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

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

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \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[(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*}

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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[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^6,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 {a c+(b c+a d) x+b d x^2}{(a+b x)^6} \, 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)^6,x]

[Out]

IntegrateAlgebraic[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^6, x]

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fricas [A]  time = 0.39, 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((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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giac [A]  time = 0.15, size = 25, normalized size = 0.66 \begin {gather*} -\frac {4 \, b d x + 3 \, b c + a d}{12 \, {\left (b x + a\right )}^{4} 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)^6,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, 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((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 = 1.16, 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((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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mupad [B]  time = 0.05, 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((a*c + x*(a*d + b*c) + b*d*x^2)/(a + b*x)^6,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 = 0.52, 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((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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