3.1768 \(\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^6} \, dx\)

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 = 0.0260423, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {24, 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[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^6,x]

[Out]

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

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 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{a c+(b c+a d) x+b d x^2}{(a+b x)^6} \, dx &=\frac{\int \frac{b^2 c+b^2 d x}{(a+b x)^5} \, dx}{b^2}\\ &=\frac{\int \left (\frac{b (b c-a d)}{(a+b x)^5}+\frac{b d}{(a+b x)^4}\right ) \, dx}{b^2}\\ &=-\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 = 0.0089455, 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[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^6,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 = 0.044, size = 35, normalized size = 0.9 \begin{align*} -{\frac{-ad+bc}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}}-{\frac{d}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \end{align*}

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)^6,x)

[Out]

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

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Maxima [A]  time = 1.04554, 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((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a)^6,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 = 1.77898, 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((a*c+(a*d+b*c)*x+b*d*x^2)/(b*x+a)^6,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 = 0.738193, 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((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a)**6,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 = 1.17497, size = 34, normalized size = 0.89 \begin{align*} -\frac{4 \, b d x + 3 \, b c + a d}{12 \,{\left (b x + a\right )}^{4} b^{2}} \end{align*}

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)^6,x, algorithm="giac")

[Out]

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