### 3.1795 $$\int \frac{(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^9} \, dx$$

Optimal. Leaf size=58 $\frac{d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac{(c+d x)^4}{5 (a+b x)^5 (b c-a d)}$

[Out]

-(c + d*x)^4/(5*(b*c - a*d)*(a + b*x)^5) + (d*(c + d*x)^4)/(20*(b*c - a*d)^2*(a + b*x)^4)

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Rubi [A]  time = 0.0189528, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.103, Rules used = {626, 45, 37} $\frac{d (c+d x)^4}{20 (a+b x)^4 (b c-a d)^2}-\frac{(c+d x)^4}{5 (a+b x)^5 (b c-a d)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^9,x]

[Out]

-(c + d*x)^4/(5*(b*c - a*d)*(a + b*x)^5) + (d*(c + d*x)^4)/(20*(b*c - a*d)^2*(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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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{\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^9} \, dx &=\int \frac{(c+d x)^3}{(a+b x)^6} \, dx\\ &=-\frac{(c+d x)^4}{5 (b c-a d) (a+b x)^5}-\frac{d \int \frac{(c+d x)^3}{(a+b x)^5} \, dx}{5 (b c-a d)}\\ &=-\frac{(c+d x)^4}{5 (b c-a d) (a+b x)^5}+\frac{d (c+d x)^4}{20 (b c-a d)^2 (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.033927, size = 97, normalized size = 1.67 $-\frac{a^2 b d^2 (2 c+5 d x)+a^3 d^3+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (15 c^2 d x+4 c^3+20 c d^2 x^2+10 d^3 x^3\right )}{20 b^4 (a+b x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^9,x]

[Out]

-(a^3*d^3 + a^2*b*d^2*(2*c + 5*d*x) + a*b^2*d*(3*c^2 + 10*c*d*x + 10*d^2*x^2) + b^3*(4*c^3 + 15*c^2*d*x + 20*c
*d^2*x^2 + 10*d^3*x^3))/(20*b^4*(a + b*x)^5)

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Maple [B]  time = 0.044, size = 121, normalized size = 2.1 \begin{align*} -{\frac{{d}^{3}}{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}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{3\,d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}+{\frac{{d}^{2} \left ( ad-bc \right ) }{{b}^{4} \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)^3/(b*x+a)^9,x)

[Out]

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

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Maxima [B]  time = 1.00864, size = 216, normalized size = 3.72 \begin{align*} -\frac{10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \,{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} 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)^9,x, algorithm="maxima")

[Out]

-1/20*(10*b^3*d^3*x^3 + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3 + 10*(2*b^3*c*d^2 + a*b^2*d^3)*x^2
+ 5*(3*b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^9*x^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5
*a^4*b^5*x + a^5*b^4)

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Fricas [B]  time = 1.55489, size = 328, normalized size = 5.66 \begin{align*} -\frac{10 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3} + 10 \,{\left (2 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 5 \,{\left (3 \, b^{3} c^{2} d + 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{20 \,{\left (b^{9} x^{5} + 5 \, a b^{8} x^{4} + 10 \, a^{2} b^{7} x^{3} + 10 \, a^{3} b^{6} x^{2} + 5 \, a^{4} b^{5} x + a^{5} 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)^9,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*d^3*x^3 + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3 + 10*(2*b^3*c*d^2 + a*b^2*d^3)*x^2
+ 5*(3*b^3*c^2*d + 2*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^9*x^5 + 5*a*b^8*x^4 + 10*a^2*b^7*x^3 + 10*a^3*b^6*x^2 + 5
*a^4*b^5*x + a^5*b^4)

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Sympy [B]  time = 5.88685, size = 170, normalized size = 2.93 \begin{align*} - \frac{a^{3} d^{3} + 2 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + 4 b^{3} c^{3} + 10 b^{3} d^{3} x^{3} + x^{2} \left (10 a b^{2} d^{3} + 20 b^{3} c d^{2}\right ) + x \left (5 a^{2} b d^{3} + 10 a b^{2} c d^{2} + 15 b^{3} c^{2} d\right )}{20 a^{5} b^{4} + 100 a^{4} b^{5} x + 200 a^{3} b^{6} x^{2} + 200 a^{2} b^{7} x^{3} + 100 a b^{8} x^{4} + 20 b^{9} x^{5}} \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)**9,x)

[Out]

-(a**3*d**3 + 2*a**2*b*c*d**2 + 3*a*b**2*c**2*d + 4*b**3*c**3 + 10*b**3*d**3*x**3 + x**2*(10*a*b**2*d**3 + 20*
b**3*c*d**2) + x*(5*a**2*b*d**3 + 10*a*b**2*c*d**2 + 15*b**3*c**2*d))/(20*a**5*b**4 + 100*a**4*b**5*x + 200*a*
*3*b**6*x**2 + 200*a**2*b**7*x**3 + 100*a*b**8*x**4 + 20*b**9*x**5)

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Giac [B]  time = 1.2378, size = 154, normalized size = 2.66 \begin{align*} -\frac{10 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c d^{2} x^{2} + 10 \, a b^{2} d^{3} x^{2} + 15 \, b^{3} c^{2} d x + 10 \, a b^{2} c d^{2} x + 5 \, a^{2} b d^{3} x + 4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}}{20 \,{\left (b x + a\right )}^{5} 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)^9,x, algorithm="giac")

[Out]

-1/20*(10*b^3*d^3*x^3 + 20*b^3*c*d^2*x^2 + 10*a*b^2*d^3*x^2 + 15*b^3*c^2*d*x + 10*a*b^2*c*d^2*x + 5*a^2*b*d^3*
x + 4*b^3*c^3 + 3*a*b^2*c^2*d + 2*a^2*b*c*d^2 + a^3*d^3)/((b*x + a)^5*b^4)