3.16.58 \(\int (a+b x)^3 (c+d x)^n \, dx\)
Optimal. Leaf size=111 \[ -\frac {3 b^2 (b c-a d) (c+d x)^{n+3}}{d^4 (n+3)}-\frac {(b c-a d)^3 (c+d x)^{n+1}}{d^4 (n+1)}+\frac {3 b (b c-a d)^2 (c+d x)^{n+2}}{d^4 (n+2)}+\frac {b^3 (c+d x)^{n+4}}{d^4 (n+4)} \]
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Rubi [A] time = 0.06, antiderivative size = 111, 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 b^2 (b c-a d) (c+d x)^{n+3}}{d^4 (n+3)}-\frac {(b c-a d)^3 (c+d x)^{n+1}}{d^4 (n+1)}+\frac {3 b (b c-a d)^2 (c+d x)^{n+2}}{d^4 (n+2)}+\frac {b^3 (c+d x)^{n+4}}{d^4 (n+4)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)^3*(c + d*x)^n,x]
[Out]
-(((b*c - a*d)^3*(c + d*x)^(1 + n))/(d^4*(1 + n))) + (3*b*(b*c - a*d)^2*(c + d*x)^(2 + n))/(d^4*(2 + n)) - (3*
b^2*(b*c - a*d)*(c + d*x)^(3 + n))/(d^4*(3 + n)) + (b^3*(c + d*x)^(4 + n))/(d^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)^3 (c+d x)^n \, dx &=\int \left (\frac {(-b c+a d)^3 (c+d x)^n}{d^3}+\frac {3 b (b c-a d)^2 (c+d x)^{1+n}}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^{2+n}}{d^3}+\frac {b^3 (c+d x)^{3+n}}{d^3}\right ) \, dx\\ &=-\frac {(b c-a d)^3 (c+d x)^{1+n}}{d^4 (1+n)}+\frac {3 b (b c-a d)^2 (c+d x)^{2+n}}{d^4 (2+n)}-\frac {3 b^2 (b c-a d) (c+d x)^{3+n}}{d^4 (3+n)}+\frac {b^3 (c+d x)^{4+n}}{d^4 (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 95, normalized size = 0.86 \begin {gather*} \frac {(c+d x)^{n+1} \left (-\frac {3 b^2 (c+d x)^2 (b c-a d)}{n+3}+\frac {3 b (c+d x) (b c-a d)^2}{n+2}-\frac {(b c-a d)^3}{n+1}+\frac {b^3 (c+d x)^3}{n+4}\right )}{d^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)^3*(c + d*x)^n,x]
[Out]
((c + d*x)^(1 + n)*(-((b*c - a*d)^3/(1 + n)) + (3*b*(b*c - a*d)^2*(c + d*x))/(2 + n) - (3*b^2*(b*c - a*d)*(c +
d*x)^2)/(3 + n) + (b^3*(c + d*x)^3)/(4 + n)))/d^4
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IntegrateAlgebraic [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^3 (c+d x)^n \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(a + b*x)^3*(c + d*x)^n,x]
[Out]
Defer[IntegrateAlgebraic][(a + b*x)^3*(c + d*x)^n, x]
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fricas [B] time = 1.04, size = 496, normalized size = 4.47 \begin {gather*} \frac {{\left (a^{3} c d^{3} n^{3} - 6 \, b^{3} c^{4} + 24 \, a b^{2} c^{3} d - 36 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3} + {\left (b^{3} d^{4} n^{3} + 6 \, b^{3} d^{4} n^{2} + 11 \, b^{3} d^{4} n + 6 \, b^{3} d^{4}\right )} x^{4} + {\left (24 \, a b^{2} d^{4} + {\left (b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} n^{3} + 3 \, {\left (b^{3} c d^{3} + 7 \, a b^{2} d^{4}\right )} n^{2} + 2 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} n\right )} x^{3} - 3 \, {\left (a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3}\right )} n^{2} + 3 \, {\left (12 \, a^{2} b d^{4} + {\left (a b^{2} c d^{3} + a^{2} b d^{4}\right )} n^{3} - {\left (b^{3} c^{2} d^{2} - 5 \, a b^{2} c d^{3} - 8 \, a^{2} b d^{4}\right )} n^{2} - {\left (b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} - 19 \, a^{2} b d^{4}\right )} n\right )} x^{2} + {\left (6 \, a b^{2} c^{3} d - 21 \, a^{2} b c^{2} d^{2} + 26 \, a^{3} c d^{3}\right )} n + {\left (24 \, a^{3} d^{4} + {\left (3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} n^{3} - 3 \, {\left (2 \, a b^{2} c^{2} d^{2} - 7 \, a^{2} b c d^{3} - 3 \, a^{3} d^{4}\right )} n^{2} + 2 \, {\left (3 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 18 \, a^{2} b c d^{3} + 13 \, a^{3} d^{4}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^n,x, algorithm="fricas")
[Out]
(a^3*c*d^3*n^3 - 6*b^3*c^4 + 24*a*b^2*c^3*d - 36*a^2*b*c^2*d^2 + 24*a^3*c*d^3 + (b^3*d^4*n^3 + 6*b^3*d^4*n^2 +
11*b^3*d^4*n + 6*b^3*d^4)*x^4 + (24*a*b^2*d^4 + (b^3*c*d^3 + 3*a*b^2*d^4)*n^3 + 3*(b^3*c*d^3 + 7*a*b^2*d^4)*n
^2 + 2*(b^3*c*d^3 + 21*a*b^2*d^4)*n)*x^3 - 3*(a^2*b*c^2*d^2 - 3*a^3*c*d^3)*n^2 + 3*(12*a^2*b*d^4 + (a*b^2*c*d^
3 + a^2*b*d^4)*n^3 - (b^3*c^2*d^2 - 5*a*b^2*c*d^3 - 8*a^2*b*d^4)*n^2 - (b^3*c^2*d^2 - 4*a*b^2*c*d^3 - 19*a^2*b
*d^4)*n)*x^2 + (6*a*b^2*c^3*d - 21*a^2*b*c^2*d^2 + 26*a^3*c*d^3)*n + (24*a^3*d^4 + (3*a^2*b*c*d^3 + a^3*d^4)*n
^3 - 3*(2*a*b^2*c^2*d^2 - 7*a^2*b*c*d^3 - 3*a^3*d^4)*n^2 + 2*(3*b^3*c^3*d - 12*a*b^2*c^2*d^2 + 18*a^2*b*c*d^3
+ 13*a^3*d^4)*n)*x)*(d*x + c)^n/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n + 24*d^4)
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giac [B] time = 0.98, size = 833, normalized size = 7.50 \begin {gather*} \frac {{\left (d x + c\right )}^{n} b^{3} d^{4} n^{3} x^{4} + {\left (d x + c\right )}^{n} b^{3} c d^{3} n^{3} x^{3} + 3 \, {\left (d x + c\right )}^{n} a b^{2} d^{4} n^{3} x^{3} + 6 \, {\left (d x + c\right )}^{n} b^{3} d^{4} n^{2} x^{4} + 3 \, {\left (d x + c\right )}^{n} a b^{2} c d^{3} n^{3} x^{2} + 3 \, {\left (d x + c\right )}^{n} a^{2} b d^{4} n^{3} x^{2} + 3 \, {\left (d x + c\right )}^{n} b^{3} c d^{3} n^{2} x^{3} + 21 \, {\left (d x + c\right )}^{n} a b^{2} d^{4} n^{2} x^{3} + 11 \, {\left (d x + c\right )}^{n} b^{3} d^{4} n x^{4} + 3 \, {\left (d x + c\right )}^{n} a^{2} b c d^{3} n^{3} x + {\left (d x + c\right )}^{n} a^{3} d^{4} n^{3} x - 3 \, {\left (d x + c\right )}^{n} b^{3} c^{2} d^{2} n^{2} x^{2} + 15 \, {\left (d x + c\right )}^{n} a b^{2} c d^{3} n^{2} x^{2} + 24 \, {\left (d x + c\right )}^{n} a^{2} b d^{4} n^{2} x^{2} + 2 \, {\left (d x + c\right )}^{n} b^{3} c d^{3} n x^{3} + 42 \, {\left (d x + c\right )}^{n} a b^{2} d^{4} n x^{3} + 6 \, {\left (d x + c\right )}^{n} b^{3} d^{4} x^{4} + {\left (d x + c\right )}^{n} a^{3} c d^{3} n^{3} - 6 \, {\left (d x + c\right )}^{n} a b^{2} c^{2} d^{2} n^{2} x + 21 \, {\left (d x + c\right )}^{n} a^{2} b c d^{3} n^{2} x + 9 \, {\left (d x + c\right )}^{n} a^{3} d^{4} n^{2} x - 3 \, {\left (d x + c\right )}^{n} b^{3} c^{2} d^{2} n x^{2} + 12 \, {\left (d x + c\right )}^{n} a b^{2} c d^{3} n x^{2} + 57 \, {\left (d x + c\right )}^{n} a^{2} b d^{4} n x^{2} + 24 \, {\left (d x + c\right )}^{n} a b^{2} d^{4} x^{3} - 3 \, {\left (d x + c\right )}^{n} a^{2} b c^{2} d^{2} n^{2} + 9 \, {\left (d x + c\right )}^{n} a^{3} c d^{3} n^{2} + 6 \, {\left (d x + c\right )}^{n} b^{3} c^{3} d n x - 24 \, {\left (d x + c\right )}^{n} a b^{2} c^{2} d^{2} n x + 36 \, {\left (d x + c\right )}^{n} a^{2} b c d^{3} n x + 26 \, {\left (d x + c\right )}^{n} a^{3} d^{4} n x + 36 \, {\left (d x + c\right )}^{n} a^{2} b d^{4} x^{2} + 6 \, {\left (d x + c\right )}^{n} a b^{2} c^{3} d n - 21 \, {\left (d x + c\right )}^{n} a^{2} b c^{2} d^{2} n + 26 \, {\left (d x + c\right )}^{n} a^{3} c d^{3} n + 24 \, {\left (d x + c\right )}^{n} a^{3} d^{4} x - 6 \, {\left (d x + c\right )}^{n} b^{3} c^{4} + 24 \, {\left (d x + c\right )}^{n} a b^{2} c^{3} d - 36 \, {\left (d x + c\right )}^{n} a^{2} b c^{2} d^{2} + 24 \, {\left (d x + c\right )}^{n} a^{3} c d^{3}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^n,x, algorithm="giac")
[Out]
((d*x + c)^n*b^3*d^4*n^3*x^4 + (d*x + c)^n*b^3*c*d^3*n^3*x^3 + 3*(d*x + c)^n*a*b^2*d^4*n^3*x^3 + 6*(d*x + c)^n
*b^3*d^4*n^2*x^4 + 3*(d*x + c)^n*a*b^2*c*d^3*n^3*x^2 + 3*(d*x + c)^n*a^2*b*d^4*n^3*x^2 + 3*(d*x + c)^n*b^3*c*d
^3*n^2*x^3 + 21*(d*x + c)^n*a*b^2*d^4*n^2*x^3 + 11*(d*x + c)^n*b^3*d^4*n*x^4 + 3*(d*x + c)^n*a^2*b*c*d^3*n^3*x
+ (d*x + c)^n*a^3*d^4*n^3*x - 3*(d*x + c)^n*b^3*c^2*d^2*n^2*x^2 + 15*(d*x + c)^n*a*b^2*c*d^3*n^2*x^2 + 24*(d*
x + c)^n*a^2*b*d^4*n^2*x^2 + 2*(d*x + c)^n*b^3*c*d^3*n*x^3 + 42*(d*x + c)^n*a*b^2*d^4*n*x^3 + 6*(d*x + c)^n*b^
3*d^4*x^4 + (d*x + c)^n*a^3*c*d^3*n^3 - 6*(d*x + c)^n*a*b^2*c^2*d^2*n^2*x + 21*(d*x + c)^n*a^2*b*c*d^3*n^2*x +
9*(d*x + c)^n*a^3*d^4*n^2*x - 3*(d*x + c)^n*b^3*c^2*d^2*n*x^2 + 12*(d*x + c)^n*a*b^2*c*d^3*n*x^2 + 57*(d*x +
c)^n*a^2*b*d^4*n*x^2 + 24*(d*x + c)^n*a*b^2*d^4*x^3 - 3*(d*x + c)^n*a^2*b*c^2*d^2*n^2 + 9*(d*x + c)^n*a^3*c*d^
3*n^2 + 6*(d*x + c)^n*b^3*c^3*d*n*x - 24*(d*x + c)^n*a*b^2*c^2*d^2*n*x + 36*(d*x + c)^n*a^2*b*c*d^3*n*x + 26*(
d*x + c)^n*a^3*d^4*n*x + 36*(d*x + c)^n*a^2*b*d^4*x^2 + 6*(d*x + c)^n*a*b^2*c^3*d*n - 21*(d*x + c)^n*a^2*b*c^2
*d^2*n + 26*(d*x + c)^n*a^3*c*d^3*n + 24*(d*x + c)^n*a^3*d^4*x - 6*(d*x + c)^n*b^3*c^4 + 24*(d*x + c)^n*a*b^2*
c^3*d - 36*(d*x + c)^n*a^2*b*c^2*d^2 + 24*(d*x + c)^n*a^3*c*d^3)/(d^4*n^4 + 10*d^4*n^3 + 35*d^4*n^2 + 50*d^4*n
+ 24*d^4)
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maple [B] time = 0.01, size = 386, normalized size = 3.48 \begin {gather*} \frac {\left (b^{3} d^{3} n^{3} x^{3}+3 a \,b^{2} d^{3} n^{3} x^{2}+6 b^{3} d^{3} n^{2} x^{3}+3 a^{2} b \,d^{3} n^{3} x +21 a \,b^{2} d^{3} n^{2} x^{2}-3 b^{3} c \,d^{2} n^{2} x^{2}+11 b^{3} d^{3} n \,x^{3}+a^{3} d^{3} n^{3}+24 a^{2} b \,d^{3} n^{2} x -6 a \,b^{2} c \,d^{2} n^{2} x +42 a \,b^{2} d^{3} n \,x^{2}-9 b^{3} c \,d^{2} n \,x^{2}+6 b^{3} d^{3} x^{3}+9 a^{3} d^{3} n^{2}-3 a^{2} b c \,d^{2} n^{2}+57 a^{2} b \,d^{3} n x -30 a \,b^{2} c \,d^{2} n x +24 a \,b^{2} d^{3} x^{2}+6 b^{3} c^{2} d n x -6 b^{3} c \,d^{2} x^{2}+26 a^{3} d^{3} n -21 a^{2} b c \,d^{2} n +36 a^{2} b \,d^{3} x +6 a \,b^{2} c^{2} d n -24 a \,b^{2} c \,d^{2} x +6 b^{3} c^{2} d x +24 a^{3} d^{3}-36 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) \left (d x +c \right )^{n +1}}{\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^3*(d*x+c)^n,x)
[Out]
(d*x+c)^(n+1)*(b^3*d^3*n^3*x^3+3*a*b^2*d^3*n^3*x^2+6*b^3*d^3*n^2*x^3+3*a^2*b*d^3*n^3*x+21*a*b^2*d^3*n^2*x^2-3*
b^3*c*d^2*n^2*x^2+11*b^3*d^3*n*x^3+a^3*d^3*n^3+24*a^2*b*d^3*n^2*x-6*a*b^2*c*d^2*n^2*x+42*a*b^2*d^3*n*x^2-9*b^3
*c*d^2*n*x^2+6*b^3*d^3*x^3+9*a^3*d^3*n^2-3*a^2*b*c*d^2*n^2+57*a^2*b*d^3*n*x-30*a*b^2*c*d^2*n*x+24*a*b^2*d^3*x^
2+6*b^3*c^2*d*n*x-6*b^3*c*d^2*x^2+26*a^3*d^3*n-21*a^2*b*c*d^2*n+36*a^2*b*d^3*x+6*a*b^2*c^2*d*n-24*a*b^2*c*d^2*
x+6*b^3*c^2*d*x+24*a^3*d^3-36*a^2*b*c*d^2+24*a*b^2*c^2*d-6*b^3*c^3)/d^4/(n^4+10*n^3+35*n^2+50*n+24)
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maxima [B] time = 1.30, size = 246, normalized size = 2.22 \begin {gather*} \frac {3 \, {\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} a^{2} b}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d x + c\right )}^{n + 1} a^{3}}{d {\left (n + 1\right )}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} a b^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} x^{2} + 6 \, c^{3} d n x - 6 \, c^{4}\right )} {\left (d x + c\right )}^{n} b^{3}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)^3*(d*x+c)^n,x, algorithm="maxima")
[Out]
3*(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*a^2*b/((n^2 + 3*n + 2)*d^2) + (d*x + c)^(n + 1)*a^3/(d*(n + 1)
) + 3*((n^2 + 3*n + 2)*d^3*x^3 + (n^2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*a*b^2/((n^3 + 6*n^2 +
11*n + 6)*d^3) + ((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n)*c^2*d^2*x^2 +
6*c^3*d*n*x - 6*c^4)*(d*x + c)^n*b^3/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4)
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mupad [B] time = 0.91, size = 478, normalized size = 4.31 \begin {gather*} \frac {x\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^4\,n^3+9\,a^3\,d^4\,n^2+26\,a^3\,d^4\,n+24\,a^3\,d^4+3\,a^2\,b\,c\,d^3\,n^3+21\,a^2\,b\,c\,d^3\,n^2+36\,a^2\,b\,c\,d^3\,n-6\,a\,b^2\,c^2\,d^2\,n^2-24\,a\,b^2\,c^2\,d^2\,n+6\,b^3\,c^3\,d\,n\right )}{d^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b^3\,x^4\,{\left (c+d\,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 {c\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^3\,n^3+9\,a^3\,d^3\,n^2+26\,a^3\,d^3\,n+24\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^2-21\,a^2\,b\,c\,d^2\,n-36\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d\,n+24\,a\,b^2\,c^2\,d-6\,b^3\,c^3\right )}{d^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,b\,x^2\,\left (n+1\right )\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^2\,n^2+7\,a^2\,d^2\,n+12\,a^2\,d^2+a\,b\,c\,d\,n^2+4\,a\,b\,c\,d\,n-b^2\,c^2\,n\right )}{d^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b^2\,x^3\,{\left (c+d\,x\right )}^n\,\left (12\,a\,d+3\,a\,d\,n+b\,c\,n\right )\,\left (n^2+3\,n+2\right )}{d\,\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)^3*(c + d*x)^n,x)
[Out]
(x*(c + d*x)^n*(24*a^3*d^4 + 26*a^3*d^4*n + 9*a^3*d^4*n^2 + a^3*d^4*n^3 + 6*b^3*c^3*d*n + 36*a^2*b*c*d^3*n - 2
4*a*b^2*c^2*d^2*n + 21*a^2*b*c*d^3*n^2 + 3*a^2*b*c*d^3*n^3 - 6*a*b^2*c^2*d^2*n^2))/(d^4*(50*n + 35*n^2 + 10*n^
3 + n^4 + 24)) + (b^3*x^4*(c + d*x)^n*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n^2 + 10*n^3 + n^4 + 24) + (c*(c +
d*x)^n*(24*a^3*d^3 - 6*b^3*c^3 + 26*a^3*d^3*n + 9*a^3*d^3*n^2 + a^3*d^3*n^3 + 24*a*b^2*c^2*d - 36*a^2*b*c*d^2
+ 6*a*b^2*c^2*d*n - 21*a^2*b*c*d^2*n - 3*a^2*b*c*d^2*n^2))/(d^4*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (3*b*x^
2*(n + 1)*(c + d*x)^n*(12*a^2*d^2 + 7*a^2*d^2*n - b^2*c^2*n + a^2*d^2*n^2 + 4*a*b*c*d*n + a*b*c*d*n^2))/(d^2*(
50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (b^2*x^3*(c + d*x)^n*(12*a*d + 3*a*d*n + b*c*n)*(3*n + n^2 + 2))/(d*(50*
n + 35*n^2 + 10*n^3 + n^4 + 24))
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sympy [A] time = 4.44, size = 4058, normalized size = 36.56
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)**3*(d*x+c)**n,x)
[Out]
Piecewise((c**n*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), Eq(d, 0)), (-2*a**3*d**3/(6*c**3*d**4
+ 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 3*a**2*b*c*d**2/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x
**2 + 6*d**7*x**3) - 9*a**2*b*d**3*x/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 6*a*b**2*
c**2*d/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 18*a*b**2*c*d**2*x/(6*c**3*d**4 + 18*c*
*2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) - 18*a*b**2*d**3*x**2/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2
+ 6*d**7*x**3) + 6*b**3*c**3*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 11*
b**3*c**3/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 18*b**3*c**2*d*x*log(c/d + x)/(6*c**
3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 27*b**3*c**2*d*x/(6*c**3*d**4 + 18*c**2*d**5*x + 18*
c*d**6*x**2 + 6*d**7*x**3) + 18*b**3*c*d**2*x**2*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 +
6*d**7*x**3) + 18*b**3*c*d**2*x**2/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3) + 6*b**3*d**
3*x**3*log(c/d + x)/(6*c**3*d**4 + 18*c**2*d**5*x + 18*c*d**6*x**2 + 6*d**7*x**3), Eq(n, -4)), (-a**3*d**3/(2*
c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 3*a**2*b*c*d**2/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 6*a**2*b*d*
*3*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 6*a*b**2*c**2*d*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**
6*x**2) + 9*a*b**2*c**2*d/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 12*a*b**2*c*d**2*x*log(c/d + x)/(2*c**2*d
**4 + 4*c*d**5*x + 2*d**6*x**2) + 12*a*b**2*c*d**2*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 6*a*b**2*d**3*
x**2*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 6*b**3*c**3*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*
x + 2*d**6*x**2) - 9*b**3*c**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*b**3*c**2*d*x*log(c/d + x)/(2*c**
2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 12*b**3*c**2*d*x/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) - 6*b**3*c*d**2
*x**2*log(c/d + x)/(2*c**2*d**4 + 4*c*d**5*x + 2*d**6*x**2) + 2*b**3*d**3*x**3/(2*c**2*d**4 + 4*c*d**5*x + 2*d
**6*x**2), Eq(n, -3)), (-2*a**3*d**3/(2*c*d**4 + 2*d**5*x) + 6*a**2*b*c*d**2*log(c/d + x)/(2*c*d**4 + 2*d**5*x
) + 6*a**2*b*c*d**2/(2*c*d**4 + 2*d**5*x) + 6*a**2*b*d**3*x*log(c/d + x)/(2*c*d**4 + 2*d**5*x) - 12*a*b**2*c**
2*d*log(c/d + x)/(2*c*d**4 + 2*d**5*x) - 12*a*b**2*c**2*d/(2*c*d**4 + 2*d**5*x) - 12*a*b**2*c*d**2*x*log(c/d +
x)/(2*c*d**4 + 2*d**5*x) + 6*a*b**2*d**3*x**2/(2*c*d**4 + 2*d**5*x) + 6*b**3*c**3*log(c/d + x)/(2*c*d**4 + 2*
d**5*x) + 6*b**3*c**3/(2*c*d**4 + 2*d**5*x) + 6*b**3*c**2*d*x*log(c/d + x)/(2*c*d**4 + 2*d**5*x) - 3*b**3*c*d*
*2*x**2/(2*c*d**4 + 2*d**5*x) + b**3*d**3*x**3/(2*c*d**4 + 2*d**5*x), Eq(n, -2)), (a**3*log(c/d + x)/d - 3*a**
2*b*c*log(c/d + x)/d**2 + 3*a**2*b*x/d + 3*a*b**2*c**2*log(c/d + x)/d**3 - 3*a*b**2*c*x/d**2 + 3*a*b**2*x**2/(
2*d) - b**3*c**3*log(c/d + x)/d**4 + b**3*c**2*x/d**3 - b**3*c*x**2/(2*d**2) + b**3*x**3/(3*d), Eq(n, -1)), (a
**3*c*d**3*n**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 9*a**3*c*d**3*n
**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 26*a**3*c*d**3*n*(c + d*x)*
*n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 24*a**3*c*d**3*(c + d*x)**n/(d**4*n**4 +
10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + a**3*d**4*n**3*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 +
35*d**4*n**2 + 50*d**4*n + 24*d**4) + 9*a**3*d**4*n**2*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**
2 + 50*d**4*n + 24*d**4) + 26*a**3*d**4*n*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n
+ 24*d**4) + 24*a**3*d**4*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 3*a
**2*b*c**2*d**2*n**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 21*a**2*b*
c**2*d**2*n*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 36*a**2*b*c**2*d**2
*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 3*a**2*b*c*d**3*n**3*x*(c + d*
x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 21*a**2*b*c*d**3*n**2*x*(c + d*x)**n/(
d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 36*a**2*b*c*d**3*n*x*(c + d*x)**n/(d**4*n**4
+ 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 3*a**2*b*d**4*n**3*x**2*(c + d*x)**n/(d**4*n**4 + 10*d*
*4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 24*a**2*b*d**4*n**2*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**
3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 57*a**2*b*d**4*n*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d*
*4*n**2 + 50*d**4*n + 24*d**4) + 36*a**2*b*d**4*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 5
0*d**4*n + 24*d**4) + 6*a*b**2*c**3*d*n*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24
*d**4) + 24*a*b**2*c**3*d*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 6*a*b
**2*c**2*d**2*n**2*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 24*a*b**2*
c**2*d**2*n*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 3*a*b**2*c*d**3*n
**3*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 15*a*b**2*c*d**3*n**2*
x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 12*a*b**2*c*d**3*n*x**2*(c
+ d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 3*a*b**2*d**4*n**3*x**3*(c + d*x)
**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 21*a*b**2*d**4*n**2*x**3*(c + d*x)**n/(d
**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 42*a*b**2*d**4*n*x**3*(c + d*x)**n/(d**4*n**4
+ 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 24*a*b**2*d**4*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n
**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) - 6*b**3*c**4*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2
+ 50*d**4*n + 24*d**4) + 6*b**3*c**3*d*n*x*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n +
24*d**4) - 3*b**3*c**2*d**2*n**2*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*
d**4) - 3*b**3*c**2*d**2*n*x**2*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) +
b**3*c*d**3*n**3*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 3*b**3*c
*d**3*n**2*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 2*b**3*c*d**3*n
*x**3*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + b**3*d**4*n**3*x**4*(c +
d*x)**n/(d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 6*b**3*d**4*n**2*x**4*(c + d*x)**n/(
d**4*n**4 + 10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 11*b**3*d**4*n*x**4*(c + d*x)**n/(d**4*n**4 +
10*d**4*n**3 + 35*d**4*n**2 + 50*d**4*n + 24*d**4) + 6*b**3*d**4*x**4*(c + d*x)**n/(d**4*n**4 + 10*d**4*n**3
+ 35*d**4*n**2 + 50*d**4*n + 24*d**4), True))
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