Optimal. Leaf size=145 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
[Out]
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Rubi [A] time = 0.116313, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^4,x]
[Out]
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Rubi in Sympy [A] time = 18.3613, size = 121, normalized size = 0.83 \[ - \frac{a b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x \left (a + b x\right )} + \frac{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} - \frac{b \left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 x^{2}} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} \]
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)/x**4,x)
[Out]
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Mathematica [A] time = 0.0329192, size = 57, normalized size = 0.39 \[ -\frac{\sqrt{(a+b x)^2} \left (a \left (2 a^2+9 a b x+18 b^2 x^2\right )-6 b^3 x^3 \log (x)\right )}{6 x^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^4,x]
[Out]
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Maple [A] time = 0.01, size = 54, normalized size = 0.4 \[{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-18\,a{b}^{2}{x}^{2}-9\,{a}^{2}bx-2\,{a}^{3}}{6\, \left ( bx+a \right ) ^{3}{x}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^(3/2)/x^4,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)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235117, size = 50, normalized size = 0.34 \[ \frac{6 \, b^{3} x^{3} \log \left (x\right ) - 18 \, a b^{2} x^{2} - 9 \, a^{2} b x - 2 \, a^{3}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/x^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**(3/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.209596, size = 80, normalized size = 0.55 \[ b^{3}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{18 \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 9 \, a^{2} b x{\rm sign}\left (b x + a\right ) + 2 \, a^{3}{\rm sign}\left (b x + a\right )}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/x^4,x, algorithm="giac")
[Out]