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

Optimal. Leaf size=110 \[ \frac{3 b \log (x) (2 A b-a B)}{a^5}-\frac{3 b (2 A b-a B) \log (a+b x)}{a^5}+\frac{3 A b-a B}{a^4 x}+\frac{b (3 A b-2 a B)}{a^4 (a+b x)}+\frac{b (A b-a B)}{2 a^3 (a+b x)^2}-\frac{A}{2 a^3 x^2} \]

[Out]

-A/(2*a^3*x^2) + (3*A*b - a*B)/(a^4*x) + (b*(A*b - a*B))/(2*a^3*(a + b*x)^2) + (
b*(3*A*b - 2*a*B))/(a^4*(a + b*x)) + (3*b*(2*A*b - a*B)*Log[x])/a^5 - (3*b*(2*A*
b - a*B)*Log[a + b*x])/a^5

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Rubi [A]  time = 0.215935, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{3 b \log (x) (2 A b-a B)}{a^5}-\frac{3 b (2 A b-a B) \log (a+b x)}{a^5}+\frac{3 A b-a B}{a^4 x}+\frac{b (3 A b-2 a B)}{a^4 (a+b x)}+\frac{b (A b-a B)}{2 a^3 (a+b x)^2}-\frac{A}{2 a^3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-A/(2*a^3*x^2) + (3*A*b - a*B)/(a^4*x) + (b*(A*b - a*B))/(2*a^3*(a + b*x)^2) + (
b*(3*A*b - 2*a*B))/(a^4*(a + b*x)) + (3*b*(2*A*b - a*B)*Log[x])/a^5 - (3*b*(2*A*
b - a*B)*Log[a + b*x])/a^5

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Rubi in Sympy [A]  time = 39.579, size = 104, normalized size = 0.95 \[ - \frac{A}{2 a^{3} x^{2}} + \frac{b \left (A b - B a\right )}{2 a^{3} \left (a + b x\right )^{2}} + \frac{b \left (3 A b - 2 B a\right )}{a^{4} \left (a + b x\right )} + \frac{3 A b - B a}{a^{4} x} + \frac{3 b \left (2 A b - B a\right ) \log{\left (x \right )}}{a^{5}} - \frac{3 b \left (2 A b - B a\right ) \log{\left (a + b x \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.175514, size = 102, normalized size = 0.93 \[ \frac{-\frac{a \left (a^3 (A+2 B x)+a^2 b x (9 B x-4 A)+6 a b^2 x^2 (B x-3 A)-12 A b^3 x^3\right )}{x^2 (a+b x)^2}+6 b \log (x) (2 A b-a B)+6 b (a B-2 A b) \log (a+b x)}{2 a^5} \]

Antiderivative was successfully verified.

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

[Out]

(-((a*(-12*A*b^3*x^3 + 6*a*b^2*x^2*(-3*A + B*x) + a^3*(A + 2*B*x) + a^2*b*x*(-4*
A + 9*B*x)))/(x^2*(a + b*x)^2)) + 6*b*(2*A*b - a*B)*Log[x] + 6*b*(-2*A*b + a*B)*
Log[a + b*x])/(2*a^5)

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Maple [A]  time = 0.017, size = 138, normalized size = 1.3 \[ -{\frac{A}{2\,{a}^{3}{x}^{2}}}+3\,{\frac{Ab}{{a}^{4}x}}-{\frac{B}{{a}^{3}x}}+6\,{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{5}}}-3\,{\frac{bB\ln \left ( x \right ) }{{a}^{4}}}-6\,{\frac{{b}^{2}\ln \left ( bx+a \right ) A}{{a}^{5}}}+3\,{\frac{b\ln \left ( bx+a \right ) B}{{a}^{4}}}+3\,{\frac{A{b}^{2}}{{a}^{4} \left ( bx+a \right ) }}-2\,{\frac{Bb}{{a}^{3} \left ( bx+a \right ) }}+{\frac{A{b}^{2}}{2\,{a}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{Bb}{2\,{a}^{2} \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^3/(b*x+a)^3,x)

[Out]

-1/2*A/a^3/x^2+3/a^4/x*A*b-1/a^3/x*B+6*b^2/a^5*ln(x)*A-3*b/a^4*ln(x)*B-6*b^2/a^5
*ln(b*x+a)*A+3*b/a^4*ln(b*x+a)*B+3/a^4*b^2/(b*x+a)*A-2/a^3*b/(b*x+a)*B+1/2/a^3*b
^2/(b*x+a)^2*A-1/2/a^2*b/(b*x+a)^2*B

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Maxima [A]  time = 1.35128, size = 177, normalized size = 1.61 \[ -\frac{A a^{3} + 6 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} x^{3} + 9 \,{\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{2} + 2 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} x}{2 \,{\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} + \frac{3 \,{\left (B a b - 2 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{5}} - \frac{3 \,{\left (B a b - 2 \, A b^{2}\right )} \log \left (x\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(A*a^3 + 6*(B*a*b^2 - 2*A*b^3)*x^3 + 9*(B*a^2*b - 2*A*a*b^2)*x^2 + 2*(B*a^3
- 2*A*a^2*b)*x)/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2) + 3*(B*a*b - 2*A*b^2)*log
(b*x + a)/a^5 - 3*(B*a*b - 2*A*b^2)*log(x)/a^5

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Fricas [A]  time = 0.209108, size = 304, normalized size = 2.76 \[ -\frac{A a^{4} + 6 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 9 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 2 \,{\left (B a^{4} - 2 \, A a^{3} b\right )} x - 6 \,{\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} +{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} +{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(A*a^4 + 6*(B*a^2*b^2 - 2*A*a*b^3)*x^3 + 9*(B*a^3*b - 2*A*a^2*b^2)*x^2 + 2*
(B*a^4 - 2*A*a^3*b)*x - 6*((B*a*b^3 - 2*A*b^4)*x^4 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x
^3 + (B*a^3*b - 2*A*a^2*b^2)*x^2)*log(b*x + a) + 6*((B*a*b^3 - 2*A*b^4)*x^4 + 2*
(B*a^2*b^2 - 2*A*a*b^3)*x^3 + (B*a^3*b - 2*A*a^2*b^2)*x^2)*log(x))/(a^5*b^2*x^4
+ 2*a^6*b*x^3 + a^7*x^2)

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Sympy [A]  time = 5.59785, size = 219, normalized size = 1.99 \[ - \frac{A a^{3} + x^{3} \left (- 12 A b^{3} + 6 B a b^{2}\right ) + x^{2} \left (- 18 A a b^{2} + 9 B a^{2} b\right ) + x \left (- 4 A a^{2} b + 2 B a^{3}\right )}{2 a^{6} x^{2} + 4 a^{5} b x^{3} + 2 a^{4} b^{2} x^{4}} - \frac{3 b \left (- 2 A b + B a\right ) \log{\left (x + \frac{- 6 A a b^{2} + 3 B a^{2} b - 3 a b \left (- 2 A b + B a\right )}{- 12 A b^{3} + 6 B a b^{2}} \right )}}{a^{5}} + \frac{3 b \left (- 2 A b + B a\right ) \log{\left (x + \frac{- 6 A a b^{2} + 3 B a^{2} b + 3 a b \left (- 2 A b + B a\right )}{- 12 A b^{3} + 6 B a b^{2}} \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*a**3 + x**3*(-12*A*b**3 + 6*B*a*b**2) + x**2*(-18*A*a*b**2 + 9*B*a**2*b) + x
*(-4*A*a**2*b + 2*B*a**3))/(2*a**6*x**2 + 4*a**5*b*x**3 + 2*a**4*b**2*x**4) - 3*
b*(-2*A*b + B*a)*log(x + (-6*A*a*b**2 + 3*B*a**2*b - 3*a*b*(-2*A*b + B*a))/(-12*
A*b**3 + 6*B*a*b**2))/a**5 + 3*b*(-2*A*b + B*a)*log(x + (-6*A*a*b**2 + 3*B*a**2*
b + 3*a*b*(-2*A*b + B*a))/(-12*A*b**3 + 6*B*a*b**2))/a**5

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GIAC/XCAS [A]  time = 0.262541, size = 167, normalized size = 1.52 \[ -\frac{3 \,{\left (B a b - 2 \, A b^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} + \frac{3 \,{\left (B a b^{2} - 2 \, A b^{3}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac{6 \, B a b^{2} x^{3} - 12 \, A b^{3} x^{3} + 9 \, B a^{2} b x^{2} - 18 \, A a b^{2} x^{2} + 2 \, B a^{3} x - 4 \, A a^{2} b x + A a^{3}}{2 \,{\left (b x^{2} + a x\right )}^{2} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(B*a*b - 2*A*b^2)*ln(abs(x))/a^5 + 3*(B*a*b^2 - 2*A*b^3)*ln(abs(b*x + a))/(a^
5*b) - 1/2*(6*B*a*b^2*x^3 - 12*A*b^3*x^3 + 9*B*a^2*b*x^2 - 18*A*a*b^2*x^2 + 2*B*
a^3*x - 4*A*a^2*b*x + A*a^3)/((b*x^2 + a*x)^2*a^4)