Optimal. Leaf size=183 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^4 (a+b x)}-\frac{b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (2 b d-3 a e)}{e^3 (a+b x)}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x)} \]
[Out]
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Rubi [A] time = 0.258136, antiderivative size = 183, 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 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^4 (a+b x) (d+e x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x)}{e^4 (a+b x)}-\frac{b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (2 b d-3 a e)}{e^3 (a+b x)}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 21.8991, size = 143, normalized size = 0.78 \[ \frac{b \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 e^{2}} + \frac{3 b \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e^{3}} + \frac{3 b \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (d + e x \right )}}{e^{4} \left (a + b x\right )} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{e \left (d + e x\right )} \]
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)**2,x)
[Out]
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Mathematica [A] time = 0.169148, size = 132, normalized size = 0.72 \[ \frac{\sqrt{(a+b x)^2} \left (-2 a^3 e^3+6 a^2 b d e^2+6 a b^2 e \left (-d^2+d e x+e^2 x^2\right )+6 b (d+e x) (b d-a e)^2 \log (d+e x)+b^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )}{2 e^4 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(d + e*x)^2,x]
[Out]
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Maple [A] time = 0.022, size = 216, normalized size = 1.2 \[{\frac{{x}^{3}{b}^{3}{e}^{3}+6\,\ln \left ( ex+d \right ) x{a}^{2}b{e}^{3}-12\,\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{2}+6\,\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}e+6\,{x}^{2}a{b}^{2}{e}^{3}-3\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ){a}^{2}bd{e}^{2}-12\,\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}e+6\,\ln \left ( ex+d \right ){b}^{3}{d}^{3}+6\,xa{b}^{2}d{e}^{2}-4\,x{b}^{3}{d}^{2}e-2\,{a}^{3}{e}^{3}+6\,{a}^{2}bd{e}^{2}-6\,a{b}^{2}{d}^{2}e+2\,{b}^{3}{d}^{3}}{2\, \left ( bx+a \right ) ^{3}{e}^{4} \left ( ex+d \right ) } \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
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)^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)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210004, size = 232, normalized size = 1.27 \[ \frac{b^{3} e^{3} x^{3} + 2 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 6 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} - 3 \,{\left (b^{3} d e^{2} - 2 \, a b^{2} e^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} d^{2} e - 3 \, a b^{2} d e^{2}\right )} x + 6 \,{\left (b^{3} d^{3} - 2 \, a b^{2} d^{2} e + a^{2} b d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (e^{5} x + d e^{4}\right )}} \]
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)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{2}}\, dx \]
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)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.212968, size = 236, normalized size = 1.29 \[ 3 \,{\left (b^{3} d^{2}{\rm sign}\left (b x + a\right ) - 2 \, a b^{2} d e{\rm sign}\left (b x + a\right ) + a^{2} b e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{3} x^{2} e^{2}{\rm sign}\left (b x + a\right ) - 4 \, b^{3} d x e{\rm sign}\left (b x + a\right ) + 6 \, a b^{2} x e^{2}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )} + \frac{{\left (b^{3} d^{3}{\rm sign}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b d e^{2}{\rm sign}\left (b x + a\right ) - a^{3} e^{3}{\rm sign}\left (b x + a\right )\right )} e^{\left (-4\right )}}{x e + d} \]
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)^2,x, algorithm="giac")
[Out]