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

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]

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

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Rubi [A]  time = 0.12, antiderivative size = 140, 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} \[ -\frac {b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac {b^2 (A b-a B)}{2 a^4 (a+b x)^2}-\frac {2 b^2 \log (x) (5 A b-3 a B)}{a^6}+\frac {2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}+\frac {3 A b-a B}{2 a^4 x^2}-\frac {3 b (2 A b-a B)}{a^5 x}-\frac {A}{3 a^3 x^3} \]

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

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)^3} \, dx &=\int \left (\frac {A}{a^3 x^4}+\frac {-3 A b+a B}{a^4 x^3}-\frac {3 b (-2 A b+a B)}{a^5 x^2}+\frac {2 b^2 (-5 A b+3 a B)}{a^6 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)^3}-\frac {b^3 (-4 A b+3 a B)}{a^5 (a+b x)^2}-\frac {2 b^3 (-5 A b+3 a B)}{a^6 (a+b x)}\right ) \, dx\\ &=-\frac {A}{3 a^3 x^3}+\frac {3 A b-a B}{2 a^4 x^2}-\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 {b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac {2 b^2 (5 A b-3 a B) \log (x)}{a^6}+\frac {2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 129, normalized size = 0.92 \[ \frac {\frac {a \left (-\left (a^4 (2 A+3 B x)\right )+a^3 b x (5 A+12 B x)+2 a^2 b^2 x^2 (27 B x-10 A)+18 a b^3 x^3 (2 B x-5 A)-60 A b^4 x^4\right )}{x^3 (a+b x)^2}+12 b^2 \log (x) (3 a B-5 A b)+12 b^2 (5 A b-3 a B) \log (a+b x)}{6 a^6} \]

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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fricas [A]  time = 0.96, size = 262, normalized size = 1.87 \[ -\frac {2 \, A a^{5} - 12 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \, {\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + {\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x + 12 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 12 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \relax (x)}{6 \, {\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(b*x+a)^3,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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giac [A]  time = 0.94, size = 158, normalized size = 1.13 \[ \frac {2 \, {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac {2 \, {\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac {2 \, A a^{5} - 12 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \, {\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + {\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x}{6 \, {\left (b x + a\right )}^{2} a^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 168, normalized size = 1.20 \[ -\frac {A \,b^{3}}{2 \left (b x +a \right )^{2} a^{4}}+\frac {B \,b^{2}}{2 \left (b x +a \right )^{2} a^{3}}-\frac {4 A \,b^{3}}{\left (b x +a \right ) a^{5}}-\frac {10 A \,b^{3} \ln \relax (x )}{a^{6}}+\frac {10 A \,b^{3} \ln \left (b x +a \right )}{a^{6}}+\frac {3 B \,b^{2}}{\left (b x +a \right ) a^{4}}+\frac {6 B \,b^{2} \ln \relax (x )}{a^{5}}-\frac {6 B \,b^{2} \ln \left (b x +a \right )}{a^{5}}-\frac {6 A \,b^{2}}{a^{5} x}+\frac {3 B b}{a^{4} x}+\frac {3 A b}{2 a^{4} x^{2}}-\frac {B}{2 a^{3} x^{2}}-\frac {A}{3 a^{3} x^{3}} \]

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*b^2/a^5/x*A+3*b/a^4/x*B-10*b^3/a^6*ln(x)*A+6*b^2/a^5*ln(x)*B-4*
b^3/a^5/(b*x+a)*A+3*b^2/a^4/(b*x+a)*B-1/2*b^3/a^4/(b*x+a)^2*A+1/2*b^2/a^3/(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 = 1.06, size = 164, normalized size = 1.17 \[ -\frac {2 \, A a^{4} - 12 \, {\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} - 18 \, {\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} - 4 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} + {\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x}{6 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} - \frac {2 \, {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{6}} + \frac {2 \, {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \relax (x)}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(b*x+a)^3,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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mupad [B]  time = 0.38, size = 164, normalized size = 1.17 \[ \frac {4\,b^2\,\mathrm {atanh}\left (\frac {2\,b^2\,\left (5\,A\,b-3\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (10\,A\,b^3-6\,B\,a\,b^2\right )}\right )\,\left (5\,A\,b-3\,B\,a\right )}{a^6}-\frac {\frac {A}{3\,a}-\frac {x\,\left (5\,A\,b-3\,B\,a\right )}{6\,a^2}+\frac {3\,b^2\,x^3\,\left (5\,A\,b-3\,B\,a\right )}{a^4}+\frac {2\,b^3\,x^4\,\left (5\,A\,b-3\,B\,a\right )}{a^5}+\frac {2\,b\,x^2\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^3}}{a^2\,x^3+2\,a\,b\,x^4+b^2\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^4*(a + b*x)^3),x)

[Out]

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

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sympy [A]  time = 1.31, 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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