3.711 \(\int \frac{A+B x}{x^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=162 \[ \frac{(a+b x) (A b-a B)}{a^2 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{2 a x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b \log (x) (a+b x) (A b-a B)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (a+b x) (A b-a B) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-(A*(a + b*x))/(2*a*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*(a + b*x))
/(a^2*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b*(A*b - a*B)*(a + b*x)*Log[x])/(a^3*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(A*b - a*B)*(a + b*x)*Log[a + b*x])/(a^3*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.240174, antiderivative size = 162, 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 \[ \frac{(a+b x) (A b-a B)}{a^2 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{2 a x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b \log (x) (a+b x) (A b-a B)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (a+b x) (A b-a B) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-(A*(a + b*x))/(2*a*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*(a + b*x))
/(a^2*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b*(A*b - a*B)*(a + b*x)*Log[x])/(a^3*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(A*b - a*B)*(a + b*x)*Log[a + b*x])/(a^3*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 30.5472, size = 153, normalized size = 0.94 \[ - \frac{A \left (2 a + 2 b x\right )}{4 a x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{b \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{3} \left (a + b x\right )} - \frac{b \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{3} \left (a + b x\right )} + \frac{\left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**3/((b*x+a)**2)**(1/2),x)

[Out]

-A*(2*a + 2*b*x)/(4*a*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + b*(A*b - B*a)*sqr
t(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a**3*(a + b*x)) - b*(A*b - B*a)*sqrt(a**2
+ 2*a*b*x + b**2*x**2)*log(a + b*x)/(a**3*(a + b*x)) + (A*b - B*a)*sqrt(a**2 + 2
*a*b*x + b**2*x**2)/(a**3*x)

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Mathematica [A]  time = 0.0958227, size = 79, normalized size = 0.49 \[ -\frac{(a+b x) \left (2 b x^2 \log (x) (a B-A b)+2 b x^2 (A b-a B) \log (a+b x)+a (a A+2 a B x-2 A b x)\right )}{2 a^3 x^2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-((a + b*x)*(a*(a*A - 2*A*b*x + 2*a*B*x) + 2*b*(-(A*b) + a*B)*x^2*Log[x] + 2*b*(
A*b - a*B)*x^2*Log[a + b*x]))/(2*a^3*x^2*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.02, size = 93, normalized size = 0.6 \[{\frac{ \left ( bx+a \right ) \left ( 2\,A\ln \left ( x \right ){x}^{2}{b}^{2}-2\,A\ln \left ( bx+a \right ){x}^{2}{b}^{2}-2\,B\ln \left ( x \right ){x}^{2}ab+2\,B\ln \left ( bx+a \right ){x}^{2}ab+2\,aAbx-2\,{a}^{2}Bx-A{a}^{2} \right ) }{2\,{x}^{2}{a}^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^3/((b*x+a)^2)^(1/2),x)

[Out]

1/2*(b*x+a)*(2*A*ln(x)*x^2*b^2-2*A*ln(b*x+a)*x^2*b^2-2*B*ln(x)*x^2*a*b+2*B*ln(b*
x+a)*x^2*a*b+2*a*A*b*x-2*a^2*B*x-A*a^2)/((b*x+a)^2)^(1/2)/x^2/a^3

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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*x + A)/(sqrt((b*x + a)^2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289896, size = 93, normalized size = 0.57 \[ \frac{2 \,{\left (B a b - A b^{2}\right )} x^{2} \log \left (b x + a\right ) - 2 \,{\left (B a b - A b^{2}\right )} x^{2} \log \left (x\right ) - A a^{2} - 2 \,{\left (B a^{2} - A a b\right )} x}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*x^3),x, algorithm="fricas")

[Out]

1/2*(2*(B*a*b - A*b^2)*x^2*log(b*x + a) - 2*(B*a*b - A*b^2)*x^2*log(x) - A*a^2 -
2*(B*a^2 - A*a*b)*x)/(a^3*x^2)

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Sympy [A]  time = 2.41901, size = 131, normalized size = 0.81 \[ - \frac{A a + x \left (- 2 A b + 2 B a\right )}{2 a^{2} x^{2}} - \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} + B a^{2} b - a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} + \frac{b \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{2} + B a^{2} b + a b \left (- A b + B a\right )}{- 2 A b^{3} + 2 B a b^{2}} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**3/((b*x+a)**2)**(1/2),x)

[Out]

-(A*a + x*(-2*A*b + 2*B*a))/(2*a**2*x**2) - b*(-A*b + B*a)*log(x + (-A*a*b**2 +
B*a**2*b - a*b*(-A*b + B*a))/(-2*A*b**3 + 2*B*a*b**2))/a**3 + b*(-A*b + B*a)*log
(x + (-A*a*b**2 + B*a**2*b + a*b*(-A*b + B*a))/(-2*A*b**3 + 2*B*a*b**2))/a**3

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GIAC/XCAS [A]  time = 0.271002, size = 158, normalized size = 0.98 \[ -\frac{{\left (B a b{\rm sign}\left (b x + a\right ) - A b^{2}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{{\left (B a b^{2}{\rm sign}\left (b x + a\right ) - A b^{3}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac{A a^{2}{\rm sign}\left (b x + a\right ) + 2 \,{\left (B a^{2}{\rm sign}\left (b x + a\right ) - A a b{\rm sign}\left (b x + a\right )\right )} x}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*x^3),x, algorithm="giac")

[Out]

-(B*a*b*sign(b*x + a) - A*b^2*sign(b*x + a))*ln(abs(x))/a^3 + (B*a*b^2*sign(b*x
+ a) - A*b^3*sign(b*x + a))*ln(abs(b*x + a))/(a^3*b) - 1/2*(A*a^2*sign(b*x + a)
+ 2*(B*a^2*sign(b*x + a) - A*a*b*sign(b*x + a))*x)/(a^3*x^2)