3.170 \(\int \frac{A+B x}{x^4 (a+b x)^3} \, dx\)

Optimal. Leaf size=140 \[ -\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}+\frac{2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}-\frac{b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 (A b-a B)}{2 a^4 (a+b x)^2}+\frac{3 A b-a B}{2 a^4 x^2}-\frac{A}{3 a^3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.287136, antiderivative size = 140, 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 \[ -\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}+\frac{2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}-\frac{b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac{3 b (2 A b-a B)}{a^5 x}-\frac{b^2 (A b-a B)}{2 a^4 (a+b x)^2}+\frac{3 A b-a B}{2 a^4 x^2}-\frac{A}{3 a^3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^4*(a + b*x)^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 53.5509, size = 134, normalized size = 0.96 \[ - \frac{A}{3 a^{3} x^{3}} - \frac{b^{2} \left (A b - B a\right )}{2 a^{4} \left (a + b x\right )^{2}} + \frac{3 A b - B a}{2 a^{4} x^{2}} - \frac{b^{2} \left (4 A b - 3 B a\right )}{a^{5} \left (a + b x\right )} - \frac{3 b \left (2 A b - B a\right )}{a^{5} x} - \frac{2 b^{2} \left (5 A b - 3 B a\right ) \log{\left (x \right )}}{a^{6}} + \frac{2 b^{2} \left (5 A b - 3 B a\right ) \log{\left (a + b x \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**4/(b*x+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.238561, size = 129, normalized size = 0.92 \[ \frac{\frac{a \left (a^4 (-(2 A+3 B x))+a^3 b x (5 A+12 B x)+2 a^2 b^2 x^2 (27 B x-10 A)+18 a b^3 x^3 (2 B x-5 A)-60 A b^4 x^4\right )}{x^3 (a+b x)^2}+12 b^2 \log (x) (3 a B-5 A b)+12 b^2 (5 A b-3 a B) \log (a+b x)}{6 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^4*(a + b*x)^3),x]

[Out]

((a*(-60*A*b^4*x^4 + 18*a*b^3*x^3*(-5*A + 2*B*x) - a^4*(2*A + 3*B*x) + a^3*b*x*(
5*A + 12*B*x) + 2*a^2*b^2*x^2*(-10*A + 27*B*x)))/(x^3*(a + b*x)^2) + 12*b^2*(-5*
A*b + 3*a*B)*Log[x] + 12*b^2*(5*A*b - 3*a*B)*Log[a + b*x])/(6*a^6)

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Maple [A]  time = 0.017, size = 168, normalized size = 1.2 \[ -{\frac{A}{3\,{a}^{3}{x}^{3}}}+{\frac{3\,Ab}{2\,{a}^{4}{x}^{2}}}-{\frac{B}{2\,{a}^{3}{x}^{2}}}-6\,{\frac{A{b}^{2}}{{a}^{5}x}}+3\,{\frac{Bb}{{a}^{4}x}}-10\,{\frac{A\ln \left ( x \right ){b}^{3}}{{a}^{6}}}+6\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{5}}}-4\,{\frac{A{b}^{3}}{{a}^{5} \left ( bx+a \right ) }}+3\,{\frac{B{b}^{2}}{{a}^{4} \left ( bx+a \right ) }}-{\frac{A{b}^{3}}{2\,{a}^{4} \left ( bx+a \right ) ^{2}}}+{\frac{B{b}^{2}}{2\,{a}^{3} \left ( bx+a \right ) ^{2}}}+10\,{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{6}}}-6\,{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^4/(b*x+a)^3,x)

[Out]

-1/3*A/a^3/x^3+3/2/a^4/x^2*A*b-1/2/a^3/x^2*B-6/a^5*b^2/x*A+3/a^4*b/x*B-10*b^3/a^
6*ln(x)*A+6*b^2/a^5*ln(x)*B-4/a^5*b^3/(b*x+a)*A+3/a^4*b^2/(b*x+a)*B-1/2/a^4*b^3/
(b*x+a)^2*A+1/2/a^3*b^2/(b*x+a)^2*B+10*b^3/a^6*ln(b*x+a)*A-6*b^2/a^5*ln(b*x+a)*B

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Maxima [A]  time = 1.36404, size = 221, normalized size = 1.58 \[ -\frac{2 \, A a^{4} - 12 \,{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} +{\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x}{6 \,{\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} - \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{6}} + \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*x^4),x, algorithm="maxima")

[Out]

-1/6*(2*A*a^4 - 12*(3*B*a*b^3 - 5*A*b^4)*x^4 - 18*(3*B*a^2*b^2 - 5*A*a*b^3)*x^3
- 4*(3*B*a^3*b - 5*A*a^2*b^2)*x^2 + (3*B*a^4 - 5*A*a^3*b)*x)/(a^5*b^2*x^5 + 2*a^
6*b*x^4 + a^7*x^3) - 2*(3*B*a*b^2 - 5*A*b^3)*log(b*x + a)/a^6 + 2*(3*B*a*b^2 - 5
*A*b^3)*log(x)/a^6

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Fricas [A]  time = 0.215402, size = 354, normalized size = 2.53 \[ -\frac{2 \, A a^{5} - 12 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x + 12 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 12 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*x^4),x, algorithm="fricas")

[Out]

-1/6*(2*A*a^5 - 12*(3*B*a^2*b^3 - 5*A*a*b^4)*x^4 - 18*(3*B*a^3*b^2 - 5*A*a^2*b^3
)*x^3 - 4*(3*B*a^4*b - 5*A*a^3*b^2)*x^2 + (3*B*a^5 - 5*A*a^4*b)*x + 12*((3*B*a*b
^4 - 5*A*b^5)*x^5 + 2*(3*B*a^2*b^3 - 5*A*a*b^4)*x^4 + (3*B*a^3*b^2 - 5*A*a^2*b^3
)*x^3)*log(b*x + a) - 12*((3*B*a*b^4 - 5*A*b^5)*x^5 + 2*(3*B*a^2*b^3 - 5*A*a*b^4
)*x^4 + (3*B*a^3*b^2 - 5*A*a^2*b^3)*x^3)*log(x))/(a^6*b^2*x^5 + 2*a^7*b*x^4 + a^
8*x^3)

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Sympy [A]  time = 6.19622, size = 262, normalized size = 1.87 \[ \frac{- 2 A a^{4} + x^{4} \left (- 60 A b^{4} + 36 B a b^{3}\right ) + x^{3} \left (- 90 A a b^{3} + 54 B a^{2} b^{2}\right ) + x^{2} \left (- 20 A a^{2} b^{2} + 12 B a^{3} b\right ) + x \left (5 A a^{3} b - 3 B a^{4}\right )}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac{2 b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (x + \frac{- 10 A a b^{3} + 6 B a^{2} b^{2} - 2 a b^{2} \left (- 5 A b + 3 B a\right )}{- 20 A b^{4} + 12 B a b^{3}} \right )}}{a^{6}} - \frac{2 b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (x + \frac{- 10 A a b^{3} + 6 B a^{2} b^{2} + 2 a b^{2} \left (- 5 A b + 3 B a\right )}{- 20 A b^{4} + 12 B a b^{3}} \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**4/(b*x+a)**3,x)

[Out]

(-2*A*a**4 + x**4*(-60*A*b**4 + 36*B*a*b**3) + x**3*(-90*A*a*b**3 + 54*B*a**2*b*
*2) + x**2*(-20*A*a**2*b**2 + 12*B*a**3*b) + x*(5*A*a**3*b - 3*B*a**4))/(6*a**7*
x**3 + 12*a**6*b*x**4 + 6*a**5*b**2*x**5) + 2*b**2*(-5*A*b + 3*B*a)*log(x + (-10
*A*a*b**3 + 6*B*a**2*b**2 - 2*a*b**2*(-5*A*b + 3*B*a))/(-20*A*b**4 + 12*B*a*b**3
))/a**6 - 2*b**2*(-5*A*b + 3*B*a)*log(x + (-10*A*a*b**3 + 6*B*a**2*b**2 + 2*a*b*
*2*(-5*A*b + 3*B*a))/(-20*A*b**4 + 12*B*a*b**3))/a**6

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GIAC/XCAS [A]  time = 0.361478, size = 213, normalized size = 1.52 \[ \frac{2 \,{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )}{\rm ln}\left ({\left | x \right |}\right )}{a^{6}} - \frac{2 \,{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac{2 \, A a^{5} - 12 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \,{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \,{\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} +{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x}{6 \,{\left (b x + a\right )}^{2} a^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^3*x^4),x, algorithm="giac")

[Out]

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