3.9.66 \(\int (a+b x)^n (c+d x)^3 \, dx\)
Optimal. Leaf size=110 \[ \frac {3 d^2 (b c-a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac {(b c-a d)^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 d (b c-a d)^2 (a+b x)^{n+2}}{b^4 (n+2)}+\frac {d^3 (a+b x)^{n+4}}{b^4 (n+4)} \]
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Rubi [A] time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used =
{43} \begin {gather*} \frac {3 d^2 (b c-a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac {(b c-a d)^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 d (b c-a d)^2 (a+b x)^{n+2}}{b^4 (n+2)}+\frac {d^3 (a+b x)^{n+4}}{b^4 (n+4)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^n*(c + d*x)^3,x]
[Out]
((b*c - a*d)^3*(a + b*x)^(1 + n))/(b^4*(1 + n)) + (3*d*(b*c - a*d)^2*(a + b*x)^(2 + n))/(b^4*(2 + n)) + (3*d^2
*(b*c - a*d)*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d^3*(a + b*x)^(4 + n))/(b^4*(4 + n))
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)^n (c+d x)^3 \, dx &=\int \left (\frac {(b c-a d)^3 (a+b x)^n}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^{1+n}}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^{2+n}}{b^3}+\frac {d^3 (a+b x)^{3+n}}{b^3}\right ) \, dx\\ &=\frac {(b c-a d)^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {3 d (b c-a d)^2 (a+b x)^{2+n}}{b^4 (2+n)}+\frac {3 d^2 (b c-a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d^3 (a+b x)^{4+n}}{b^4 (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 94, normalized size = 0.85 \begin {gather*} \frac {(a+b x)^{n+1} \left (\frac {3 d^2 (a+b x)^2 (b c-a d)}{n+3}+\frac {3 d (a+b x) (b c-a d)^2}{n+2}+\frac {(b c-a d)^3}{n+1}+\frac {d^3 (a+b x)^3}{n+4}\right )}{b^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^n*(c + d*x)^3,x]
[Out]
((a + b*x)^(1 + n)*((b*c - a*d)^3/(1 + n) + (3*d*(b*c - a*d)^2*(a + b*x))/(2 + n) + (3*d^2*(b*c - a*d)*(a + b*
x)^2)/(3 + n) + (d^3*(a + b*x)^3)/(4 + n)))/b^4
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IntegrateAlgebraic [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^n (c+d x)^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(a + b*x)^n*(c + d*x)^3,x]
[Out]
Defer[IntegrateAlgebraic][(a + b*x)^n*(c + d*x)^3, x]
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fricas [B] time = 1.93, size = 497, normalized size = 4.52 \begin {gather*} \frac {{\left (a b^{3} c^{3} n^{3} + 24 \, a b^{3} c^{3} - 36 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (b^{4} d^{3} n^{3} + 6 \, b^{4} d^{3} n^{2} + 11 \, b^{4} d^{3} n + 6 \, b^{4} d^{3}\right )} x^{4} + {\left (24 \, b^{4} c d^{2} + {\left (3 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n^{3} + 3 \, {\left (7 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n^{2} + 2 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )} n^{2} + 3 \, {\left (12 \, b^{4} c^{2} d + {\left (b^{4} c^{2} d + a b^{3} c d^{2}\right )} n^{3} + {\left (8 \, b^{4} c^{2} d + 5 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} n^{2} + {\left (19 \, b^{4} c^{2} d + 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} n\right )} x^{2} + {\left (26 \, a b^{3} c^{3} - 21 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2}\right )} n + {\left (24 \, b^{4} c^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d\right )} n^{3} + 3 \, {\left (3 \, b^{4} c^{3} + 7 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2}\right )} n^{2} + 2 \, {\left (13 \, b^{4} c^{3} + 18 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x+c)^3,x, algorithm="fricas")
[Out]
(a*b^3*c^3*n^3 + 24*a*b^3*c^3 - 36*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 - 6*a^4*d^3 + (b^4*d^3*n^3 + 6*b^4*d^3*n^2 +
11*b^4*d^3*n + 6*b^4*d^3)*x^4 + (24*b^4*c*d^2 + (3*b^4*c*d^2 + a*b^3*d^3)*n^3 + 3*(7*b^4*c*d^2 + a*b^3*d^3)*n
^2 + 2*(21*b^4*c*d^2 + a*b^3*d^3)*n)*x^3 + 3*(3*a*b^3*c^3 - a^2*b^2*c^2*d)*n^2 + 3*(12*b^4*c^2*d + (b^4*c^2*d
+ a*b^3*c*d^2)*n^3 + (8*b^4*c^2*d + 5*a*b^3*c*d^2 - a^2*b^2*d^3)*n^2 + (19*b^4*c^2*d + 4*a*b^3*c*d^2 - a^2*b^2
*d^3)*n)*x^2 + (26*a*b^3*c^3 - 21*a^2*b^2*c^2*d + 6*a^3*b*c*d^2)*n + (24*b^4*c^3 + (b^4*c^3 + 3*a*b^3*c^2*d)*n
^3 + 3*(3*b^4*c^3 + 7*a*b^3*c^2*d - 2*a^2*b^2*c*d^2)*n^2 + 2*(13*b^4*c^3 + 18*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 +
3*a^3*b*d^3)*n)*x)*(b*x + a)^n/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)
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giac [B] time = 1.31, size = 833, normalized size = 7.57 \begin {gather*} \frac {{\left (b x + a\right )}^{n} b^{4} d^{3} n^{3} x^{4} + 3 \, {\left (b x + a\right )}^{n} b^{4} c d^{2} n^{3} x^{3} + {\left (b x + a\right )}^{n} a b^{3} d^{3} n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d^{3} n^{2} x^{4} + 3 \, {\left (b x + a\right )}^{n} b^{4} c^{2} d n^{3} x^{2} + 3 \, {\left (b x + a\right )}^{n} a b^{3} c d^{2} n^{3} x^{2} + 21 \, {\left (b x + a\right )}^{n} b^{4} c d^{2} n^{2} x^{3} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d^{3} n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d^{3} n x^{4} + {\left (b x + a\right )}^{n} b^{4} c^{3} n^{3} x + 3 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} d n^{3} x + 24 \, {\left (b x + a\right )}^{n} b^{4} c^{2} d n^{2} x^{2} + 15 \, {\left (b x + a\right )}^{n} a b^{3} c d^{2} n^{2} x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d^{3} n^{2} x^{2} + 42 \, {\left (b x + a\right )}^{n} b^{4} c d^{2} n x^{3} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d^{3} n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d^{3} x^{4} + {\left (b x + a\right )}^{n} a b^{3} c^{3} n^{3} + 9 \, {\left (b x + a\right )}^{n} b^{4} c^{3} n^{2} x + 21 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} d n^{2} x - 6 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c d^{2} n^{2} x + 57 \, {\left (b x + a\right )}^{n} b^{4} c^{2} d n x^{2} + 12 \, {\left (b x + a\right )}^{n} a b^{3} c d^{2} n x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d^{3} n x^{2} + 24 \, {\left (b x + a\right )}^{n} b^{4} c d^{2} x^{3} + 9 \, {\left (b x + a\right )}^{n} a b^{3} c^{3} n^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} d n^{2} + 26 \, {\left (b x + a\right )}^{n} b^{4} c^{3} n x + 36 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} d n x - 24 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c d^{2} n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d^{3} n x + 36 \, {\left (b x + a\right )}^{n} b^{4} c^{2} d x^{2} + 26 \, {\left (b x + a\right )}^{n} a b^{3} c^{3} n - 21 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} d n + 6 \, {\left (b x + a\right )}^{n} a^{3} b c d^{2} n + 24 \, {\left (b x + a\right )}^{n} b^{4} c^{3} x + 24 \, {\left (b x + a\right )}^{n} a b^{3} c^{3} - 36 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} d + 24 \, {\left (b x + a\right )}^{n} a^{3} b c d^{2} - 6 \, {\left (b x + a\right )}^{n} a^{4} d^{3}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x+c)^3,x, algorithm="giac")
[Out]
((b*x + a)^n*b^4*d^3*n^3*x^4 + 3*(b*x + a)^n*b^4*c*d^2*n^3*x^3 + (b*x + a)^n*a*b^3*d^3*n^3*x^3 + 6*(b*x + a)^n
*b^4*d^3*n^2*x^4 + 3*(b*x + a)^n*b^4*c^2*d*n^3*x^2 + 3*(b*x + a)^n*a*b^3*c*d^2*n^3*x^2 + 21*(b*x + a)^n*b^4*c*
d^2*n^2*x^3 + 3*(b*x + a)^n*a*b^3*d^3*n^2*x^3 + 11*(b*x + a)^n*b^4*d^3*n*x^4 + (b*x + a)^n*b^4*c^3*n^3*x + 3*(
b*x + a)^n*a*b^3*c^2*d*n^3*x + 24*(b*x + a)^n*b^4*c^2*d*n^2*x^2 + 15*(b*x + a)^n*a*b^3*c*d^2*n^2*x^2 - 3*(b*x
+ a)^n*a^2*b^2*d^3*n^2*x^2 + 42*(b*x + a)^n*b^4*c*d^2*n*x^3 + 2*(b*x + a)^n*a*b^3*d^3*n*x^3 + 6*(b*x + a)^n*b^
4*d^3*x^4 + (b*x + a)^n*a*b^3*c^3*n^3 + 9*(b*x + a)^n*b^4*c^3*n^2*x + 21*(b*x + a)^n*a*b^3*c^2*d*n^2*x - 6*(b*
x + a)^n*a^2*b^2*c*d^2*n^2*x + 57*(b*x + a)^n*b^4*c^2*d*n*x^2 + 12*(b*x + a)^n*a*b^3*c*d^2*n*x^2 - 3*(b*x + a)
^n*a^2*b^2*d^3*n*x^2 + 24*(b*x + a)^n*b^4*c*d^2*x^3 + 9*(b*x + a)^n*a*b^3*c^3*n^2 - 3*(b*x + a)^n*a^2*b^2*c^2*
d*n^2 + 26*(b*x + a)^n*b^4*c^3*n*x + 36*(b*x + a)^n*a*b^3*c^2*d*n*x - 24*(b*x + a)^n*a^2*b^2*c*d^2*n*x + 6*(b*
x + a)^n*a^3*b*d^3*n*x + 36*(b*x + a)^n*b^4*c^2*d*x^2 + 26*(b*x + a)^n*a*b^3*c^3*n - 21*(b*x + a)^n*a^2*b^2*c^
2*d*n + 6*(b*x + a)^n*a^3*b*c*d^2*n + 24*(b*x + a)^n*b^4*c^3*x + 24*(b*x + a)^n*a*b^3*c^3 - 36*(b*x + a)^n*a^2
*b^2*c^2*d + 24*(b*x + a)^n*a^3*b*c*d^2 - 6*(b*x + a)^n*a^4*d^3)/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n
+ 24*b^4)
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maple [B] time = 0.01, size = 389, normalized size = 3.54 \begin {gather*} -\frac {\left (-b^{3} d^{3} n^{3} x^{3}-3 b^{3} c \,d^{2} n^{3} x^{2}-6 b^{3} d^{3} n^{2} x^{3}+3 a \,b^{2} d^{3} n^{2} x^{2}-3 b^{3} c^{2} d \,n^{3} x -21 b^{3} c \,d^{2} n^{2} x^{2}-11 b^{3} d^{3} n \,x^{3}+6 a \,b^{2} c \,d^{2} n^{2} x +9 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}-24 b^{3} c^{2} d \,n^{2} x -42 b^{3} c \,d^{2} n \,x^{2}-6 b^{3} d^{3} x^{3}-6 a^{2} b \,d^{3} n x +3 a \,b^{2} c^{2} d \,n^{2}+30 a \,b^{2} c \,d^{2} n x +6 a \,b^{2} d^{3} x^{2}-9 b^{3} c^{3} n^{2}-57 b^{3} c^{2} d n x -24 b^{3} c \,d^{2} x^{2}-6 a^{2} b c \,d^{2} n -6 a^{2} b \,d^{3} x +21 a \,b^{2} c^{2} d n +24 a \,b^{2} c \,d^{2} x -26 b^{3} c^{3} n -36 b^{3} c^{2} d x +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) \left (b x +a \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^n*(d*x+c)^3,x)
[Out]
-(b*x+a)^(n+1)*(-b^3*d^3*n^3*x^3-3*b^3*c*d^2*n^3*x^2-6*b^3*d^3*n^2*x^3+3*a*b^2*d^3*n^2*x^2-3*b^3*c^2*d*n^3*x-2
1*b^3*c*d^2*n^2*x^2-11*b^3*d^3*n*x^3+6*a*b^2*c*d^2*n^2*x+9*a*b^2*d^3*n*x^2-b^3*c^3*n^3-24*b^3*c^2*d*n^2*x-42*b
^3*c*d^2*n*x^2-6*b^3*d^3*x^3-6*a^2*b*d^3*n*x+3*a*b^2*c^2*d*n^2+30*a*b^2*c*d^2*n*x+6*a*b^2*d^3*x^2-9*b^3*c^3*n^
2-57*b^3*c^2*d*n*x-24*b^3*c*d^2*x^2-6*a^2*b*c*d^2*n-6*a^2*b*d^3*x+21*a*b^2*c^2*d*n+24*a*b^2*c*d^2*x-26*b^3*c^3
*n-36*b^3*c^2*d*x+6*a^3*d^3-24*a^2*b*c*d^2+36*a*b^2*c^2*d-24*b^3*c^3)/b^4/(n^4+10*n^3+35*n^2+50*n+24)
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maxima [B] time = 0.66, size = 246, normalized size = 2.24 \begin {gather*} \frac {3 \, {\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{n + 1} c^{3}}{b {\left (n + 1\right )}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^n*(d*x+c)^3,x, algorithm="maxima")
[Out]
3*(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2*d/((n^2 + 3*n + 2)*b^2) + (b*x + a)^(n + 1)*c^3/(b*(n + 1)
) + 3*((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c*d^2/((n^3 + 6*n^2 +
11*n + 6)*b^3) + ((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 +
6*a^3*b*n*x - 6*a^4)*(b*x + a)^n*d^3/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4)
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mupad [B] time = 1.45, size = 478, normalized size = 4.35 \begin {gather*} \frac {x\,{\left (a+b\,x\right )}^n\,\left (6\,a^3\,b\,d^3\,n-6\,a^2\,b^2\,c\,d^2\,n^2-24\,a^2\,b^2\,c\,d^2\,n+3\,a\,b^3\,c^2\,d\,n^3+21\,a\,b^3\,c^2\,d\,n^2+36\,a\,b^3\,c^2\,d\,n+b^4\,c^3\,n^3+9\,b^4\,c^3\,n^2+26\,b^4\,c^3\,n+24\,b^4\,c^3\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d^3\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}+\frac {a\,{\left (a+b\,x\right )}^n\,\left (-6\,a^3\,d^3+6\,a^2\,b\,c\,d^2\,n+24\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d\,n^2-21\,a\,b^2\,c^2\,d\,n-36\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+9\,b^3\,c^3\,n^2+26\,b^3\,c^3\,n+24\,b^3\,c^3\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,d\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-a^2\,d^2\,n+a\,b\,c\,d\,n^2+4\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d^2\,x^3\,{\left (a+b\,x\right )}^n\,\left (12\,b\,c+a\,d\,n+3\,b\,c\,n\right )\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)^n*(c + d*x)^3,x)
[Out]
(x*(a + b*x)^n*(24*b^4*c^3 + 26*b^4*c^3*n + 9*b^4*c^3*n^2 + b^4*c^3*n^3 + 6*a^3*b*d^3*n + 36*a*b^3*c^2*d*n - 2
4*a^2*b^2*c*d^2*n + 21*a*b^3*c^2*d*n^2 + 3*a*b^3*c^2*d*n^3 - 6*a^2*b^2*c*d^2*n^2))/(b^4*(50*n + 35*n^2 + 10*n^
3 + n^4 + 24)) + (d^3*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n^2 + 10*n^3 + n^4 + 24) + (a*(a +
b*x)^n*(24*b^3*c^3 - 6*a^3*d^3 + 26*b^3*c^3*n + 9*b^3*c^3*n^2 + b^3*c^3*n^3 - 36*a*b^2*c^2*d + 24*a^2*b*c*d^2
- 21*a*b^2*c^2*d*n + 6*a^2*b*c*d^2*n - 3*a*b^2*c^2*d*n^2))/(b^4*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (3*d*x^
2*(n + 1)*(a + b*x)^n*(12*b^2*c^2 - a^2*d^2*n + 7*b^2*c^2*n + b^2*c^2*n^2 + 4*a*b*c*d*n + a*b*c*d*n^2))/(b^2*(
50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (d^2*x^3*(a + b*x)^n*(12*b*c + a*d*n + 3*b*c*n)*(3*n + n^2 + 2))/(b*(50*
n + 35*n^2 + 10*n^3 + n^4 + 24))
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sympy [A] time = 4.89, size = 4058, normalized size = 36.89
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**n*(d*x+c)**3,x)
[Out]
Piecewise((a**n*(c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4), Eq(b, 0)), (6*a**3*d**3*log(a/b + x)/(
6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 11*a**3*d**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18
*a*b**6*x**2 + 6*b**7*x**3) - 6*a**2*b*c*d**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) +
18*a**2*b*d**3*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 27*a**2*b*d**3*x
/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*a*b**2*c**2*d/(6*a**3*b**4 + 18*a**2*b**5*x
+ 18*a*b**6*x**2 + 6*b**7*x**3) - 18*a*b**2*c*d**2*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*
x**3) + 18*a*b**2*d**3*x**2*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*
b**2*d**3*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*b**3*c**3/(6*a**3*b**4 + 18*a
**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 9*b**3*c**2*d*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 +
6*b**7*x**3) - 18*b**3*c*d**2*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3*d**3
*x**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq(n, -4)), (-6*a**3*d**3*lo
g(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 6
*a**2*b*c*d**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 9*a**2*b*c*d**2/(2*a**2*b**4 + 4*a*b**5
*x + 2*b**6*x**2) - 12*a**2*b*d**3*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d**3*x/
(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 3*a*b**2*c**2*d/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 12*a*b**
2*c*d**2*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 12*a*b**2*c*d**2*x/(2*a**2*b**4 + 4*a*b**5*
x + 2*b**6*x**2) - 6*a*b**2*d**3*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - b**3*c**3/(2*a**
2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 6*b**3*c**2*d*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 6*b**3*c*d**2*
x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*d**3*x**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b*
*6*x**2), Eq(n, -3)), (6*a**3*d**3*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a**3*d**3/(2*a*b**4 + 2*b**5*x) - 12
*a**2*b*c*d**2*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 12*a**2*b*c*d**2/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*d**3*x*l
og(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a*b**2*c**2*d*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*a*b**2*c**2*d/(2*a*
b**4 + 2*b**5*x) - 12*a*b**2*c*d**2*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) - 3*a*b**2*d**3*x**2/(2*a*b**4 + 2*b*
*5*x) - 2*b**3*c**3/(2*a*b**4 + 2*b**5*x) + 6*b**3*c**2*d*x*log(a/b + x)/(2*a*b**4 + 2*b**5*x) + 6*b**3*c*d**2
*x**2/(2*a*b**4 + 2*b**5*x) + b**3*d**3*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*d**3*log(a/b + x)/b**4
+ 3*a**2*c*d**2*log(a/b + x)/b**3 + a**2*d**3*x/b**3 - 3*a*c**2*d*log(a/b + x)/b**2 - 3*a*c*d**2*x/b**2 - a*d*
*3*x**2/(2*b**2) + c**3*log(a/b + x)/b + 3*c**2*d*x/b + 3*c*d**2*x**2/(2*b) + d**3*x**3/(3*b), Eq(n, -1)), (-6
*a**4*d**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*a**3*b*c*d**2*n*(a
+ b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*a**3*b*c*d**2*(a + b*x)**n/(b*
*4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*a**3*b*d**3*n*x*(a + b*x)**n/(b**4*n**4 + 10*
b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*c**2*d*n**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n*
*3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 21*a**2*b**2*c**2*d*n*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b
**4*n**2 + 50*b**4*n + 24*b**4) - 36*a**2*b**2*c**2*d*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 +
50*b**4*n + 24*b**4) - 6*a**2*b**2*c*d**2*n**2*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b*
*4*n + 24*b**4) - 24*a**2*b**2*c*d**2*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n +
24*b**4) - 3*a**2*b**2*d**3*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b
**4) - 3*a**2*b**2*d**3*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) +
a*b**3*c**3*n**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 9*a*b**3*c**3*
n**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 26*a*b**3*c**3*n*(a + b*x)
**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*a*b**3*c**3*(a + b*x)**n/(b**4*n**4 +
10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*c**2*d*n**3*x*(a + b*x)**n/(b**4*n**4 + 10*b**4
*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 21*a*b**3*c**2*d*n**2*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 +
35*b**4*n**2 + 50*b**4*n + 24*b**4) + 36*a*b**3*c**2*d*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n
**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*c*d**2*n**3*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 +
50*b**4*n + 24*b**4) + 15*a*b**3*c*d**2*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*
b**4*n + 24*b**4) + 12*a*b**3*c*d**2*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n
+ 24*b**4) + a*b**3*d**3*n**3*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4
) + 3*a*b**3*d**3*n**2*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a
*b**3*d**3*n*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*c**3*n**
3*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 9*b**4*c**3*n**2*x*(a + b*x
)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 26*b**4*c**3*n*x*(a + b*x)**n/(b**4*n**
4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*b**4*c**3*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3
+ 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*b**4*c**2*d*n**3*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b
**4*n**2 + 50*b**4*n + 24*b**4) + 24*b**4*c**2*d*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n*
*2 + 50*b**4*n + 24*b**4) + 57*b**4*c**2*d*n*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b
**4*n + 24*b**4) + 36*b**4*c**2*d*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*
b**4) + 3*b**4*c*d**2*n**3*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) +
21*b**4*c*d**2*n**2*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 42*b*
*4*c*d**2*n*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*b**4*c*d**2
*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*d**3*n**3*x**4*(a +
b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d**3*n**2*x**4*(a + b*x)**n/(
b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 11*b**4*d**3*n*x**4*(a + b*x)**n/(b**4*n**4 +
10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d**3*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3
+ 35*b**4*n**2 + 50*b**4*n + 24*b**4), True))
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