3.2.99 \(\int \frac {A+B x}{x^2 (a+b x)^3} \, dx\) [199]

Optimal. Leaf size=88 \[ -\frac {A}{a^3 x}-\frac {A b-a B}{2 a^2 (a+b x)^2}-\frac {2 A b-a B}{a^3 (a+b x)}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log (a+b x)}{a^4} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 88, 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 = {78} \begin {gather*} -\frac {\log (x) (3 A b-a B)}{a^4}+\frac {(3 A b-a B) \log (a+b x)}{a^4}-\frac {2 A b-a B}{a^3 (a+b x)}-\frac {A}{a^3 x}-\frac {A b-a B}{2 a^2 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^2*(a + b*x)^3),x]

[Out]

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

Rule 78

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

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Mathematica [A]
time = 0.05, size = 81, normalized size = 0.92 \begin {gather*} \frac {-\frac {2 a A}{x}+\frac {a^2 (-A b+a B)}{(a+b x)^2}+\frac {2 a (-2 A b+a B)}{a+b x}+2 (-3 A b+a B) \log (x)+2 (3 A b-a B) \log (a+b x)}{2 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^2*(a + b*x)^3),x]

[Out]

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

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Maple [A]
time = 0.07, size = 85, normalized size = 0.97

method	result	size
default	\(-\frac {2 A b -B a}{a^{3} \left (b x +a \right )}+\frac {\left (3 A b -B a \right ) \ln \left (b x +a \right )}{a^{4}}-\frac {A b -B a}{2 a^{2} \left (b x +a \right )^{2}}-\frac {A}{a^{3} x}+\frac {\left (-3 A b +B a \right ) \ln \left (x \right )}{a^{4}}\)	\(85\)
norman	\(\frac {-\frac {A}{a}+\frac {2 b \left (3 A b -B a \right ) x^{2}}{a^{3}}+\frac {b^{2} \left (9 A b -3 B a \right ) x^{3}}{2 a^{4}}}{x \left (b x +a \right )^{2}}+\frac {\left (3 A b -B a \right ) \ln \left (b x +a \right )}{a^{4}}-\frac {\left (3 A b -B a \right ) \ln \left (x \right )}{a^{4}}\)	\(93\)
risch	\(\frac {-\frac {b \left (3 A b -B a \right ) x^{2}}{a^{3}}-\frac {3 \left (3 A b -B a \right ) x}{2 a^{2}}-\frac {A}{a}}{x \left (b x +a \right )^{2}}-\frac {3 \ln \left (x \right ) A b}{a^{4}}+\frac {\ln \left (x \right ) B}{a^{3}}+\frac {3 \ln \left (-b x -a \right ) A b}{a^{4}}-\frac {\ln \left (-b x -a \right ) B}{a^{3}}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^2/(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 100, normalized size = 1.14 \begin {gather*} -\frac {2 \, A a^{2} - 2 \, {\left (B a b - 3 \, A b^{2}\right )} x^{2} - 3 \, {\left (B a^{2} - 3 \, A a b\right )} x}{2 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} - \frac {{\left (B a - 3 \, A b\right )} \log \left (b x + a\right )}{a^{4}} + \frac {{\left (B a - 3 \, A b\right )} \log \left (x\right )}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^2/(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (82) = 164\).
time = 0.92, size = 187, normalized size = 2.12 \begin {gather*} -\frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x + 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{3} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} x\right )} \log \left (x\right )}{2 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^2/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (76) = 152\).
time = 0.32, size = 168, normalized size = 1.91 \begin {gather*} \frac {- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 2 B a b\right ) + x \left (- 9 A a b + 3 B a^{2}\right )}{2 a^{5} x + 4 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac {\left (- 3 A b + B a\right ) \log {\left (x + \frac {- 3 A a b + B a^{2} - a \left (- 3 A b + B a\right )}{- 6 A b^{2} + 2 B a b} \right )}}{a^{4}} - \frac {\left (- 3 A b + B a\right ) \log {\left (x + \frac {- 3 A a b + B a^{2} + a \left (- 3 A b + B a\right )}{- 6 A b^{2} + 2 B a b} \right )}}{a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**2/(b*x+a)**3,x)

[Out]

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

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Giac [A]
time = 1.39, size = 99, normalized size = 1.12 \begin {gather*} \frac {{\left (B a - 3 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac {2 \, A a^{3} - 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} x}{2 \, {\left (b x + a\right )}^{2} a^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^2/(b*x+a)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.10, size = 87, normalized size = 0.99 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )\,\left (3\,A\,b-B\,a\right )}{a^4}-\frac {\frac {A}{a}+\frac {3\,x\,\left (3\,A\,b-B\,a\right )}{2\,a^2}+\frac {b\,x^2\,\left (3\,A\,b-B\,a\right )}{a^3}}{a^2\,x+2\,a\,b\,x^2+b^2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^2*(a + b*x)^3),x)

[Out]

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

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