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

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

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Rubi [A]  time = 0.09, antiderivative size = 110, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 {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 {A}{2 a^3 x^2} \end {gather*}

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

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

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Mathematica [A]  time = 0.11, size = 102, normalized size = 0.93 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{x^3 (a+b x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.26, size = 225, normalized size = 2.05 \begin {gather*} -\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 \relax (x)}{2 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^3/(b*x+a)^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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giac [A]  time = 1.03, size = 124, normalized size = 1.13 \begin {gather*} -\frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {3 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} \log \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 138, normalized size = 1.25 \begin {gather*} \frac {A \,b^{2}}{2 \left (b x +a \right )^{2} a^{3}}-\frac {B b}{2 \left (b x +a \right )^{2} a^{2}}+\frac {3 A \,b^{2}}{\left (b x +a \right ) a^{4}}+\frac {6 A \,b^{2} \ln \relax (x )}{a^{5}}-\frac {6 A \,b^{2} \ln \left (b x +a \right )}{a^{5}}-\frac {2 B b}{\left (b x +a \right ) a^{3}}-\frac {3 B b \ln \relax (x )}{a^{4}}+\frac {3 B b \ln \left (b x +a \right )}{a^{4}}+\frac {3 A b}{a^{4} x}-\frac {B}{a^{3} x}-\frac {A}{2 a^{3} x^{2}} \end {gather*}

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

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maxima [A]  time = 1.02, size = 131, normalized size = 1.19 \begin {gather*} -\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 \relax (x)}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^3/(b*x+a)^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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mupad [B]  time = 0.12, size = 136, normalized size = 1.24 \begin {gather*} \frac {\frac {x\,\left (2\,A\,b-B\,a\right )}{a^2}-\frac {A}{2\,a}+\frac {3\,b^2\,x^3\,\left (2\,A\,b-B\,a\right )}{a^4}+\frac {9\,b\,x^2\,\left (2\,A\,b-B\,a\right )}{2\,a^3}}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4}-\frac {6\,b\,\mathrm {atanh}\left (\frac {3\,b\,\left (2\,A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (6\,A\,b^2-3\,B\,a\,b\right )}\right )\,\left (2\,A\,b-B\,a\right )}{a^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^3*(a + b*x)^3),x)

[Out]

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

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sympy [B]  time = 0.79, size = 219, normalized size = 1.99 \begin {gather*} \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}} \end {gather*}

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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