Optimal. Leaf size=114 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^5 (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]
[Out]
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Rubi [A] time = 0.15752, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^5 (a+b x)}-\frac{a A \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^7,x]
[Out]
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Rubi in Sympy [A] time = 18.6441, size = 112, normalized size = 0.98 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 a x^{6}} - \frac{\left (\frac{A b}{30} - \frac{B a}{20}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x^{5} \left (a + b x\right )} + \frac{\left (2 A b - 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**7,x)
[Out]
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Mathematica [A] time = 0.0262594, size = 49, normalized size = 0.43 \[ -\frac{\sqrt{(a+b x)^2} (2 a (5 A+6 B x)+3 b x (4 A+5 B x))}{60 x^6 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^7,x]
[Out]
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Maple [A] time = 0.007, size = 44, normalized size = 0.4 \[ -{\frac{15\,Bb{x}^{2}+12\,Abx+12\,aBx+10\,aA}{60\,{x}^{6} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*((b*x+a)^2)^(1/2)/x^7,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^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.280447, size = 36, normalized size = 0.32 \[ -\frac{15 \, B b x^{2} + 10 \, A a + 12 \,{\left (B a + A b\right )} x}{60 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/x^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.65453, size = 31, normalized size = 0.27 \[ - \frac{10 A a + 15 B b x^{2} + x \left (12 A b + 12 B a\right )}{60 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**7,x)
[Out]
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GIAC/XCAS [A] time = 0.272161, size = 104, normalized size = 0.91 \[ \frac{{\left (3 \, B a b^{5} - 2 \, A b^{6}\right )}{\rm sign}\left (b x + a\right )}{60 \, a^{5}} - \frac{15 \, B b x^{2}{\rm sign}\left (b x + a\right ) + 12 \, B a x{\rm sign}\left (b x + a\right ) + 12 \, A b x{\rm sign}\left (b x + a\right ) + 10 \, A a{\rm sign}\left (b x + a\right )}{60 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/x^7,x, algorithm="giac")
[Out]