Optimal. Leaf size=75 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac{A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]
[Out]
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Rubi [A] time = 0.150538, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac{A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^4,x]
[Out]
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Rubi in Sympy [A] time = 26.6283, size = 73, normalized size = 0.97 \[ - \frac{A \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a^{2} x^{3}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0291988, size = 46, normalized size = 0.61 \[ -\frac{\sqrt{(a+b x)^2} (a (2 A+3 B x)+3 b x (A+2 B x))}{6 x^3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^4,x]
[Out]
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Maple [A] time = 0.007, size = 44, normalized size = 0.6 \[ -{\frac{6\,Bb{x}^{2}+3\,Abx+3\,aBx+2\,aA}{6\,{x}^{3} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*((b*x+a)^2)^(1/2)/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(sqrt((b*x + a)^2)*(B*x + A)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283428, size = 36, normalized size = 0.48 \[ -\frac{6 \, B b x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.81732, size = 31, normalized size = 0.41 \[ - \frac{2 A a + 6 B b x^{2} + x \left (3 A b + 3 B a\right )}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.268335, size = 104, normalized size = 1.39 \[ -\frac{{\left (3 \, B a b^{2} - A b^{3}\right )}{\rm sign}\left (b x + a\right )}{6 \, a^{2}} - \frac{6 \, B b x^{2}{\rm sign}\left (b x + a\right ) + 3 \, B a x{\rm sign}\left (b x + a\right ) + 3 \, A b x{\rm sign}\left (b x + a\right ) + 2 \, A a{\rm sign}\left (b x + a\right )}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/x^4,x, algorithm="giac")
[Out]