3.183 \(\int \frac {A+B x}{x^5 (a+b x)} \, dx\)

Optimal. Leaf size=106 \[ \frac {b^3 \log (x) (A b-a B)}{a^5}-\frac {b^3 (A b-a B) \log (a+b x)}{a^5}+\frac {b^2 (A b-a B)}{a^4 x}-\frac {b (A b-a B)}{2 a^3 x^2}+\frac {A b-a B}{3 a^2 x^3}-\frac {A}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 106, 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} \[ \frac {b^2 (A b-a B)}{a^4 x}+\frac {b^3 \log (x) (A b-a B)}{a^5}-\frac {b^3 (A b-a B) \log (a+b x)}{a^5}-\frac {b (A b-a B)}{2 a^3 x^2}+\frac {A b-a B}{3 a^2 x^3}-\frac {A}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^5*(a + b*x)),x]

[Out]

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

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Mathematica [A]  time = 0.08, size = 100, normalized size = 0.94 \[ \frac {\frac {a \left (-\left (a^3 (3 A+4 B x)\right )+2 a^2 b x (2 A+3 B x)-6 a b^2 x^2 (A+2 B x)+12 A b^3 x^3\right )}{x^4}+12 b^3 \log (x) (A b-a B)-12 b^3 (A b-a B) \log (a+b x)}{12 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^5*(a + b*x)),x]

[Out]

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

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fricas [A]  time = 0.93, size = 117, normalized size = 1.10 \[ \frac {12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \relax (x) - 3 \, A a^{4} - 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \, {\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(b*x+a),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.12, size = 122, normalized size = 1.15 \[ -\frac {{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {{\left (B a b^{4} - A b^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {3 \, A a^{4} + 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 4 \, {\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(b*x+a),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 125, normalized size = 1.18 \[ \frac {A \,b^{4} \ln \relax (x )}{a^{5}}-\frac {A \,b^{4} \ln \left (b x +a \right )}{a^{5}}-\frac {B \,b^{3} \ln \relax (x )}{a^{4}}+\frac {B \,b^{3} \ln \left (b x +a \right )}{a^{4}}+\frac {A \,b^{3}}{a^{4} x}-\frac {B \,b^{2}}{a^{3} x}-\frac {A \,b^{2}}{2 a^{3} x^{2}}+\frac {B b}{2 a^{2} x^{2}}+\frac {A b}{3 a^{2} x^{3}}-\frac {B}{3 a \,x^{3}}-\frac {A}{4 a \,x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^5/(b*x+a),x)

[Out]

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

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maxima [A]  time = 1.05, size = 112, normalized size = 1.06 \[ \frac {{\left (B a b^{3} - A b^{4}\right )} \log \left (b x + a\right )}{a^{5}} - \frac {{\left (B a b^{3} - A b^{4}\right )} \log \relax (x)}{a^{5}} - \frac {3 \, A a^{3} + 12 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 4 \, {\left (B a^{3} - A a^{2} b\right )} x}{12 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^5/(b*x+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.35, size = 116, normalized size = 1.09 \[ -\frac {\frac {A}{4\,a}-\frac {x\,\left (A\,b-B\,a\right )}{3\,a^2}-\frac {b^2\,x^3\,\left (A\,b-B\,a\right )}{a^4}+\frac {b\,x^2\,\left (A\,b-B\,a\right )}{2\,a^3}}{x^4}-\frac {2\,b^3\,\mathrm {atanh}\left (\frac {b^3\,\left (A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (A\,b^4-B\,a\,b^3\right )}\right )\,\left (A\,b-B\,a\right )}{a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^5*(a + b*x)),x)

[Out]

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

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sympy [B]  time = 0.65, size = 189, normalized size = 1.78 \[ \frac {- 3 A a^{3} + x^{3} \left (12 A b^{3} - 12 B a b^{2}\right ) + x^{2} \left (- 6 A a b^{2} + 6 B a^{2} b\right ) + x \left (4 A a^{2} b - 4 B a^{3}\right )}{12 a^{4} x^{4}} - \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} + B a^{2} b^{3} - a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} + \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} + B a^{2} b^{3} + a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**5/(b*x+a),x)

[Out]

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

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