3.15.52 \(\int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx\)

Optimal. Leaf size=28 \[ -\frac {(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {24, 37} \begin {gather*} -\frac {(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^4,x]

[Out]

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

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.93 \begin {gather*} -\frac {a d+b (c+2 d x)}{2 b^2 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^4,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^4,x]

[Out]

IntegrateAlgebraic[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^4, x]

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fricas [A]  time = 0.39, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a)^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 24, normalized size = 0.86 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (b x + a\right )}^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a)^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 35, normalized size = 1.25 \begin {gather*} -\frac {d}{\left (b x +a \right ) b^{2}}-\frac {-a d +b c}{2 \left (b x +a \right )^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a)^4,x)

[Out]

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

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maxima [A]  time = 1.18, size = 38, normalized size = 1.36 \begin {gather*} -\frac {2 \, b d x + b c + a d}{2 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a)^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.55, size = 39, normalized size = 1.39 \begin {gather*} -\frac {\frac {a\,d+b\,c}{2\,b^2}+\frac {d\,x}{b}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c + x*(a*d + b*c) + b*d*x^2)/(a + b*x)^4,x)

[Out]

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

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sympy [A]  time = 0.30, size = 39, normalized size = 1.39 \begin {gather*} \frac {- a d - b c - 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a)**4,x)

[Out]

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

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