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

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

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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}{4 b^2 (a+b x)^4}-\frac {d}{3 b^2 (a+b x)^3} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.71 \begin {gather*} -\frac {a d+3 b c+4 b d x}{12 b^2 (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + b*x)^5,x]

[Out]

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

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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)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(c + d*x)/(a + b*x)^5,x]

[Out]

IntegrateAlgebraic[(c + d*x)/(a + b*x)^5, x]

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fricas [A]  time = 1.20, size = 61, normalized size = 1.61 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.92, size = 41, normalized size = 1.08 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.00, size = 35, normalized size = 0.92 \begin {gather*} -\frac {d}{3 \left (b x +a \right )^{3} b^{2}}-\frac {-a d +b c}{4 \left (b x +a \right )^{4} b^{2}} \end {gather*}

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 = 1.39, size = 61, normalized size = 1.61 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.04, size = 63, normalized size = 1.66 \begin {gather*} -\frac {\frac {a\,d+3\,b\,c}{12\,b^2}+\frac {d\,x}{3\,b}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a + b*x)^5,x)

[Out]

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

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sympy [B]  time = 0.43, size = 65, normalized size = 1.71 \begin {gather*} \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 {gather*}

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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