### 3.1794 $$\int \frac{(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^8} \, dx$$

Optimal. Leaf size=28 $-\frac{(c+d x)^4}{4 (a+b x)^4 (b c-a d)}$

[Out]

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

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Rubi [A]  time = 0.0116204, 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} $-\frac{(c+d x)^4}{4 (a+b x)^4 (b c-a d)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^8,x]

[Out]

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

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 )^3}{(a+b x)^8} \, dx &=\int \frac{(c+d x)^3}{(a+b x)^5} \, dx\\ &=-\frac{(c+d x)^4}{4 (b c-a d) (a+b x)^4}\\ \end{align*}

Mathematica [B]  time = 0.0294395, size = 91, normalized size = 3.25 $-\frac{a^2 b d^2 (c+4 d x)+a^3 d^3+a b^2 d \left (c^2+4 c d x+6 d^2 x^2\right )+b^3 \left (4 c^2 d x+c^3+6 c d^2 x^2+4 d^3 x^3\right )}{4 b^4 (a+b x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^8,x]

[Out]

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

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Maple [B]  time = 0.045, size = 122, normalized size = 4.4 \begin{align*}{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{-{a}^{3}{d}^{3}+3\,cb{a}^{2}{d}^{2}-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{{d}^{3}}{{b}^{4} \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)^3/(b*x+a)^8,x)

[Out]

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

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Maxima [B]  time = 1.23457, size = 193, normalized size = 6.89 \begin{align*} -\frac{4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\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)^3/(b*x+a)^8,x, algorithm="maxima")

[Out]

-1/4*(4*b^3*d^3*x^3 + b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3 + 6*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c
^2*d + a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^8*x^4 + 4*a*b^7*x^3 + 6*a^2*b^6*x^2 + 4*a^3*b^5*x + a^4*b^4)

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Fricas [B]  time = 1.46081, size = 284, normalized size = 10.14 \begin{align*} -\frac{4 \, b^{3} d^{3} x^{3} + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d + a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\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)^3/(b*x+a)^8,x, algorithm="fricas")

[Out]

-1/4*(4*b^3*d^3*x^3 + b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3 + 6*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c
^2*d + a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^8*x^4 + 4*a*b^7*x^3 + 6*a^2*b^6*x^2 + 4*a^3*b^5*x + a^4*b^4)

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Sympy [B]  time = 2.90295, size = 153, normalized size = 5.46 \begin{align*} - \frac{a^{3} d^{3} + a^{2} b c d^{2} + a b^{2} c^{2} d + b^{3} c^{3} + 4 b^{3} d^{3} x^{3} + x^{2} \left (6 a b^{2} d^{3} + 6 b^{3} c d^{2}\right ) + x \left (4 a^{2} b d^{3} + 4 a b^{2} c d^{2} + 4 b^{3} c^{2} d\right )}{4 a^{4} b^{4} + 16 a^{3} b^{5} x + 24 a^{2} b^{6} x^{2} + 16 a b^{7} x^{3} + 4 b^{8} 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)**3/(b*x+a)**8,x)

[Out]

-(a**3*d**3 + a**2*b*c*d**2 + a*b**2*c**2*d + b**3*c**3 + 4*b**3*d**3*x**3 + x**2*(6*a*b**2*d**3 + 6*b**3*c*d*
*2) + x*(4*a**2*b*d**3 + 4*a*b**2*c*d**2 + 4*b**3*c**2*d))/(4*a**4*b**4 + 16*a**3*b**5*x + 24*a**2*b**6*x**2 +
16*a*b**7*x**3 + 4*b**8*x**4)

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Giac [B]  time = 1.16293, size = 150, normalized size = 5.36 \begin{align*} -\frac{4 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, a b^{2} d^{3} x^{2} + 4 \, b^{3} c^{2} d x + 4 \, a b^{2} c d^{2} x + 4 \, a^{2} b d^{3} x + b^{3} c^{3} + a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}}{4 \,{\left (b x + a\right )}^{4} b^{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)^3/(b*x+a)^8,x, algorithm="giac")

[Out]

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