,{\left (2 \, b^{2} c^{3} d - b^{3} c^{2} e\right )}}{c^{4}}\right )} x - \frac{15 \,{\left (2 \, b^{3} c^{2} d - b^{4} c e\right )}}{c^{4}}\right )} - \frac{3 \,{\left (2 \, b^{4} c d - b^{5} e\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(e*x + d),x, algorithm="giac")
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