3.20.33 \(\int (A+B x) (a c+b c x)^m (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=58 \[ \frac {(A b-a B) (a c+b c x)^{m+7}}{b^2 c^7 (m+7)}+\frac {B (a c+b c x)^{m+8}}{b^2 c^8 (m+8)} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00,
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} \begin {gather*} \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)} \end {gather*}
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 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 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 = 0.04, size = 48, normalized size = 0.83 \begin {gather*} \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)} \end {gather*}
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))
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (a c+b c x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(A + B*x)*(a*c + b*c*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
Defer[IntegrateAlgebraic][(A + B*x)*(a*c + b*c*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3, x]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 350, normalized size = 6.03 \begin {gather*} \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 {gather*}
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)
________________________________________________________________________________________
giac [B] time = 0.28, size = 695, normalized size = 11.98 \begin {gather*} \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 {gather*}
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)
________________________________________________________________________________________
maple [A] time = 0.05, size = 71, normalized size = 1.22 \begin {gather*} \frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3} \left (B b m x +A b m +7 B b x +8 A b -B a \right ) \left (b x +a \right ) \left (b c x +a c \right )^{m}}{\left (m^{2}+15 m +56\right ) b^{2}} \end {gather*}
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)
________________________________________________________________________________________
maxima [B] time = 1.01, size = 2113, normalized size = 36.43
result too large to display
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]
(b^2*c^m*(m + 1)*x^2 + a*b*c^m*m*x - a^2*c^m)*(b*x + a)^m*B*a^6/((m^2 + 3*m + 2)*b^2) + 6*(b^2*c^m*(m + 1)*x^2
+ a*b*c^m*m*x - a^2*c^m)*(b*x + a)^m*A*a^5/((m^2 + 3*m + 2)*b) + 6*((m^2 + 3*m + 2)*b^3*c^m*x^3 + (m^2 + m)*a
*b^2*c^m*x^2 - 2*a^2*b*c^m*m*x + 2*a^3*c^m)*(b*x + a)^m*B*a^5/((m^3 + 6*m^2 + 11*m + 6)*b^2) + 15*((m^2 + 3*m
+ 2)*b^3*c^m*x^3 + (m^2 + m)*a*b^2*c^m*x^2 - 2*a^2*b*c^m*m*x + 2*a^3*c^m)*(b*x + a)^m*A*a^4/((m^3 + 6*m^2 + 11
*m + 6)*b) + (b*c*x + a*c)^(m + 1)*A*a^6/(b*c*(m + 1)) + 15*((m^3 + 6*m^2 + 11*m + 6)*b^4*c^m*x^4 + (m^3 + 3*m
^2 + 2*m)*a*b^3*c^m*x^3 - 3*(m^2 + m)*a^2*b^2*c^m*x^2 + 6*a^3*b*c^m*m*x - 6*a^4*c^m)*(b*x + a)^m*B*a^4/((m^4 +
10*m^3 + 35*m^2 + 50*m + 24)*b^2) + 20*((m^3 + 6*m^2 + 11*m + 6)*b^4*c^m*x^4 + (m^3 + 3*m^2 + 2*m)*a*b^3*c^m*
x^3 - 3*(m^2 + m)*a^2*b^2*c^m*x^2 + 6*a^3*b*c^m*m*x - 6*a^4*c^m)*(b*x + a)^m*A*a^3/((m^4 + 10*m^3 + 35*m^2 + 5
0*m + 24)*b) + 20*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^5*c^m*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a*b^4*c^m*x^
4 - 4*(m^3 + 3*m^2 + 2*m)*a^2*b^3*c^m*x^3 + 12*(m^2 + m)*a^3*b^2*c^m*x^2 - 24*a^4*b*c^m*m*x + 24*a^5*c^m)*(b*x
+ a)^m*B*a^3/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^2) + 15*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*
b^5*c^m*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a*b^4*c^m*x^4 - 4*(m^3 + 3*m^2 + 2*m)*a^2*b^3*c^m*x^3 + 12*(m^2 + m
)*a^3*b^2*c^m*x^2 - 24*a^4*b*c^m*m*x + 24*a^5*c^m)*(b*x + a)^m*A*a^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m
+ 120)*b) + 15*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^6*c^m*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2
+ 24*m)*a*b^5*c^m*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^2*b^4*c^m*x^4 + 20*(m^3 + 3*m^2 + 2*m)*a^3*b^3*c^m*x
^3 - 60*(m^2 + m)*a^4*b^2*c^m*x^2 + 120*a^5*b*c^m*m*x - 120*a^6*c^m)*(b*x + a)^m*B*a^2/((m^6 + 21*m^5 + 175*m^
4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b^2) + 6*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^6*c^m*x^6 +
(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a*b^5*c^m*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^2*b^4*c^m*x^4 + 20*(
m^3 + 3*m^2 + 2*m)*a^3*b^3*c^m*x^3 - 60*(m^2 + m)*a^4*b^2*c^m*x^2 + 120*a^5*b*c^m*m*x - 120*a^6*c^m)*(b*x + a)
^m*A*a/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b) + 6*((m^6 + 21*m^5 + 175*m^4 + 735*m^3
+ 1624*m^2 + 1764*m + 720)*b^7*c^m*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*a*b^6*c^m*x^6 -
6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^2*b^5*c^m*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*a^3*b^4*c^m*x^4 -
120*(m^3 + 3*m^2 + 2*m)*a^4*b^3*c^m*x^3 + 360*(m^2 + m)*a^5*b^2*c^m*x^2 - 720*a^6*b*c^m*m*x + 720*a^7*c^m)*(b*
x + a)^m*B*a/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*b^2) + ((m^6 + 21*m^
5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b^7*c^m*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 12
0*m)*a*b^6*c^m*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^2*b^5*c^m*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*
m)*a^3*b^4*c^m*x^4 - 120*(m^3 + 3*m^2 + 2*m)*a^4*b^3*c^m*x^3 + 360*(m^2 + m)*a^5*b^2*c^m*x^2 - 720*a^6*b*c^m*m
*x + 720*a^7*c^m)*(b*x + a)^m*A/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*b
) + ((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*b^8*c^m*x^8 + (m^7 + 21*m^6 +
175*m^5 + 735*m^4 + 1624*m^3 + 1764*m^2 + 720*m)*a*b^7*c^m*x^7 - 7*(m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2
+ 120*m)*a^2*b^6*c^m*x^6 + 42*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^3*b^5*c^m*x^5 - 210*(m^4 + 6*m^3 + 11
*m^2 + 6*m)*a^4*b^4*c^m*x^4 + 840*(m^3 + 3*m^2 + 2*m)*a^5*b^3*c^m*x^3 - 2520*(m^2 + m)*a^6*b^2*c^m*x^2 + 5040*
a^7*b*c^m*m*x - 5040*a^8*c^m)*(b*x + a)^m*B/((m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 1181
24*m^2 + 109584*m + 40320)*b^2)
________________________________________________________________________________________
mupad [B] time = 2.55, size = 319, normalized size = 5.50 \begin {gather*} {\left (a\,c+b\,c\,x\right )}^m\,\left (\frac {a^7\,\left (8\,A\,b-B\,a+A\,b\,m\right )}{b^2\,\left (m^2+15\,m+56\right )}+\frac {7\,a^5\,x^2\,\left (24\,A\,b+4\,B\,a+3\,A\,b\,m+B\,a\,m\right )}{m^2+15\,m+56}+\frac {b^5\,x^7\,\left (8\,A\,b+48\,B\,a+A\,b\,m+7\,B\,a\,m\right )}{m^2+15\,m+56}+\frac {35\,a^3\,b^2\,x^4\,\left (8\,A\,b+6\,B\,a+A\,b\,m+B\,a\,m\right )}{m^2+15\,m+56}+\frac {7\,a^2\,b^3\,x^5\,\left (24\,A\,b+32\,B\,a+3\,A\,b\,m+5\,B\,a\,m\right )}{m^2+15\,m+56}+\frac {B\,b^6\,x^8\,\left (m+7\right )}{m^2+15\,m+56}+\frac {a^6\,x\,\left (56\,A\,b+7\,A\,b\,m+B\,a\,m\right )}{b\,\left (m^2+15\,m+56\right )}+\frac {7\,a\,b^4\,x^6\,\left (8\,A\,b+20\,B\,a+A\,b\,m+3\,B\,a\,m\right )}{m^2+15\,m+56}+\frac {7\,a^4\,b\,x^3\,\left (40\,A\,b+16\,B\,a+5\,A\,b\,m+3\,B\,a\,m\right )}{m^2+15\,m+56}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*c + b*c*x)^m*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
[Out]
(a*c + b*c*x)^m*((a^7*(8*A*b - B*a + A*b*m))/(b^2*(15*m + m^2 + 56)) + (7*a^5*x^2*(24*A*b + 4*B*a + 3*A*b*m +
B*a*m))/(15*m + m^2 + 56) + (b^5*x^7*(8*A*b + 48*B*a + A*b*m + 7*B*a*m))/(15*m + m^2 + 56) + (35*a^3*b^2*x^4*(
8*A*b + 6*B*a + A*b*m + B*a*m))/(15*m + m^2 + 56) + (7*a^2*b^3*x^5*(24*A*b + 32*B*a + 3*A*b*m + 5*B*a*m))/(15*
m + m^2 + 56) + (B*b^6*x^8*(m + 7))/(15*m + m^2 + 56) + (a^6*x*(56*A*b + 7*A*b*m + B*a*m))/(b*(15*m + m^2 + 56
)) + (7*a*b^4*x^6*(8*A*b + 20*B*a + A*b*m + 3*B*a*m))/(15*m + m^2 + 56) + (7*a^4*b*x^3*(40*A*b + 16*B*a + 5*A*
b*m + 3*B*a*m))/(15*m + m^2 + 56))
________________________________________________________________________________________
sympy [A] time = 7.82, size = 1488, normalized size = 25.66
result too large to display
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)), (-A*b/(a*b**2*c**8 + b**3*c**8*x) + B*a*log(a/b + x)/(a*
b**2*c**8 + b**3*c**8*x) + B*a/(a*b**2*c**8 + b**3*c**8*x) + B*b*x*log(a/b + x)/(a*b**2*c**8 + b**3*c**8*x), E
q(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))
________________________________________________________________________________________