3.1244 \(\int \frac{c+d x}{(a+b x)^4} \, dx\)

Optimal. Leaf size=38 \[ -\frac{b c-a d}{3 b^2 (a+b x)^3}-\frac{d}{2 b^2 (a+b x)^2} \]

[Out]

-(b*c - a*d)/(3*b^2*(a + b*x)^3) - d/(2*b^2*(a + b*x)^2)

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Rubi [A]  time = 0.0199422, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ -\frac{b c-a d}{3 b^2 (a+b x)^3}-\frac{d}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a + b*x)^4,x]

[Out]

-(b*c - a*d)/(3*b^2*(a + b*x)^3) - d/(2*b^2*(a + b*x)^2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0087701, size = 27, normalized size = 0.71 \[ -\frac{a d+2 b c+3 b d x}{6 b^2 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + b*x)^4,x]

[Out]

-(2*b*c + a*d + 3*b*d*x)/(6*b^2*(a + b*x)^3)

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Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \begin{align*} -{\frac{d}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{-ad+bc}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/(b*x+a)^4,x)

[Out]

-1/2*d/b^2/(b*x+a)^2-1/3*(-a*d+b*c)/b^2/(b*x+a)^3

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Maxima [A]  time = 0.956878, size = 68, normalized size = 1.79 \begin{align*} -\frac{3 \, b d x + 2 \, b c + a d}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*x+a)^4,x, algorithm="maxima")

[Out]

-1/6*(3*b*d*x + 2*b*c + a*d)/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)

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Fricas [A]  time = 1.94994, size = 105, normalized size = 2.76 \begin{align*} -\frac{3 \, b d x + 2 \, b c + a d}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*x+a)^4,x, algorithm="fricas")

[Out]

-1/6*(3*b*d*x + 2*b*c + a*d)/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)

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Sympy [A]  time = 0.501469, size = 53, normalized size = 1.39 \begin{align*} - \frac{a d + 2 b c + 3 b d x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*x+a)**4,x)

[Out]

-(a*d + 2*b*c + 3*b*d*x)/(6*a**3*b**2 + 18*a**2*b**3*x + 18*a*b**4*x**2 + 6*b**5*x**3)

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Giac [A]  time = 1.07809, size = 34, normalized size = 0.89 \begin{align*} -\frac{3 \, b d x + 2 \, b c + a d}{6 \,{\left (b x + a\right )}^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*x+a)^4,x, algorithm="giac")

[Out]

-1/6*(3*b*d*x + 2*b*c + a*d)/((b*x + a)^3*b^2)