Optimal. Leaf size=214 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac{b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}+\frac{7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \]
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Rubi [A] time = 0.408209, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{b^2 (b+2 c x) \sqrt{b x+c x^2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{512 c^4}+\frac{(b+2 c x) \left (b x+c x^2\right )^{3/2} \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right )}{192 c^3}+\frac{b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{9/2}}+\frac{7 e \left (b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac{e \left (b x+c x^2\right )^{5/2} (d+e x)}{6 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 36.2419, size = 207, normalized size = 0.97 \[ \frac{b^{4} \left (7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{512 c^{\frac{9}{2}}} - \frac{b^{2} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right )}{512 c^{4}} + \frac{e \left (d + e x\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{6 c} - \frac{7 e \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}}}{60 c^{2}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (7 b^{2} e^{2} - 24 b c d e + 24 c^{2} d^{2}\right )}{192 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.382251, size = 226, normalized size = 1.06 \[ \frac{\sqrt{x (b+c x)} \left (\frac{15 b^4 \left (7 b^2 e^2-24 b c d e+24 c^2 d^2\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-105 b^5 e^2+10 b^4 c e (36 d+7 e x)-8 b^3 c^2 \left (45 d^2+30 d e x+7 e^2 x^2\right )+48 b^2 c^3 x \left (5 d^2+4 d e x+e^2 x^2\right )+64 b c^4 x^2 \left (45 d^2+66 d e x+26 e^2 x^2\right )+128 c^5 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )\right )}{7680 c^{9/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.013, size = 420, normalized size = 2. \[{\frac{{d}^{2}x}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{2}b}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}{d}^{2}x}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{d}^{2}{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{d}^{2}{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{e}^{2}x}{6\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{e}^{2}b}{60\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{7\,{b}^{2}{e}^{2}x}{96\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{3}{e}^{2}}{192\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{e}^{2}{b}^{4}x}{256\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{7\,{e}^{2}{b}^{5}}{512\,{c}^{4}}\sqrt{c{x}^{2}+bx}}+{\frac{7\,{e}^{2}{b}^{6}}{1024}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}}+{\frac{2\,de}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{bdex}{4\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}de}{8\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,de{b}^{3}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,de{b}^{4}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{3\,de{b}^{5}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(c*x^2+b*x)^(3/2),x)
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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)^2,x, algorithm="maxima")
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Fricas [A] time = 0.231582, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, c^{5} e^{2} x^{5} - 360 \, b^{3} c^{2} d^{2} + 360 \, b^{4} c d e - 105 \, b^{5} e^{2} + 128 \,{\left (24 \, c^{5} d e + 13 \, b c^{4} e^{2}\right )} x^{4} + 48 \,{\left (40 \, c^{5} d^{2} + 88 \, b c^{4} d e + b^{2} c^{3} e^{2}\right )} x^{3} + 8 \,{\left (360 \, b c^{4} d^{2} + 24 \, b^{2} c^{3} d e - 7 \, b^{3} c^{2} e^{2}\right )} x^{2} + 10 \,{\left (24 \, b^{2} c^{3} d^{2} - 24 \, b^{3} c^{2} d e + 7 \, b^{4} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} + 15 \,{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{15360 \, c^{\frac{9}{2}}}, \frac{{\left (1280 \, c^{5} e^{2} x^{5} - 360 \, b^{3} c^{2} d^{2} + 360 \, b^{4} c d e - 105 \, b^{5} e^{2} + 128 \,{\left (24 \, c^{5} d e + 13 \, b c^{4} e^{2}\right )} x^{4} + 48 \,{\left (40 \, c^{5} d^{2} + 88 \, b c^{4} d e + b^{2} c^{3} e^{2}\right )} x^{3} + 8 \,{\left (360 \, b c^{4} d^{2} + 24 \, b^{2} c^{3} d e - 7 \, b^{3} c^{2} e^{2}\right )} x^{2} + 10 \,{\left (24 \, b^{2} c^{3} d^{2} - 24 \, b^{3} c^{2} d e + 7 \, b^{4} c e^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 15 \,{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{7680 \, \sqrt{-c} c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(e*x + d)^2,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(c*x**2+b*x)**(3/2),x)
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GIAC/XCAS [A] time = 0.221905, size = 354, normalized size = 1.65 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x e^{2} + \frac{24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac{3 \,{\left (40 \, c^{6} d^{2} + 88 \, b c^{5} d e + b^{2} c^{4} e^{2}\right )}}{c^{5}}\right )} x + \frac{360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e - 7 \, b^{3} c^{3} e^{2}}{c^{5}}\right )} x + \frac{5 \,{\left (24 \, b^{2} c^{4} d^{2} - 24 \, b^{3} c^{3} d e + 7 \, b^{4} c^{2} e^{2}\right )}}{c^{5}}\right )} x - \frac{15 \,{\left (24 \, b^{3} c^{3} d^{2} - 24 \, b^{4} c^{2} d e + 7 \, b^{5} c e^{2}\right )}}{c^{5}}\right )} - \frac{{\left (24 \, b^{4} c^{2} d^{2} - 24 \, b^{5} c d e + 7 \, b^{6} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(e*x + d)^2,x, algorithm="giac")
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