3.1778 \(\int \frac{(a c+(b c+a d) x+b d x^2)^2}{(a+b x)^6} \, dx\)

Optimal. Leaf size=28 \[ -\frac{(c+d x)^3}{3 (a+b x)^3 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0112407, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 37} \[ -\frac{(c+d x)^3}{3 (a+b x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

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{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^6} \, dx &=\int \frac{(c+d x)^2}{(a+b x)^4} \, dx\\ &=-\frac{(c+d x)^3}{3 (b c-a d) (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.022367, size = 53, normalized size = 1.89 \[ -\frac{a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )}{3 b^3 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.043, size = 70, normalized size = 2.5 \begin{align*}{\frac{ \left ( ad-bc \right ) d}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{{a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2}}{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{2}}{{b}^{3} \left ( bx+a \right ) }} \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)^2/(b*x+a)^6,x)

[Out]

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

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Maxima [B]  time = 1.18399, size = 113, normalized size = 4.04 \begin{align*} -\frac{3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \,{\left (b^{2} c d + a b d^{2}\right )} x}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\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)^2/(b*x+a)^6,x, algorithm="maxima")

[Out]

-1/3*(3*b^2*d^2*x^2 + b^2*c^2 + a*b*c*d + a^2*d^2 + 3*(b^2*c*d + a*b*d^2)*x)/(b^6*x^3 + 3*a*b^5*x^2 + 3*a^2*b^
4*x + a^3*b^3)

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Fricas [B]  time = 1.58046, size = 170, normalized size = 6.07 \begin{align*} -\frac{3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \,{\left (b^{2} c d + a b d^{2}\right )} x}{3 \,{\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\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)^2/(b*x+a)^6,x, algorithm="fricas")

[Out]

-1/3*(3*b^2*d^2*x^2 + b^2*c^2 + a*b*c*d + a^2*d^2 + 3*(b^2*c*d + a*b*d^2)*x)/(b^6*x^3 + 3*a*b^5*x^2 + 3*a^2*b^
4*x + a^3*b^3)

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Sympy [B]  time = 1.26452, size = 88, normalized size = 3.14 \begin{align*} - \frac{a^{2} d^{2} + a b c d + b^{2} c^{2} + 3 b^{2} d^{2} x^{2} + x \left (3 a b d^{2} + 3 b^{2} c d\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \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)**2/(b*x+a)**6,x)

[Out]

-(a**2*d**2 + a*b*c*d + b**2*c**2 + 3*b**2*d**2*x**2 + x*(3*a*b*d**2 + 3*b**2*c*d))/(3*a**3*b**3 + 9*a**2*b**4
*x + 9*a*b**5*x**2 + 3*b**6*x**3)

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Giac [B]  time = 1.18049, size = 80, normalized size = 2.86 \begin{align*} -\frac{3 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c d x + 3 \, a b d^{2} x + b^{2} c^{2} + a b c d + a^{2} d^{2}}{3 \,{\left (b x + a\right )}^{3} b^{3}} \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)^2/(b*x+a)^6,x, algorithm="giac")

[Out]

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