Optimal. Leaf size=172 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{2 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)^3}{4 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4} \]
[Out]
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Rubi [A] time = 0.325979, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{2 b^4}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4 (b d-a e)^2}{5 b^4}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)^3}{4 b^4}+\frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{7 b^4} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 25.5751, size = 163, normalized size = 0.95 \[ \frac{\left (d + e x\right )^{4} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{7 e} + \frac{\left (3 a + 3 b x\right ) \left (d + e x\right )^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{42 e^{2}} + \frac{\left (d + e x\right )^{4} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{35 e^{3}} + \frac{\left (d + e x\right )^{4} \left (a e - b d\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{140 e^{4} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.114952, size = 171, normalized size = 0.99 \[ \frac{x \sqrt{(a+b x)^2} \left (35 a^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+21 a^2 b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+7 a b^2 x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+b^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right )}{140 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 206, normalized size = 1.2 \[{\frac{x \left ( 20\,{b}^{3}{e}^{3}{x}^{6}+70\,{x}^{5}a{b}^{2}{e}^{3}+70\,{x}^{5}{b}^{3}d{e}^{2}+84\,{x}^{4}{a}^{2}b{e}^{3}+252\,{x}^{4}a{b}^{2}d{e}^{2}+84\,{x}^{4}{b}^{3}{d}^{2}e+35\,{x}^{3}{a}^{3}{e}^{3}+315\,{x}^{3}{a}^{2}bd{e}^{2}+315\,{x}^{3}a{b}^{2}{d}^{2}e+35\,{x}^{3}{b}^{3}{d}^{3}+140\,{a}^{3}d{e}^{2}{x}^{2}+420\,{a}^{2}b{d}^{2}e{x}^{2}+140\,a{b}^{2}{d}^{3}{x}^{2}+210\,x{a}^{3}{d}^{2}e+210\,x{a}^{2}b{d}^{3}+140\,{a}^{3}{d}^{3} \right ) }{140\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(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((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205934, size = 225, normalized size = 1.31 \[ \frac{1}{7} \, b^{3} e^{3} x^{7} + a^{3} d^{3} x + \frac{1}{2} \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{6} + \frac{3}{5} \,{\left (b^{3} d^{2} e + 3 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} d^{3} + 9 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} x^{4} +{\left (a b^{2} d^{3} + 3 \, a^{2} b d^{2} e + a^{3} d e^{2}\right )} x^{3} + \frac{3}{2} \,{\left (a^{2} b d^{3} + a^{3} d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212825, size = 378, normalized size = 2.2 \[ \frac{1}{7} \, b^{3} x^{7} e^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b^{3} d x^{6} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, b^{3} d^{2} x^{5} e{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, b^{3} d^{3} x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a b^{2} x^{6} e^{3}{\rm sign}\left (b x + a\right ) + \frac{9}{5} \, a b^{2} d x^{5} e^{2}{\rm sign}\left (b x + a\right ) + \frac{9}{4} \, a b^{2} d^{2} x^{4} e{\rm sign}\left (b x + a\right ) + a b^{2} d^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{5} \, a^{2} b x^{5} e^{3}{\rm sign}\left (b x + a\right ) + \frac{9}{4} \, a^{2} b d x^{4} e^{2}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b d^{2} x^{3} e{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b d^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, a^{3} x^{4} e^{3}{\rm sign}\left (b x + a\right ) + a^{3} d x^{3} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{3} d^{2} x^{2} e{\rm sign}\left (b x + a\right ) + a^{3} d^{3} x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^3,x, algorithm="giac")
[Out]