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

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {b c-a d}{3 b^2 (a+b x)^3}-\frac {d}{2 b^2 (a+b x)^2} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {a d+2 b c+3 b d x}{6 b^2 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + b*x)^4,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x}{(a+b x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)/(a + b*x)^4,x]

[Out]

IntegrateAlgebraic[(c + d*x)/(a + b*x)^4, x]

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fricas [A]  time = 1.75, size = 50, normalized size = 1.32 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.75, size = 25, normalized size = 0.66 \begin {gather*} -\frac {3 \, b d x + 2 \, b c + a d}{6 \, {\left (b x + a\right )}^{3} b^{2}} \end {gather*}

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)

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maple [A]  time = 0.01, size = 35, normalized size = 0.92 \begin {gather*} -\frac {d}{2 \left (b x +a \right )^{2} b^{2}}-\frac {-a d +b c}{3 \left (b x +a \right )^{3} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 50, normalized size = 1.32 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.17, size = 52, normalized size = 1.37 \begin {gather*} -\frac {\frac {a\,d+2\,b\,c}{6\,b^2}+\frac {d\,x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a + b*x)^4,x)

[Out]

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

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sympy [A]  time = 0.34, size = 53, normalized size = 1.39 \begin {gather*} \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 {gather*}

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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