3.719 \(\int \frac{A+B x}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=140 \[ \frac{A b-a B}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

A/(a^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(2*a*b*(a + b*x)*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (A*(a + b*x)*Log[x])/(a^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(A*(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 = 0.241587, antiderivative size = 140, 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 \[ \frac{A b-a B}{2 a b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

A/(a^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(2*a*b*(a + b*x)*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (A*(a + b*x)*Log[x])/(a^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(A*(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 = 37.8816, size = 136, normalized size = 0.97 \[ \frac{A}{a^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{A \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{3} \left (a + b x\right )} - \frac{A \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{3} \left (a + b x\right )} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right )}{4 a b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0743554, size = 80, normalized size = 0.57 \[ \frac{a \left (a^2 (-B)+3 a A b+2 A b^2 x\right )+2 A b \log (x) (a+b x)^2-2 A b (a+b x)^2 \log (a+b x)}{2 a^3 b (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 117, normalized size = 0.8 \[{\frac{ \left ( 2\,A\ln \left ( x \right ){x}^{2}{b}^{3}-2\,A\ln \left ( bx+a \right ){x}^{2}{b}^{3}+4\,A\ln \left ( x \right ) xa{b}^{2}-4\,A\ln \left ( bx+a \right ) xa{b}^{2}+2\,A\ln \left ( x \right ){a}^{2}b-2\,A\ln \left ( bx+a \right ){a}^{2}b+2\,Axa{b}^{2}+3\,A{a}^{2}b-B{a}^{3} \right ) \left ( bx+a \right ) }{2\,{a}^{3}b} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*A*ln(x)*x^2*b^3-2*A*ln(b*x+a)*x^2*b^3+4*A*ln(x)*x*a*b^2-4*A*ln(b*x+a)*x*a
*b^2+2*A*ln(x)*a^2*b-2*A*ln(b*x+a)*a^2*b+2*A*x*a*b^2+3*A*a^2*b-B*a^3)*(b*x+a)/b/
a^3/((b*x+a)^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295363, size = 147, normalized size = 1.05 \[ \frac{2 \, A a b^{2} x - B a^{3} + 3 \, A a^{2} b - 2 \,{\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (b x + a\right ) + 2 \,{\left (A b^{3} x^{2} + 2 \, A a b^{2} x + A a^{2} b\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*A*a*b^2*x - B*a^3 + 3*A*a^2*b - 2*(A*b^3*x^2 + 2*A*a*b^2*x + A*a^2*b)*log
(b*x + a) + 2*(A*b^3*x^2 + 2*A*a*b^2*x + A*a^2*b)*log(x))/(a^3*b^3*x^2 + 2*a^4*b
^2*x + a^5*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/(x*((a + b*x)**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.596954, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x