3.1798 \(\int \frac{(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^{12}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac{3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac{(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac{d^3}{5 b^4 (a+b x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0551401, 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} \[ -\frac{d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac{3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac{(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac{d^3}{5 b^4 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^12,x]

[Out]

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

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)^{12}} \, dx &=\int \frac{(c+d x)^3}{(a+b x)^9} \, dx\\ &=\int \left (\frac{(b c-a d)^3}{b^3 (a+b x)^9}+\frac{3 d (b c-a d)^2}{b^3 (a+b x)^8}+\frac{3 d^2 (b c-a d)}{b^3 (a+b x)^7}+\frac{d^3}{b^3 (a+b x)^6}\right ) \, dx\\ &=-\frac{(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac{3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac{d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac{d^3}{5 b^4 (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0354081, size = 97, normalized size = 1.05 \[ -\frac{a^2 b d^2 (5 c+8 d x)+a^3 d^3+a b^2 d \left (15 c^2+40 c d x+28 d^2 x^2\right )+b^3 \left (120 c^2 d x+35 c^3+140 c d^2 x^2+56 d^3 x^3\right )}{280 b^4 (a+b x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^12,x]

[Out]

-(a^3*d^3 + a^2*b*d^2*(5*c + 8*d*x) + a*b^2*d*(15*c^2 + 40*c*d*x + 28*d^2*x^2) + b^3*(35*c^3 + 120*c^2*d*x + 1
40*c*d^2*x^2 + 56*d^3*x^3))/(280*b^4*(a + b*x)^8)

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Maple [A]  time = 0.044, size = 122, normalized size = 1.3 \begin{align*} -{\frac{{d}^{3}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{3\,d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{7\,{b}^{4} \left ( bx+a \right ) ^{7}}}-{\frac{-{a}^{3}{d}^{3}+3\,cb{a}^{2}{d}^{2}-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{8\,{b}^{4} \left ( bx+a \right ) ^{8}}}+{\frac{{d}^{2} \left ( ad-bc \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)^12,x)

[Out]

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

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Maxima [B]  time = 1.0898, size = 261, normalized size = 2.84 \begin{align*} -\frac{56 \, b^{3} d^{3} x^{3} + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3} + 28 \,{\left (5 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 8 \,{\left (15 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{280 \,{\left (b^{12} x^{8} + 8 \, a b^{11} x^{7} + 28 \, a^{2} b^{10} x^{6} + 56 \, a^{3} b^{9} x^{5} + 70 \, a^{4} b^{8} x^{4} + 56 \, a^{5} b^{7} x^{3} + 28 \, a^{6} b^{6} x^{2} + 8 \, a^{7} b^{5} x + a^{8} 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)^12,x, algorithm="maxima")

[Out]

-1/280*(56*b^3*d^3*x^3 + 35*b^3*c^3 + 15*a*b^2*c^2*d + 5*a^2*b*c*d^2 + a^3*d^3 + 28*(5*b^3*c*d^2 + a*b^2*d^3)*
x^2 + 8*(15*b^3*c^2*d + 5*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^12*x^8 + 8*a*b^11*x^7 + 28*a^2*b^10*x^6 + 56*a^3*b^9*
x^5 + 70*a^4*b^8*x^4 + 56*a^5*b^7*x^3 + 28*a^6*b^6*x^2 + 8*a^7*b^5*x + a^8*b^4)

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Fricas [B]  time = 1.62724, size = 406, normalized size = 4.41 \begin{align*} -\frac{56 \, b^{3} d^{3} x^{3} + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3} + 28 \,{\left (5 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 8 \,{\left (15 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{280 \,{\left (b^{12} x^{8} + 8 \, a b^{11} x^{7} + 28 \, a^{2} b^{10} x^{6} + 56 \, a^{3} b^{9} x^{5} + 70 \, a^{4} b^{8} x^{4} + 56 \, a^{5} b^{7} x^{3} + 28 \, a^{6} b^{6} x^{2} + 8 \, a^{7} b^{5} x + a^{8} 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)^12,x, algorithm="fricas")

[Out]

-1/280*(56*b^3*d^3*x^3 + 35*b^3*c^3 + 15*a*b^2*c^2*d + 5*a^2*b*c*d^2 + a^3*d^3 + 28*(5*b^3*c*d^2 + a*b^2*d^3)*
x^2 + 8*(15*b^3*c^2*d + 5*a*b^2*c*d^2 + a^2*b*d^3)*x)/(b^12*x^8 + 8*a*b^11*x^7 + 28*a^2*b^10*x^6 + 56*a^3*b^9*
x^5 + 70*a^4*b^8*x^4 + 56*a^5*b^7*x^3 + 28*a^6*b^6*x^2 + 8*a^7*b^5*x + a^8*b^4)

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Sympy [B]  time = 25.5739, size = 206, normalized size = 2.24 \begin{align*} - \frac{a^{3} d^{3} + 5 a^{2} b c d^{2} + 15 a b^{2} c^{2} d + 35 b^{3} c^{3} + 56 b^{3} d^{3} x^{3} + x^{2} \left (28 a b^{2} d^{3} + 140 b^{3} c d^{2}\right ) + x \left (8 a^{2} b d^{3} + 40 a b^{2} c d^{2} + 120 b^{3} c^{2} d\right )}{280 a^{8} b^{4} + 2240 a^{7} b^{5} x + 7840 a^{6} b^{6} x^{2} + 15680 a^{5} b^{7} x^{3} + 19600 a^{4} b^{8} x^{4} + 15680 a^{3} b^{9} x^{5} + 7840 a^{2} b^{10} x^{6} + 2240 a b^{11} x^{7} + 280 b^{12} x^{8}} \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)**12,x)

[Out]

-(a**3*d**3 + 5*a**2*b*c*d**2 + 15*a*b**2*c**2*d + 35*b**3*c**3 + 56*b**3*d**3*x**3 + x**2*(28*a*b**2*d**3 + 1
40*b**3*c*d**2) + x*(8*a**2*b*d**3 + 40*a*b**2*c*d**2 + 120*b**3*c**2*d))/(280*a**8*b**4 + 2240*a**7*b**5*x +
7840*a**6*b**6*x**2 + 15680*a**5*b**7*x**3 + 19600*a**4*b**8*x**4 + 15680*a**3*b**9*x**5 + 7840*a**2*b**10*x**
6 + 2240*a*b**11*x**7 + 280*b**12*x**8)

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Giac [A]  time = 1.29392, size = 154, normalized size = 1.67 \begin{align*} -\frac{56 \, b^{3} d^{3} x^{3} + 140 \, b^{3} c d^{2} x^{2} + 28 \, a b^{2} d^{3} x^{2} + 120 \, b^{3} c^{2} d x + 40 \, a b^{2} c d^{2} x + 8 \, a^{2} b d^{3} x + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}}{280 \,{\left (b x + a\right )}^{8} 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)^12,x, algorithm="giac")

[Out]

-1/280*(56*b^3*d^3*x^3 + 140*b^3*c*d^2*x^2 + 28*a*b^2*d^3*x^2 + 120*b^3*c^2*d*x + 40*a*b^2*c*d^2*x + 8*a^2*b*d
^3*x + 35*b^3*c^3 + 15*a*b^2*c^2*d + 5*a^2*b*c*d^2 + a^3*d^3)/((b*x + a)^8*b^4)