3.2170 \(\int (a+b x) (a c+b c x)^m (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=24 \[ \frac{(a c+b c x)^{m+8}}{b c^8 (m+8)} \]

[Out]

(a*c + b*c*x)^(8 + m)/(b*c^8*(8 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.016604, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {21, 27, 32} \[ \frac{(a c+b c x)^{m+8}}{b c^8 (m+8)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c + b*c*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a*c + b*c*x)^(8 + m)/(b*c^8*(8 + m))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int (a+b x) (a c+b c x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\frac{\int (a c+b c x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx}{c}\\ &=\frac{\int (a+b x)^6 (a c+b c x)^{1+m} \, dx}{c}\\ &=\frac{\int (a c+b c x)^{7+m} \, dx}{c^7}\\ &=\frac{(a c+b c x)^{8+m}}{b c^8 (8+m)}\\ \end{align*}

Mathematica [A]  time = 0.0222249, size = 25, normalized size = 1.04 \[ \frac{(a+b x)^8 (c (a+b x))^m}{b (m+8)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a*c + b*c*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((a + b*x)^8*(c*(a + b*x))^m)/(b*(8 + m))

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 45, normalized size = 1.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{2} \left ( bcx+ac \right ) ^{m} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{3}}{b \left ( 8+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b*c*x+a*c)^m*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

(b*x+a)^2/b/(8+m)*(b*c*x+a*c)^m*(b^2*x^2+2*a*b*x+a^2)^3

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b*c*x+a*c)^m*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.62922, size = 211, normalized size = 8.79 \begin{align*} \frac{{\left (b^{8} x^{8} + 8 \, a b^{7} x^{7} + 28 \, a^{2} b^{6} x^{6} + 56 \, a^{3} b^{5} x^{5} + 70 \, a^{4} b^{4} x^{4} + 56 \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} x^{2} + 8 \, a^{7} b x + a^{8}\right )}{\left (b c x + a c\right )}^{m}}{b m + 8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b*c*x+a*c)^m*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

(b^8*x^8 + 8*a*b^7*x^7 + 28*a^2*b^6*x^6 + 56*a^3*b^5*x^5 + 70*a^4*b^4*x^4 + 56*a^5*b^3*x^3 + 28*a^6*b^2*x^2 +
8*a^7*b*x + a^8)*(b*c*x + a*c)^m/(b*m + 8*b)

________________________________________________________________________________________

Sympy [A]  time = 3.80708, size = 270, normalized size = 11.25 \begin{align*} \begin{cases} \frac{x}{a c^{8}} & \text{for}\: b = 0 \wedge m = -8 \\a^{7} x \left (a c\right )^{m} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + x \right )}}{b c^{8}} & \text{for}\: m = -8 \\\frac{a^{8} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac{8 a^{7} b x \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac{28 a^{6} b^{2} x^{2} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac{56 a^{5} b^{3} x^{3} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac{70 a^{4} b^{4} x^{4} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac{56 a^{3} b^{5} x^{5} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac{28 a^{2} b^{6} x^{6} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac{8 a b^{7} x^{7} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac{b^{8} x^{8} \left (a c + b c x\right )^{m}}{b m + 8 b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b*c*x+a*c)**m*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Piecewise((x/(a*c**8), Eq(b, 0) & Eq(m, -8)), (a**7*x*(a*c)**m, Eq(b, 0)), (log(a/b + x)/(b*c**8), Eq(m, -8)),
(a**8*(a*c + b*c*x)**m/(b*m + 8*b) + 8*a**7*b*x*(a*c + b*c*x)**m/(b*m + 8*b) + 28*a**6*b**2*x**2*(a*c + b*c*x
)**m/(b*m + 8*b) + 56*a**5*b**3*x**3*(a*c + b*c*x)**m/(b*m + 8*b) + 70*a**4*b**4*x**4*(a*c + b*c*x)**m/(b*m +
8*b) + 56*a**3*b**5*x**5*(a*c + b*c*x)**m/(b*m + 8*b) + 28*a**2*b**6*x**6*(a*c + b*c*x)**m/(b*m + 8*b) + 8*a*b
**7*x**7*(a*c + b*c*x)**m/(b*m + 8*b) + b**8*x**8*(a*c + b*c*x)**m/(b*m + 8*b), True))

________________________________________________________________________________________

Giac [B]  time = 1.16301, size = 247, normalized size = 10.29 \begin{align*} \frac{{\left (b c x + a c\right )}^{m} b^{8} x^{8} + 8 \,{\left (b c x + a c\right )}^{m} a b^{7} x^{7} + 28 \,{\left (b c x + a c\right )}^{m} a^{2} b^{6} x^{6} + 56 \,{\left (b c x + a c\right )}^{m} a^{3} b^{5} x^{5} + 70 \,{\left (b c x + a c\right )}^{m} a^{4} b^{4} x^{4} + 56 \,{\left (b c x + a c\right )}^{m} a^{5} b^{3} x^{3} + 28 \,{\left (b c x + a c\right )}^{m} a^{6} b^{2} x^{2} + 8 \,{\left (b c x + a c\right )}^{m} a^{7} b x +{\left (b c x + a c\right )}^{m} a^{8}}{b m + 8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b*c*x+a*c)^m*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

((b*c*x + a*c)^m*b^8*x^8 + 8*(b*c*x + a*c)^m*a*b^7*x^7 + 28*(b*c*x + a*c)^m*a^2*b^6*x^6 + 56*(b*c*x + a*c)^m*a
^3*b^5*x^5 + 70*(b*c*x + a*c)^m*a^4*b^4*x^4 + 56*(b*c*x + a*c)^m*a^5*b^3*x^3 + 28*(b*c*x + a*c)^m*a^6*b^2*x^2
+ 8*(b*c*x + a*c)^m*a^7*b*x + (b*c*x + a*c)^m*a^8)/(b*m + 8*b)