### 3.1781 $$\int \frac{(a c+(b c+a d) x+b d x^2)^2}{(a+b x)^9} \, dx$$

Optimal. Leaf size=65 $-\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{d^2}{4 b^3 (a+b x)^4}$

[Out]

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

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Rubi [A]  time = 0.0411441, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 43} $-\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{d^2}{4 b^3 (a+b x)^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b*c - a*d)^2/(6*b^3*(a + b*x)^6) - (2*d*(b*c - a*d))/(5*b^3*(a + b*x)^5) - d^2/(4*b^3*(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 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{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx &=\int \frac{(c+d x)^2}{(a+b x)^7} \, dx\\ &=\int \left (\frac{(b c-a d)^2}{b^2 (a+b x)^7}+\frac{2 d (b c-a d)}{b^2 (a+b x)^6}+\frac{d^2}{b^2 (a+b x)^5}\right ) \, dx\\ &=-\frac{(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac{2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac{d^2}{4 b^3 (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0201183, size = 58, normalized size = 0.89 $-\frac{a^2 d^2+2 a b d (2 c+3 d x)+b^2 \left (10 c^2+24 c d x+15 d^2 x^2\right )}{60 b^3 (a+b x)^6}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a^2*d^2 + 2*a*b*d*(2*c + 3*d*x) + b^2*(10*c^2 + 24*c*d*x + 15*d^2*x^2))/(60*b^3*(a + b*x)^6)

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Maple [A]  time = 0.044, size = 71, normalized size = 1.1 \begin{align*}{\frac{ \left ( 2\,ad-2\,bc \right ) d}{5\,{b}^{3} \left ( bx+a \right ) ^{5}}}-{\frac{{d}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}}-{\frac{{a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2}}{6\,{b}^{3} \left ( bx+a \right ) ^{6}}} \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)^9,x)

[Out]

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

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Maxima [B]  time = 1.03486, size = 162, normalized size = 2.49 \begin{align*} -\frac{15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \,{\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} 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)^9,x, algorithm="maxima")

[Out]

-1/60*(15*b^2*d^2*x^2 + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2 + 6*(4*b^2*c*d + a*b*d^2)*x)/(b^9*x^6 + 6*a*b^8*x^5 +
15*a^2*b^7*x^4 + 20*a^3*b^6*x^3 + 15*a^4*b^5*x^2 + 6*a^5*b^4*x + a^6*b^3)

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Fricas [B]  time = 1.53198, size = 251, normalized size = 3.86 \begin{align*} -\frac{15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \,{\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \,{\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} 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)^9,x, algorithm="fricas")

[Out]

-1/60*(15*b^2*d^2*x^2 + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2 + 6*(4*b^2*c*d + a*b*d^2)*x)/(b^9*x^6 + 6*a*b^8*x^5 +
15*a^2*b^7*x^4 + 20*a^3*b^6*x^3 + 15*a^4*b^5*x^2 + 6*a^5*b^4*x + a^6*b^3)

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Sympy [B]  time = 1.9433, size = 128, normalized size = 1.97 \begin{align*} - \frac{a^{2} d^{2} + 4 a b c d + 10 b^{2} c^{2} + 15 b^{2} d^{2} x^{2} + x \left (6 a b d^{2} + 24 b^{2} c d\right )}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \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)**9,x)

[Out]

-(a**2*d**2 + 4*a*b*c*d + 10*b**2*c**2 + 15*b**2*d**2*x**2 + x*(6*a*b*d**2 + 24*b**2*c*d))/(60*a**6*b**3 + 360
*a**5*b**4*x + 900*a**4*b**5*x**2 + 1200*a**3*b**6*x**3 + 900*a**2*b**7*x**4 + 360*a*b**8*x**5 + 60*b**9*x**6)

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Giac [A]  time = 1.23679, size = 82, normalized size = 1.26 \begin{align*} -\frac{15 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 6 \, a b d^{2} x + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}}{60 \,{\left (b x + a\right )}^{6} 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)^9,x, algorithm="giac")

[Out]

-1/60*(15*b^2*d^2*x^2 + 24*b^2*c*d*x + 6*a*b*d^2*x + 10*b^2*c^2 + 4*a*b*c*d + a^2*d^2)/((b*x + a)^6*b^3)