3.712 \(\int \frac{A+B x}{x^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=211 \[ \frac{(a+b x) (A b-a B)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 \log (x) (a+b x) (A b-a B)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (a+b x) (A b-a B) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (a+b x) (A b-a B)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-(A*(a + b*x))/(3*a*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*(a + b*x))
/(2*a^2*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(A*b - a*B)*(a + b*x))/(a^3*x*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) - (b^2*(A*b - a*B)*(a + b*x)*Log[x])/(a^4*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) + (b^2*(A*b - a*B)*(a + b*x)*Log[a + b*x])/(a^4*Sqrt[a^2 +
2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.287408, antiderivative size = 211, 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 \[ \frac{(a+b x) (A b-a B)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 \log (x) (a+b x) (A b-a B)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (a+b x) (A b-a B) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b (a+b x) (A b-a B)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-(A*(a + b*x))/(3*a*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*(a + b*x))
/(2*a^2*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*(A*b - a*B)*(a + b*x))/(a^3*x*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) - (b^2*(A*b - a*B)*(a + b*x)*Log[x])/(a^4*Sqrt[a^2
+ 2*a*b*x + b^2*x^2]) + (b^2*(A*b - a*B)*(a + b*x)*Log[a + b*x])/(a^4*Sqrt[a^2 +
2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 39.544, size = 202, normalized size = 0.96 \[ - \frac{A \left (2 a + 2 b x\right )}{6 a x^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right )}{4 a^{2} x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{b^{2} \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{4} \left (a + b x\right )} + \frac{b^{2} \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{4} \left (a + b x\right )} - \frac{b \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**4/((b*x+a)**2)**(1/2),x)

[Out]

-A*(2*a + 2*b*x)/(6*a*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + (2*a + 2*b*x)*(A*
b - B*a)/(4*a**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - b**2*(A*b - B*a)*sqrt(
a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a**4*(a + b*x)) + b**2*(A*b - B*a)*sqrt(a**2
+ 2*a*b*x + b**2*x**2)*log(a + b*x)/(a**4*(a + b*x)) - b*(A*b - B*a)*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/(a**4*x)

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Mathematica [A]  time = 0.0969117, size = 102, normalized size = 0.48 \[ -\frac{(a+b x) \left (a \left (a^2 (2 A+3 B x)-3 a b x (A+2 B x)+6 A b^2 x^2\right )+6 b^2 x^3 \log (x) (A b-a B)+6 b^2 x^3 (a B-A b) \log (a+b x)\right )}{6 a^4 x^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-((a + b*x)*(a*(6*A*b^2*x^2 - 3*a*b*x*(A + 2*B*x) + a^2*(2*A + 3*B*x)) + 6*b^2*(
A*b - a*B)*x^3*Log[x] + 6*b^2*(-(A*b) + a*B)*x^3*Log[a + b*x]))/(6*a^4*x^3*Sqrt[
(a + b*x)^2])

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Maple [A]  time = 0.02, size = 119, normalized size = 0.6 \[ -{\frac{ \left ( bx+a \right ) \left ( 6\,A\ln \left ( x \right ){x}^{3}{b}^{3}-6\,A\ln \left ( bx+a \right ){x}^{3}{b}^{3}-6\,B\ln \left ( x \right ){x}^{3}a{b}^{2}+6\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{2}+6\,A{x}^{2}a{b}^{2}-6\,B{x}^{2}{a}^{2}b-3\,A{a}^{2}bx+3\,{a}^{3}Bx+2\,A{a}^{3} \right ) }{6\,{x}^{3}{a}^{4}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^4/((b*x+a)^2)^(1/2),x)

[Out]

-1/6*(b*x+a)*(6*A*ln(x)*x^3*b^3-6*A*ln(b*x+a)*x^3*b^3-6*B*ln(x)*x^3*a*b^2+6*B*ln
(b*x+a)*x^3*a*b^2+6*A*x^2*a*b^2-6*B*x^2*a^2*b-3*A*a^2*b*x+3*a^3*B*x+2*A*a^3)/((b
*x+a)^2)^(1/2)/x^3/a^4

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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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30362, size = 127, normalized size = 0.6 \[ -\frac{6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (b x + a\right ) - 6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (x\right ) + 2 \, A a^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*x^4),x, algorithm="fricas")

[Out]

-1/6*(6*(B*a*b^2 - A*b^3)*x^3*log(b*x + a) - 6*(B*a*b^2 - A*b^3)*x^3*log(x) + 2*
A*a^3 - 6*(B*a^2*b - A*a*b^2)*x^2 + 3*(B*a^3 - A*a^2*b)*x)/(a^4*x^3)

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Sympy [A]  time = 2.711, size = 165, normalized size = 0.78 \[ \frac{- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 6 B a b\right ) + x \left (3 A a b - 3 B a^{2}\right )}{6 a^{3} x^{3}} + \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} - a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} - \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} + a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**4/((b*x+a)**2)**(1/2),x)

[Out]

(-2*A*a**2 + x**2*(-6*A*b**2 + 6*B*a*b) + x*(3*A*a*b - 3*B*a**2))/(6*a**3*x**3)
+ b**2*(-A*b + B*a)*log(x + (-A*a*b**3 + B*a**2*b**2 - a*b**2*(-A*b + B*a))/(-2*
A*b**4 + 2*B*a*b**3))/a**4 - b**2*(-A*b + B*a)*log(x + (-A*a*b**3 + B*a**2*b**2
+ a*b**2*(-A*b + B*a))/(-2*A*b**4 + 2*B*a*b**3))/a**4

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GIAC/XCAS [A]  time = 0.271049, size = 207, normalized size = 0.98 \[ \frac{{\left (B a b^{2}{\rm sign}\left (b x + a\right ) - A b^{3}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (B a b^{3}{\rm sign}\left (b x + a\right ) - A b^{4}{\rm sign}\left (b x + a\right )\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, A a^{3}{\rm sign}\left (b x + a\right ) - 6 \,{\left (B a^{2} b{\rm sign}\left (b x + a\right ) - A a b^{2}{\rm sign}\left (b x + a\right )\right )} x^{2} + 3 \,{\left (B a^{3}{\rm sign}\left (b x + a\right ) - A a^{2} b{\rm sign}\left (b x + a\right )\right )} x}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*x^4),x, algorithm="giac")

[Out]

(B*a*b^2*sign(b*x + a) - A*b^3*sign(b*x + a))*ln(abs(x))/a^4 - (B*a*b^3*sign(b*x
+ a) - A*b^4*sign(b*x + a))*ln(abs(b*x + a))/(a^4*b) - 1/6*(2*A*a^3*sign(b*x +
a) - 6*(B*a^2*b*sign(b*x + a) - A*a*b^2*sign(b*x + a))*x^2 + 3*(B*a^3*sign(b*x +
a) - A*a^2*b*sign(b*x + a))*x)/(a^4*x^3)