3.2163 \(\int (A+B x) (a c+b c x)^m (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=58 \[ \frac{(A b-a B) (a c+b c x)^{m+7}}{b^2 c^7 (m+7)}+\frac{B (a c+b c x)^{m+8}}{b^2 c^8 (m+8)} \]

[Out]

((A*b - a*B)*(a*c + b*c*x)^(7 + m))/(b^2*c^7*(7 + m)) + (B*(a*c + b*c*x)^(8 + m))/(b^2*c^8*(8 + m))

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Rubi [A]  time = 0.0416603, antiderivative size = 58, 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 = {27, 21, 43} \[ \frac{(A b-a B) (a c+b c x)^{m+7}}{b^2 c^7 (m+7)}+\frac{B (a c+b c x)^{m+8}}{b^2 c^8 (m+8)} \]

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*b - a*B)*(a*c + b*c*x)^(7 + m))/(b^2*c^7*(7 + m)) + (B*(a*c + b*c*x)^(8 + m))/(b^2*c^8*(8 + m))

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

Mathematica [A]  time = 0.0472329, size = 48, normalized size = 0.83 \[ \frac{(a+b x)^7 (c (a+b x))^m (-a B+A b (m+8)+b B (m+7) x)}{b^2 (m+7) (m+8)} \]

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)^7*(c*(a + b*x))^m*(-(a*B) + A*b*(8 + m) + b*B*(7 + m)*x))/(b^2*(7 + m)*(8 + m))

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Maple [A]  time = 0.004, size = 71, normalized size = 1.2 \begin{align*}{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{3} \left ( bcx+ac \right ) ^{m} \left ( Bbmx+Abm+7\,bBx+8\,Ab-aB \right ) \left ( bx+a \right ) }{{b}^{2} \left ({m}^{2}+15\,m+56 \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^2*x^2+2*a*b*x+a^2)^3*(b*c*x+a*c)^m*(B*b*m*x+A*b*m+7*B*b*x+8*A*b-B*a)*(b*x+a)/b^2/(m^2+15*m+56)

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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

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Fricas [B]  time = 1.60313, size = 729, normalized size = 12.57 \begin{align*} \frac{{\left (A a^{7} b m - B a^{8} + 8 \, A a^{7} b +{\left (B b^{8} m + 7 \, B b^{8}\right )} x^{8} +{\left (48 \, B a b^{7} + 8 \, A b^{8} +{\left (7 \, B a b^{7} + A b^{8}\right )} m\right )} x^{7} + 7 \,{\left (20 \, B a^{2} b^{6} + 8 \, A a b^{7} +{\left (3 \, B a^{2} b^{6} + A a b^{7}\right )} m\right )} x^{6} + 7 \,{\left (32 \, B a^{3} b^{5} + 24 \, A a^{2} b^{6} +{\left (5 \, B a^{3} b^{5} + 3 \, A a^{2} b^{6}\right )} m\right )} x^{5} + 35 \,{\left (6 \, B a^{4} b^{4} + 8 \, A a^{3} b^{5} +{\left (B a^{4} b^{4} + A a^{3} b^{5}\right )} m\right )} x^{4} + 7 \,{\left (16 \, B a^{5} b^{3} + 40 \, A a^{4} b^{4} +{\left (3 \, B a^{5} b^{3} + 5 \, A a^{4} b^{4}\right )} m\right )} x^{3} + 7 \,{\left (4 \, B a^{6} b^{2} + 24 \, A a^{5} b^{3} +{\left (B a^{6} b^{2} + 3 \, A a^{5} b^{3}\right )} m\right )} x^{2} +{\left (56 \, A a^{6} b^{2} +{\left (B a^{7} b + 7 \, A a^{6} b^{2}\right )} m\right )} x\right )}{\left (b c x + a c\right )}^{m}}{b^{2} m^{2} + 15 \, b^{2} m + 56 \, b^{2}} \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]

(A*a^7*b*m - B*a^8 + 8*A*a^7*b + (B*b^8*m + 7*B*b^8)*x^8 + (48*B*a*b^7 + 8*A*b^8 + (7*B*a*b^7 + A*b^8)*m)*x^7
+ 7*(20*B*a^2*b^6 + 8*A*a*b^7 + (3*B*a^2*b^6 + A*a*b^7)*m)*x^6 + 7*(32*B*a^3*b^5 + 24*A*a^2*b^6 + (5*B*a^3*b^5
+ 3*A*a^2*b^6)*m)*x^5 + 35*(6*B*a^4*b^4 + 8*A*a^3*b^5 + (B*a^4*b^4 + A*a^3*b^5)*m)*x^4 + 7*(16*B*a^5*b^3 + 40
*A*a^4*b^4 + (3*B*a^5*b^3 + 5*A*a^4*b^4)*m)*x^3 + 7*(4*B*a^6*b^2 + 24*A*a^5*b^3 + (B*a^6*b^2 + 3*A*a^5*b^3)*m)
*x^2 + (56*A*a^6*b^2 + (B*a^7*b + 7*A*a^6*b^2)*m)*x)*(b*c*x + a*c)^m/(b^2*m^2 + 15*b^2*m + 56*b^2)

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Sympy [A]  time = 6.25277, size = 1593, normalized size = 27.47 \begin{align*} \text{result too large to display} \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((a**6*(a*c)**m*(A*x + B*x**2/2), Eq(b, 0)), (-3*A*a*b/(7*a**2*b**2*c**8 + 7*a*b**3*c**8*x) + 4*A*b**
2*x/(7*a**2*b**2*c**8 + 7*a*b**3*c**8*x) + 7*B*a**2*log(a/b + x)/(7*a**2*b**2*c**8 + 7*a*b**3*c**8*x) + 3*B*a*
*2/(7*a**2*b**2*c**8 + 7*a*b**3*c**8*x) + 7*B*a*b*x*log(a/b + x)/(7*a**2*b**2*c**8 + 7*a*b**3*c**8*x) - 4*B*a*
b*x/(7*a**2*b**2*c**8 + 7*a*b**3*c**8*x), Eq(m, -8)), (A*log(a/b + x)/(b*c**7) - B*a*log(a/b + x)/(b**2*c**7)
+ B*x/(b*c**7), Eq(m, -7)), (A*a**7*b*m*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 8*A*a**7*b*(a*c +
b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 7*A*a**6*b**2*m*x*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*
b**2) + 56*A*a**6*b**2*x*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 21*A*a**5*b**3*m*x**2*(a*c + b*c
*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 168*A*a**5*b**3*x**2*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b
**2) + 35*A*a**4*b**4*m*x**3*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 280*A*a**4*b**4*x**3*(a*c +
b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 35*A*a**3*b**5*m*x**4*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m +
56*b**2) + 280*A*a**3*b**5*x**4*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 21*A*a**2*b**6*m*x**5*(a*
c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 168*A*a**2*b**6*x**5*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m
+ 56*b**2) + 7*A*a*b**7*m*x**6*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 56*A*a*b**7*x**6*(a*c + b
*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + A*b**8*m*x**7*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2)
+ 8*A*b**8*x**7*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) - B*a**8*(a*c + b*c*x)**m/(b**2*m**2 + 15*b
**2*m + 56*b**2) + B*a**7*b*m*x*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 7*B*a**6*b**2*m*x**2*(a*c
+ b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 28*B*a**6*b**2*x**2*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m +
56*b**2) + 21*B*a**5*b**3*m*x**3*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 112*B*a**5*b**3*x**3*(a
*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 35*B*a**4*b**4*m*x**4*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2
*m + 56*b**2) + 210*B*a**4*b**4*x**4*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 35*B*a**3*b**5*m*x**
5*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 224*B*a**3*b**5*x**5*(a*c + b*c*x)**m/(b**2*m**2 + 15*b
**2*m + 56*b**2) + 21*B*a**2*b**6*m*x**6*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 140*B*a**2*b**6*
x**6*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 7*B*a*b**7*m*x**7*(a*c + b*c*x)**m/(b**2*m**2 + 15*b
**2*m + 56*b**2) + 48*B*a*b**7*x**7*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + B*b**8*m*x**8*(a*c +
b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2) + 7*B*b**8*x**8*(a*c + b*c*x)**m/(b**2*m**2 + 15*b**2*m + 56*b**2)
, True))

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Giac [B]  time = 1.19284, size = 938, normalized size = 16.17 \begin{align*} \frac{{\left (b c x + a c\right )}^{m} B b^{8} m x^{8} + 7 \,{\left (b c x + a c\right )}^{m} B a b^{7} m x^{7} +{\left (b c x + a c\right )}^{m} A b^{8} m x^{7} + 7 \,{\left (b c x + a c\right )}^{m} B b^{8} x^{8} + 21 \,{\left (b c x + a c\right )}^{m} B a^{2} b^{6} m x^{6} + 7 \,{\left (b c x + a c\right )}^{m} A a b^{7} m x^{6} + 48 \,{\left (b c x + a c\right )}^{m} B a b^{7} x^{7} + 8 \,{\left (b c x + a c\right )}^{m} A b^{8} x^{7} + 35 \,{\left (b c x + a c\right )}^{m} B a^{3} b^{5} m x^{5} + 21 \,{\left (b c x + a c\right )}^{m} A a^{2} b^{6} m x^{5} + 140 \,{\left (b c x + a c\right )}^{m} B a^{2} b^{6} x^{6} + 56 \,{\left (b c x + a c\right )}^{m} A a b^{7} x^{6} + 35 \,{\left (b c x + a c\right )}^{m} B a^{4} b^{4} m x^{4} + 35 \,{\left (b c x + a c\right )}^{m} A a^{3} b^{5} m x^{4} + 224 \,{\left (b c x + a c\right )}^{m} B a^{3} b^{5} x^{5} + 168 \,{\left (b c x + a c\right )}^{m} A a^{2} b^{6} x^{5} + 21 \,{\left (b c x + a c\right )}^{m} B a^{5} b^{3} m x^{3} + 35 \,{\left (b c x + a c\right )}^{m} A a^{4} b^{4} m x^{3} + 210 \,{\left (b c x + a c\right )}^{m} B a^{4} b^{4} x^{4} + 280 \,{\left (b c x + a c\right )}^{m} A a^{3} b^{5} x^{4} + 7 \,{\left (b c x + a c\right )}^{m} B a^{6} b^{2} m x^{2} + 21 \,{\left (b c x + a c\right )}^{m} A a^{5} b^{3} m x^{2} + 112 \,{\left (b c x + a c\right )}^{m} B a^{5} b^{3} x^{3} + 280 \,{\left (b c x + a c\right )}^{m} A a^{4} b^{4} x^{3} +{\left (b c x + a c\right )}^{m} B a^{7} b m x + 7 \,{\left (b c x + a c\right )}^{m} A a^{6} b^{2} m x + 28 \,{\left (b c x + a c\right )}^{m} B a^{6} b^{2} x^{2} + 168 \,{\left (b c x + a c\right )}^{m} A a^{5} b^{3} x^{2} +{\left (b c x + a c\right )}^{m} A a^{7} b m + 56 \,{\left (b c x + a c\right )}^{m} A a^{6} b^{2} x -{\left (b c x + a c\right )}^{m} B a^{8} + 8 \,{\left (b c x + a c\right )}^{m} A a^{7} b}{b^{2} m^{2} + 15 \, b^{2} m + 56 \, b^{2}} \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*b^8*m*x^8 + 7*(b*c*x + a*c)^m*B*a*b^7*m*x^7 + (b*c*x + a*c)^m*A*b^8*m*x^7 + 7*(b*c*x + a*c)
^m*B*b^8*x^8 + 21*(b*c*x + a*c)^m*B*a^2*b^6*m*x^6 + 7*(b*c*x + a*c)^m*A*a*b^7*m*x^6 + 48*(b*c*x + a*c)^m*B*a*b
^7*x^7 + 8*(b*c*x + a*c)^m*A*b^8*x^7 + 35*(b*c*x + a*c)^m*B*a^3*b^5*m*x^5 + 21*(b*c*x + a*c)^m*A*a^2*b^6*m*x^5
+ 140*(b*c*x + a*c)^m*B*a^2*b^6*x^6 + 56*(b*c*x + a*c)^m*A*a*b^7*x^6 + 35*(b*c*x + a*c)^m*B*a^4*b^4*m*x^4 + 3
5*(b*c*x + a*c)^m*A*a^3*b^5*m*x^4 + 224*(b*c*x + a*c)^m*B*a^3*b^5*x^5 + 168*(b*c*x + a*c)^m*A*a^2*b^6*x^5 + 21
*(b*c*x + a*c)^m*B*a^5*b^3*m*x^3 + 35*(b*c*x + a*c)^m*A*a^4*b^4*m*x^3 + 210*(b*c*x + a*c)^m*B*a^4*b^4*x^4 + 28
0*(b*c*x + a*c)^m*A*a^3*b^5*x^4 + 7*(b*c*x + a*c)^m*B*a^6*b^2*m*x^2 + 21*(b*c*x + a*c)^m*A*a^5*b^3*m*x^2 + 112
*(b*c*x + a*c)^m*B*a^5*b^3*x^3 + 280*(b*c*x + a*c)^m*A*a^4*b^4*x^3 + (b*c*x + a*c)^m*B*a^7*b*m*x + 7*(b*c*x +
a*c)^m*A*a^6*b^2*m*x + 28*(b*c*x + a*c)^m*B*a^6*b^2*x^2 + 168*(b*c*x + a*c)^m*A*a^5*b^3*x^2 + (b*c*x + a*c)^m*
A*a^7*b*m + 56*(b*c*x + a*c)^m*A*a^6*b^2*x - (b*c*x + a*c)^m*B*a^8 + 8*(b*c*x + a*c)^m*A*a^7*b)/(b^2*m^2 + 15*
b^2*m + 56*b^2)