### 3.1797 $$\int \frac{(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^{11}} \, dx$$

Optimal. Leaf size=92 $-\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d^3}{4 b^4 (a+b x)^4}$

[Out]

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

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Rubi [A]  time = 0.0576906, antiderivative size = 92, 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} $-\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d^3}{4 b^4 (a+b x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^11,x]

[Out]

-(b*c - a*d)^3/(7*b^4*(a + b*x)^7) - (d*(b*c - a*d)^2)/(2*b^4*(a + b*x)^6) - (3*d^2*(b*c - a*d))/(5*b^4*(a + b
*x)^5) - d^3/(4*b^4*(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 )^3}{(a+b x)^{11}} \, dx &=\int \frac{(c+d x)^3}{(a+b x)^8} \, dx\\ &=\int \left (\frac{(b c-a d)^3}{b^3 (a+b x)^8}+\frac{3 d (b c-a d)^2}{b^3 (a+b x)^7}+\frac{3 d^2 (b c-a d)}{b^3 (a+b x)^6}+\frac{d^3}{b^3 (a+b x)^5}\right ) \, dx\\ &=-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d^3}{4 b^4 (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.03055, size = 97, normalized size = 1.05 $-\frac{a^2 b d^2 (4 c+7 d x)+a^3 d^3+a b^2 d \left (10 c^2+28 c d x+21 d^2 x^2\right )+b^3 \left (70 c^2 d x+20 c^3+84 c d^2 x^2+35 d^3 x^3\right )}{140 b^4 (a+b x)^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^11,x]

[Out]

-(a^3*d^3 + a^2*b*d^2*(4*c + 7*d*x) + a*b^2*d*(10*c^2 + 28*c*d*x + 21*d^2*x^2) + b^3*(20*c^3 + 70*c^2*d*x + 84
*c*d^2*x^2 + 35*d^3*x^3))/(140*b^4*(a + b*x)^7)

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Maple [A]  time = 0.045, size = 122, normalized size = 1.3 \begin{align*}{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{{d}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{-{a}^{3}{d}^{3}+3\,cb{a}^{2}{d}^{2}-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{7\,{b}^{4} \left ( bx+a \right ) ^{7}}}-{\frac{d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{2\,{b}^{4} \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)^3/(b*x+a)^11,x)

[Out]

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

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Maxima [B]  time = 1.11019, size = 246, normalized size = 2.67 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \,{\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \,{\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} 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)^11,x, algorithm="maxima")

[Out]

-1/140*(35*b^3*d^3*x^3 + 20*b^3*c^3 + 10*a*b^2*c^2*d + 4*a^2*b*c*d^2 + a^3*d^3 + 21*(4*b^3*c*d^2 + a*b^2*d^3)*
x^2 + 7*(10*b^3*c^2*d + 4*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^11*x^7 + 7*a*b^10*x^6 + 21*a^2*b^9*x^5 + 35*a^3*b^8*x
^4 + 35*a^4*b^7*x^3 + 21*a^5*b^6*x^2 + 7*a^6*b^5*x + a^7*b^4)

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Fricas [B]  time = 1.52746, size = 382, normalized size = 4.15 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \,{\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \,{\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} 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)^11,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*d^3*x^3 + 20*b^3*c^3 + 10*a*b^2*c^2*d + 4*a^2*b*c*d^2 + a^3*d^3 + 21*(4*b^3*c*d^2 + a*b^2*d^3)*
x^2 + 7*(10*b^3*c^2*d + 4*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^11*x^7 + 7*a*b^10*x^6 + 21*a^2*b^9*x^5 + 35*a^3*b^8*x
^4 + 35*a^4*b^7*x^3 + 21*a^5*b^6*x^2 + 7*a^6*b^5*x + a^7*b^4)

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Sympy [B]  time = 13.7032, size = 194, normalized size = 2.11 \begin{align*} - \frac{a^{3} d^{3} + 4 a^{2} b c d^{2} + 10 a b^{2} c^{2} d + 20 b^{3} c^{3} + 35 b^{3} d^{3} x^{3} + x^{2} \left (21 a b^{2} d^{3} + 84 b^{3} c d^{2}\right ) + x \left (7 a^{2} b d^{3} + 28 a b^{2} c d^{2} + 70 b^{3} c^{2} d\right )}{140 a^{7} b^{4} + 980 a^{6} b^{5} x + 2940 a^{5} b^{6} x^{2} + 4900 a^{4} b^{7} x^{3} + 4900 a^{3} b^{8} x^{4} + 2940 a^{2} b^{9} x^{5} + 980 a b^{10} x^{6} + 140 b^{11} x^{7}} \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)**11,x)

[Out]

-(a**3*d**3 + 4*a**2*b*c*d**2 + 10*a*b**2*c**2*d + 20*b**3*c**3 + 35*b**3*d**3*x**3 + x**2*(21*a*b**2*d**3 + 8
4*b**3*c*d**2) + x*(7*a**2*b*d**3 + 28*a*b**2*c*d**2 + 70*b**3*c**2*d))/(140*a**7*b**4 + 980*a**6*b**5*x + 294
0*a**5*b**6*x**2 + 4900*a**4*b**7*x**3 + 4900*a**3*b**8*x**4 + 2940*a**2*b**9*x**5 + 980*a*b**10*x**6 + 140*b*
*11*x**7)

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Giac [A]  time = 1.26302, size = 154, normalized size = 1.67 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + 84 \, b^{3} c d^{2} x^{2} + 21 \, a b^{2} d^{3} x^{2} + 70 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 7 \, a^{2} b d^{3} x + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3}}{140 \,{\left (b x + a\right )}^{7} 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)^11,x, algorithm="giac")

[Out]

-1/140*(35*b^3*d^3*x^3 + 84*b^3*c*d^2*x^2 + 21*a*b^2*d^3*x^2 + 70*b^3*c^2*d*x + 28*a*b^2*c*d^2*x + 7*a^2*b*d^3
*x + 20*b^3*c^3 + 10*a*b^2*c^2*d + 4*a^2*b*c*d^2 + a^3*d^3)/((b*x + a)^7*b^4)