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

Optimal. Leaf size=69 \[ \frac{(a+b x) (A b-a B) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.068142, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{(a+b x) (A b-a B) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B \sqrt{a^2+2 a b x+b^2 x^2}}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.1521, size = 66, normalized size = 0.96 \[ \frac{B \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{b^{2}} + \frac{\left (a + b x\right ) \left (A b - B a\right ) \log{\left (a + b x \right )}}{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0335835, size = 40, normalized size = 0.58 \[ \frac{(a+b x) ((A b-a B) \log (a+b x)+b B x)}{b^2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 43, normalized size = 0.6 \[{\frac{ \left ( bx+a \right ) \left ( A\ln \left ( bx+a \right ) b-B\ln \left ( bx+a \right ) a+xBb \right ) }{{b}^{2}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.683419, size = 80, normalized size = 1.16 \[ A \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right ) - \frac{B a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} B}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.27871, size = 34, normalized size = 0.49 \[ \frac{B b x -{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.32167, size = 20, normalized size = 0.29 \[ \frac{B x}{b} - \frac{\left (- A b + B a\right ) \log{\left (a + b x \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.27152, size = 61, normalized size = 0.88 \[ \frac{B x{\rm sign}\left (b x + a\right )}{b} - \frac{{\left (B a{\rm sign}\left (b x + a\right ) - A b{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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