3.1245 \(\int \frac{c+d x}{(a+b x)^5} \, dx\)

Optimal. Leaf size=38 \[ -\frac{b c-a d}{4 b^2 (a+b x)^4}-\frac{d}{3 b^2 (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.019943, 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} \[ -\frac{b c-a d}{4 b^2 (a+b x)^4}-\frac{d}{3 b^2 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a + b*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0097732, size = 27, normalized size = 0.71 \[ -\frac{a d+3 b c+4 b d x}{12 b^2 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + b*x)^5,x]

[Out]

-(3*b*c + a*d + 4*b*d*x)/(12*b^2*(a + b*x)^4)

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Maple [A]  time = 0.005, size = 35, normalized size = 0.9 \begin{align*} -{\frac{d}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{-ad+bc}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)/(b*x+a)^5,x)

[Out]

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

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Maxima [A]  time = 0.948785, size = 82, normalized size = 2.16 \begin{align*} -\frac{4 \, b d x + 3 \, b c + a d}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*b*d*x + 3*b*c + a*d)/(b^6*x^4 + 4*a*b^5*x^3 + 6*a^2*b^4*x^2 + 4*a^3*b^3*x + a^4*b^2)

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Fricas [A]  time = 2.01209, size = 128, normalized size = 3.37 \begin{align*} -\frac{4 \, b d x + 3 \, b c + a d}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*b*d*x + 3*b*c + a*d)/(b^6*x^4 + 4*a*b^5*x^3 + 6*a^2*b^4*x^2 + 4*a^3*b^3*x + a^4*b^2)

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Sympy [B]  time = 0.593464, size = 65, normalized size = 1.71 \begin{align*} - \frac{a d + 3 b c + 4 b d x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)/(b*x+a)**5,x)

[Out]

-(a*d + 3*b*c + 4*b*d*x)/(12*a**4*b**2 + 48*a**3*b**3*x + 72*a**2*b**4*x**2 + 48*a*b**5*x**3 + 12*b**6*x**4)

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Giac [A]  time = 1.05095, size = 55, normalized size = 1.45 \begin{align*} -\frac{c}{4 \,{\left (b x + a\right )}^{4} b} - \frac{d}{3 \,{\left (b x + a\right )}^{3} b^{2}} + \frac{a d}{4 \,{\left (b x + a\right )}^{4} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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