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

Optimal. Leaf size=113 \[ -\frac{b^2 (A b-a B)}{a^4 (a+b x)}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac{b^2 (4 A b-3 a B) \log (a+b x)}{a^5}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{A}{3 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0884852, antiderivative size = 113, 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, Rules used = {77} \[ -\frac{b^2 (A b-a B)}{a^4 (a+b x)}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}+\frac{b^2 (4 A b-3 a B) \log (a+b x)}{a^5}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{A}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^4*(a + b*x)^2),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{x^4 (a+b x)^2} \, dx &=\int \left (\frac{A}{a^2 x^4}+\frac{-2 A b+a B}{a^3 x^3}-\frac{b (-3 A b+2 a B)}{a^4 x^2}+\frac{b^2 (-4 A b+3 a B)}{a^5 x}-\frac{b^3 (-A b+a B)}{a^4 (a+b x)^2}-\frac{b^3 (-4 A b+3 a B)}{a^5 (a+b x)}\right ) \, dx\\ &=-\frac{A}{3 a^2 x^3}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{b^2 (A b-a B)}{a^4 (a+b x)}-\frac{b^2 (4 A b-3 a B) \log (x)}{a^5}+\frac{b^2 (4 A b-3 a B) \log (a+b x)}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.0848411, size = 106, normalized size = 0.94 \[ \frac{-\frac{3 a^2 (a B-2 A b)}{x^2}-\frac{2 a^3 A}{x^3}+\frac{6 a b^2 (a B-A b)}{a+b x}+6 b^2 \log (x) (3 a B-4 A b)+6 b^2 (4 A b-3 a B) \log (a+b x)+\frac{6 a b (2 a B-3 A b)}{x}}{6 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 134, normalized size = 1.2 \begin{align*} -{\frac{A}{3\,{a}^{2}{x}^{3}}}+{\frac{Ab}{{a}^{3}{x}^{2}}}-{\frac{B}{2\,{a}^{2}{x}^{2}}}-3\,{\frac{A{b}^{2}}{{a}^{4}x}}+2\,{\frac{Bb}{{a}^{3}x}}-4\,{\frac{A\ln \left ( x \right ){b}^{3}}{{a}^{5}}}+3\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{4}}}+4\,{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{5}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{4}}}-{\frac{A{b}^{3}}{{a}^{4} \left ( bx+a \right ) }}+{\frac{B{b}^{2}}{{a}^{3} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^4/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.07478, size = 173, normalized size = 1.53 \begin{align*} -\frac{2 \, A a^{3} - 6 \,{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{2} +{\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} x}{6 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} - \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{5}} + \frac{{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (x\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.05791, size = 378, normalized size = 3.35 \begin{align*} -\frac{2 \, A a^{4} - 6 \,{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} - 3 \,{\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{2} +{\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x + 6 \,{\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} +{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} +{\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b x^{4} + a^{6} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.11655, size = 219, normalized size = 1.94 \begin{align*} \frac{- 2 A a^{3} + x^{3} \left (- 24 A b^{3} + 18 B a b^{2}\right ) + x^{2} \left (- 12 A a b^{2} + 9 B a^{2} b\right ) + x \left (4 A a^{2} b - 3 B a^{3}\right )}{6 a^{5} x^{3} + 6 a^{4} b x^{4}} + \frac{b^{2} \left (- 4 A b + 3 B a\right ) \log{\left (x + \frac{- 4 A a b^{3} + 3 B a^{2} b^{2} - a b^{2} \left (- 4 A b + 3 B a\right )}{- 8 A b^{4} + 6 B a b^{3}} \right )}}{a^{5}} - \frac{b^{2} \left (- 4 A b + 3 B a\right ) \log{\left (x + \frac{- 4 A a b^{3} + 3 B a^{2} b^{2} + a b^{2} \left (- 4 A b + 3 B a\right )}{- 8 A b^{4} + 6 B a b^{3}} \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**4/(b*x+a)**2,x)

[Out]

(-2*A*a**3 + x**3*(-24*A*b**3 + 18*B*a*b**2) + x**2*(-12*A*a*b**2 + 9*B*a**2*b) + x*(4*A*a**2*b - 3*B*a**3))/(
6*a**5*x**3 + 6*a**4*b*x**4) + b**2*(-4*A*b + 3*B*a)*log(x + (-4*A*a*b**3 + 3*B*a**2*b**2 - a*b**2*(-4*A*b + 3
*B*a))/(-8*A*b**4 + 6*B*a*b**3))/a**5 - b**2*(-4*A*b + 3*B*a)*log(x + (-4*A*a*b**3 + 3*B*a**2*b**2 + a*b**2*(-
4*A*b + 3*B*a))/(-8*A*b**4 + 6*B*a*b**3))/a**5

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Giac [A]  time = 1.18018, size = 217, normalized size = 1.92 \begin{align*} \frac{{\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} \log \left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{5} b} + \frac{\frac{B a b^{6}}{b x + a} - \frac{A b^{7}}{b x + a}}{a^{4} b^{4}} - \frac{15 \, B a b^{2} - 26 \, A b^{3} - \frac{3 \,{\left (11 \, B a^{2} b^{3} - 20 \, A a b^{4}\right )}}{{\left (b x + a\right )} b} + \frac{18 \,{\left (B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \, a^{5}{\left (\frac{a}{b x + a} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*B*a*b^3 - 4*A*b^4)*log(abs(-a/(b*x + a) + 1))/(a^5*b) + (B*a*b^6/(b*x + a) - A*b^7/(b*x + a))/(a^4*b^4) - 1
/6*(15*B*a*b^2 - 26*A*b^3 - 3*(11*B*a^2*b^3 - 20*A*a*b^4)/((b*x + a)*b) + 18*(B*a^3*b^4 - 2*A*a^2*b^5)/((b*x +
a)^2*b^2))/(a^5*(a/(b*x + a) - 1)^3)