Optimal. Leaf size=48 \[ \frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (d+e x)^4 (b d-a e)} \]
[Out]
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Rubi [A] time = 0.0701883, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (d+e x)^4 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 10.2688, size = 44, normalized size = 0.92 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 \left (d + e x\right )^{4} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**5,x)
[Out]
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Mathematica [B] time = 0.0833553, size = 109, normalized size = 2.27 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )}{4 e^4 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^5,x]
[Out]
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Maple [B] time = 0.011, size = 128, normalized size = 2.7 \[ -{\frac{4\,{x}^{3}{b}^{3}{e}^{3}+6\,{x}^{2}a{b}^{2}{e}^{3}+6\,{x}^{2}{b}^{3}d{e}^{2}+4\,x{a}^{2}b{e}^{3}+4\,xa{b}^{2}d{e}^{2}+4\,x{b}^{3}{d}^{2}e+{a}^{3}{e}^{3}+{a}^{2}bd{e}^{2}+a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{4\, \left ( ex+d \right ) ^{4}{e}^{4} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^(3/2)/(e*x+d)^5,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)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206709, size = 193, normalized size = 4.02 \[ -\frac{4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \]
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)^5,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.214259, size = 224, normalized size = 4.67 \[ -\frac{{\left (4 \, b^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 6 \, b^{3} d x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 4 \, b^{3} d^{2} x e{\rm sign}\left (b x + a\right ) + b^{3} d^{3}{\rm sign}\left (b x + a\right ) + 6 \, a b^{2} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 4 \, a b^{2} d x e^{2}{\rm sign}\left (b x + a\right ) + a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 4 \, a^{2} b x e^{3}{\rm sign}\left (b x + a\right ) + a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) + a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{4 \,{\left (x e + d\right )}^{4}} \]
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)^5,x, algorithm="giac")
[Out]