3.10.30 \(\int x^2 (a+b x)^n (c+d x)^3 \, dx\) [930]
Optimal. Leaf size=212 \[ \frac {a^2 (b c-a d)^3 (a+b x)^{1+n}}{b^6 (1+n)}-\frac {a (2 b c-5 a d) (b c-a d)^2 (a+b x)^{2+n}}{b^6 (2+n)}+\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {d \left (3 b^2 c^2-12 a b c d+10 a^2 d^2\right ) (a+b x)^{4+n}}{b^6 (4+n)}+\frac {d^2 (3 b c-5 a d) (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^3 (a+b x)^{6+n}}{b^6 (6+n)} \]
[Out]
a^2*(-a*d+b*c)^3*(b*x+a)^(1+n)/b^6/(1+n)-a*(-5*a*d+2*b*c)*(-a*d+b*c)^2*(b*x+a)^(2+n)/b^6/(2+n)+(-a*d+b*c)*(10*
a^2*d^2-8*a*b*c*d+b^2*c^2)*(b*x+a)^(3+n)/b^6/(3+n)+d*(10*a^2*d^2-12*a*b*c*d+3*b^2*c^2)*(b*x+a)^(4+n)/b^6/(4+n)
+d^2*(-5*a*d+3*b*c)*(b*x+a)^(5+n)/b^6/(5+n)+d^3*(b*x+a)^(6+n)/b^6/(6+n)
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Rubi [A]
time = 0.08, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90}
\begin {gather*} \frac {a^2 (b c-a d)^3 (a+b x)^{n+1}}{b^6 (n+1)}+\frac {(b c-a d) \left (10 a^2 d^2-8 a b c d+b^2 c^2\right ) (a+b x)^{n+3}}{b^6 (n+3)}+\frac {d \left (10 a^2 d^2-12 a b c d+3 b^2 c^2\right ) (a+b x)^{n+4}}{b^6 (n+4)}+\frac {d^2 (3 b c-5 a d) (a+b x)^{n+5}}{b^6 (n+5)}-\frac {a (2 b c-5 a d) (b c-a d)^2 (a+b x)^{n+2}}{b^6 (n+2)}+\frac {d^3 (a+b x)^{n+6}}{b^6 (n+6)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2*(a + b*x)^n*(c + d*x)^3,x]
[Out]
(a^2*(b*c - a*d)^3*(a + b*x)^(1 + n))/(b^6*(1 + n)) - (a*(2*b*c - 5*a*d)*(b*c - a*d)^2*(a + b*x)^(2 + n))/(b^6
*(2 + n)) + ((b*c - a*d)*(b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (d*(3*b^2*c^2 -
12*a*b*c*d + 10*a^2*d^2)*(a + b*x)^(4 + n))/(b^6*(4 + n)) + (d^2*(3*b*c - 5*a*d)*(a + b*x)^(5 + n))/(b^6*(5 +
n)) + (d^3*(a + b*x)^(6 + n))/(b^6*(6 + n))
Rule 90
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))
Rubi steps
\begin {align*} \int x^2 (a+b x)^n (c+d x)^3 \, dx &=\int \left (-\frac {a^2 (-b c+a d)^3 (a+b x)^n}{b^5}+\frac {a (-b c+a d)^2 (-2 b c+5 a d) (a+b x)^{1+n}}{b^5}+\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) (a+b x)^{2+n}}{b^5}+\frac {d \left (3 b^2 c^2-12 a b c d+10 a^2 d^2\right ) (a+b x)^{3+n}}{b^5}+\frac {d^2 (3 b c-5 a d) (a+b x)^{4+n}}{b^5}+\frac {d^3 (a+b x)^{5+n}}{b^5}\right ) \, dx\\ &=\frac {a^2 (b c-a d)^3 (a+b x)^{1+n}}{b^6 (1+n)}-\frac {a (2 b c-5 a d) (b c-a d)^2 (a+b x)^{2+n}}{b^6 (2+n)}+\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) (a+b x)^{3+n}}{b^6 (3+n)}+\frac {d \left (3 b^2 c^2-12 a b c d+10 a^2 d^2\right ) (a+b x)^{4+n}}{b^6 (4+n)}+\frac {d^2 (3 b c-5 a d) (a+b x)^{5+n}}{b^6 (5+n)}+\frac {d^3 (a+b x)^{6+n}}{b^6 (6+n)}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 185, normalized size = 0.87 \begin {gather*} \frac {(a+b x)^{1+n} \left (\frac {a^2 (b c-a d)^3}{1+n}+\frac {a (b c-a d)^2 (-2 b c+5 a d) (a+b x)}{2+n}+\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) (a+b x)^2}{3+n}+\frac {d \left (3 b^2 c^2-12 a b c d+10 a^2 d^2\right ) (a+b x)^3}{4+n}+\frac {d^2 (3 b c-5 a d) (a+b x)^4}{5+n}+\frac {d^3 (a+b x)^5}{6+n}\right )}{b^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2*(a + b*x)^n*(c + d*x)^3,x]
[Out]
((a + b*x)^(1 + n)*((a^2*(b*c - a*d)^3)/(1 + n) + (a*(b*c - a*d)^2*(-2*b*c + 5*a*d)*(a + b*x))/(2 + n) + ((b*c
- a*d)*(b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*(a + b*x)^2)/(3 + n) + (d*(3*b^2*c^2 - 12*a*b*c*d + 10*a^2*d^2)*(a
+ b*x)^3)/(4 + n) + (d^2*(3*b*c - 5*a*d)*(a + b*x)^4)/(5 + n) + (d^3*(a + b*x)^5)/(6 + n)))/b^6
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(746\) vs.
\(2(212)=424\).
time = 0.08, size = 747, normalized size = 3.52 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(b*x+a)^n*(d*x+c)^3,x,method=_RETURNVERBOSE)
[Out]
d^3/(6+n)*x^6*exp(n*ln(b*x+a))+(3*a*b^2*c^2*d*n^3+b^3*c^3*n^3-12*a^2*b*c*d^2*n^2+33*a*b^2*c^2*d*n^2+15*b^3*c^3
*n^2+20*a^3*d^3*n-72*a^2*b*c*d^2*n+90*a*b^2*c^2*d*n+74*b^3*c^3*n+120*b^3*c^3)/b^3/(n^4+18*n^3+119*n^2+342*n+36
0)*x^3*exp(n*ln(b*x+a))+d^2*(a*d*n+3*b*c*n+18*b*c)/b/(n^2+11*n+30)*x^5*exp(n*ln(b*x+a))-2*a^3*(-b^3*c^3*n^3+9*
a*b^2*c^2*d*n^2-15*b^3*c^3*n^2-36*a^2*b*c*d^2*n+99*a*b^2*c^2*d*n-74*b^3*c^3*n+60*a^3*d^3-216*a^2*b*c*d^2+270*a
*b^2*c^2*d-120*b^3*c^3)/b^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)*exp(n*ln(b*x+a))-(-3*a*b*c*d*n^2-
3*b^2*c^2*n^2+5*a^2*d^2*n-18*a*b*c*d*n-33*b^2*c^2*n-90*b^2*c^2)*d/b^2/(n^3+15*n^2+74*n+120)*x^4*exp(n*ln(b*x+a
))+2/b^5*n*a^2*(-b^3*c^3*n^3+9*a*b^2*c^2*d*n^2-15*b^3*c^3*n^2-36*a^2*b*c*d^2*n+99*a*b^2*c^2*d*n-74*b^3*c^3*n+6
0*a^3*d^3-216*a^2*b*c*d^2+270*a*b^2*c^2*d-120*b^3*c^3)/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)*x*exp(
n*ln(b*x+a))-(-b^3*c^3*n^3+9*a*b^2*c^2*d*n^2-15*b^3*c^3*n^2-36*a^2*b*c*d^2*n+99*a*b^2*c^2*d*n-74*b^3*c^3*n+60*
a^3*d^3-216*a^2*b*c*d^2+270*a*b^2*c^2*d-120*b^3*c^3)*a/b^4*n/(n^5+20*n^4+155*n^3+580*n^2+1044*n+720)*x^2*exp(n
*ln(b*x+a))
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 507 vs.
\(2 (212) = 424\).
time = 0.30, size = 507, normalized size = 2.39 \begin {gather*} \frac {{\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^{3}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {3 \, {\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} c^{2} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {3 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x+c)^3,x, algorithm="maxima")
[Out]
((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^3/((n^3 + 6*n^2 + 11*n + 6
)*b^3) + 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*c^2*d/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4) + 3*((n^4 + 10*n^3 + 35*n^2 + 50*n
+ 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3
*b^2*x^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*c*d^2/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^5) + ((
n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a*b^5*x^5 - 5
*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^3*x^3 - 60*(n^2 + n)*a^4*b^2*x^2 + 12
0*a^5*b*n*x - 120*a^6)*(b*x + a)^n*d^3/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^6)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1075 vs.
\(2 (212) = 424\).
time = 0.42, size = 1075, normalized size = 5.07 \begin {gather*} \frac {{\left (2 \, a^{3} b^{3} c^{3} n^{3} + 240 \, a^{3} b^{3} c^{3} - 540 \, a^{4} b^{2} c^{2} d + 432 \, a^{5} b c d^{2} - 120 \, a^{6} d^{3} + {\left (b^{6} d^{3} n^{5} + 15 \, b^{6} d^{3} n^{4} + 85 \, b^{6} d^{3} n^{3} + 225 \, b^{6} d^{3} n^{2} + 274 \, b^{6} d^{3} n + 120 \, b^{6} d^{3}\right )} x^{6} + {\left (432 \, b^{6} c d^{2} + {\left (3 \, b^{6} c d^{2} + a b^{5} d^{3}\right )} n^{5} + 2 \, {\left (24 \, b^{6} c d^{2} + 5 \, a b^{5} d^{3}\right )} n^{4} + 5 \, {\left (57 \, b^{6} c d^{2} + 7 \, a b^{5} d^{3}\right )} n^{3} + 10 \, {\left (78 \, b^{6} c d^{2} + 5 \, a b^{5} d^{3}\right )} n^{2} + 12 \, {\left (81 \, b^{6} c d^{2} + 2 \, a b^{5} d^{3}\right )} n\right )} x^{5} + {\left (540 \, b^{6} c^{2} d + 3 \, {\left (b^{6} c^{2} d + a b^{5} c d^{2}\right )} n^{5} + {\left (51 \, b^{6} c^{2} d + 36 \, a b^{5} c d^{2} - 5 \, a^{2} b^{4} d^{3}\right )} n^{4} + 3 \, {\left (107 \, b^{6} c^{2} d + 47 \, a b^{5} c d^{2} - 10 \, a^{2} b^{4} d^{3}\right )} n^{3} + {\left (921 \, b^{6} c^{2} d + 216 \, a b^{5} c d^{2} - 55 \, a^{2} b^{4} d^{3}\right )} n^{2} + 6 \, {\left (198 \, b^{6} c^{2} d + 18 \, a b^{5} c d^{2} - 5 \, a^{2} b^{4} d^{3}\right )} n\right )} x^{4} + {\left (240 \, b^{6} c^{3} + {\left (b^{6} c^{3} + 3 \, a b^{5} c^{2} d\right )} n^{5} + 6 \, {\left (3 \, b^{6} c^{3} + 7 \, a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2}\right )} n^{4} + {\left (121 \, b^{6} c^{3} + 195 \, a b^{5} c^{2} d - 108 \, a^{2} b^{4} c d^{2} + 20 \, a^{3} b^{3} d^{3}\right )} n^{3} + 12 \, {\left (31 \, b^{6} c^{3} + 28 \, a b^{5} c^{2} d - 20 \, a^{2} b^{4} c d^{2} + 5 \, a^{3} b^{3} d^{3}\right )} n^{2} + 4 \, {\left (127 \, b^{6} c^{3} + 45 \, a b^{5} c^{2} d - 36 \, a^{2} b^{4} c d^{2} + 10 \, a^{3} b^{3} d^{3}\right )} n\right )} x^{3} + 6 \, {\left (5 \, a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d\right )} n^{2} + {\left (a b^{5} c^{3} n^{5} + {\left (16 \, a b^{5} c^{3} - 9 \, a^{2} b^{4} c^{2} d\right )} n^{4} + {\left (89 \, a b^{5} c^{3} - 108 \, a^{2} b^{4} c^{2} d + 36 \, a^{3} b^{3} c d^{2}\right )} n^{3} + {\left (194 \, a b^{5} c^{3} - 369 \, a^{2} b^{4} c^{2} d + 252 \, a^{3} b^{3} c d^{2} - 60 \, a^{4} b^{2} d^{3}\right )} n^{2} + 6 \, {\left (20 \, a b^{5} c^{3} - 45 \, a^{2} b^{4} c^{2} d + 36 \, a^{3} b^{3} c d^{2} - 10 \, a^{4} b^{2} d^{3}\right )} n\right )} x^{2} + 2 \, {\left (74 \, a^{3} b^{3} c^{3} - 99 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2}\right )} n - 2 \, {\left (a^{2} b^{4} c^{3} n^{4} + 3 \, {\left (5 \, a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d\right )} n^{3} + {\left (74 \, a^{2} b^{4} c^{3} - 99 \, a^{3} b^{3} c^{2} d + 36 \, a^{4} b^{2} c d^{2}\right )} n^{2} + 6 \, {\left (20 \, a^{2} b^{4} c^{3} - 45 \, a^{3} b^{3} c^{2} d + 36 \, a^{4} b^{2} c d^{2} - 10 \, a^{5} b d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x+c)^3,x, algorithm="fricas")
[Out]
(2*a^3*b^3*c^3*n^3 + 240*a^3*b^3*c^3 - 540*a^4*b^2*c^2*d + 432*a^5*b*c*d^2 - 120*a^6*d^3 + (b^6*d^3*n^5 + 15*b
^6*d^3*n^4 + 85*b^6*d^3*n^3 + 225*b^6*d^3*n^2 + 274*b^6*d^3*n + 120*b^6*d^3)*x^6 + (432*b^6*c*d^2 + (3*b^6*c*d
^2 + a*b^5*d^3)*n^5 + 2*(24*b^6*c*d^2 + 5*a*b^5*d^3)*n^4 + 5*(57*b^6*c*d^2 + 7*a*b^5*d^3)*n^3 + 10*(78*b^6*c*d
^2 + 5*a*b^5*d^3)*n^2 + 12*(81*b^6*c*d^2 + 2*a*b^5*d^3)*n)*x^5 + (540*b^6*c^2*d + 3*(b^6*c^2*d + a*b^5*c*d^2)*
n^5 + (51*b^6*c^2*d + 36*a*b^5*c*d^2 - 5*a^2*b^4*d^3)*n^4 + 3*(107*b^6*c^2*d + 47*a*b^5*c*d^2 - 10*a^2*b^4*d^3
)*n^3 + (921*b^6*c^2*d + 216*a*b^5*c*d^2 - 55*a^2*b^4*d^3)*n^2 + 6*(198*b^6*c^2*d + 18*a*b^5*c*d^2 - 5*a^2*b^4
*d^3)*n)*x^4 + (240*b^6*c^3 + (b^6*c^3 + 3*a*b^5*c^2*d)*n^5 + 6*(3*b^6*c^3 + 7*a*b^5*c^2*d - 2*a^2*b^4*c*d^2)*
n^4 + (121*b^6*c^3 + 195*a*b^5*c^2*d - 108*a^2*b^4*c*d^2 + 20*a^3*b^3*d^3)*n^3 + 12*(31*b^6*c^3 + 28*a*b^5*c^2
*d - 20*a^2*b^4*c*d^2 + 5*a^3*b^3*d^3)*n^2 + 4*(127*b^6*c^3 + 45*a*b^5*c^2*d - 36*a^2*b^4*c*d^2 + 10*a^3*b^3*d
^3)*n)*x^3 + 6*(5*a^3*b^3*c^3 - 3*a^4*b^2*c^2*d)*n^2 + (a*b^5*c^3*n^5 + (16*a*b^5*c^3 - 9*a^2*b^4*c^2*d)*n^4 +
(89*a*b^5*c^3 - 108*a^2*b^4*c^2*d + 36*a^3*b^3*c*d^2)*n^3 + (194*a*b^5*c^3 - 369*a^2*b^4*c^2*d + 252*a^3*b^3*
c*d^2 - 60*a^4*b^2*d^3)*n^2 + 6*(20*a*b^5*c^3 - 45*a^2*b^4*c^2*d + 36*a^3*b^3*c*d^2 - 10*a^4*b^2*d^3)*n)*x^2 +
2*(74*a^3*b^3*c^3 - 99*a^4*b^2*c^2*d + 36*a^5*b*c*d^2)*n - 2*(a^2*b^4*c^3*n^4 + 3*(5*a^2*b^4*c^3 - 3*a^3*b^3*
c^2*d)*n^3 + (74*a^2*b^4*c^3 - 99*a^3*b^3*c^2*d + 36*a^4*b^2*c*d^2)*n^2 + 6*(20*a^2*b^4*c^3 - 45*a^3*b^3*c^2*d
+ 36*a^4*b^2*c*d^2 - 10*a^5*b*d^3)*n)*x)*(b*x + a)^n/(b^6*n^6 + 21*b^6*n^5 + 175*b^6*n^4 + 735*b^6*n^3 + 1624
*b^6*n^2 + 1764*b^6*n + 720*b^6)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 13352 vs.
\(2 (197) = 394\).
time = 3.81, size = 13352, normalized size = 62.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*(b*x+a)**n*(d*x+c)**3,x)
[Out]
Piecewise((a**n*(c**3*x**3/3 + 3*c**2*d*x**4/4 + 3*c*d**2*x**5/5 + d**3*x**6/6), Eq(b, 0)), (60*a**5*d**3*log(
a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**
11*x**5) + 137*a**5*d**3/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**
10*x**4 + 60*b**11*x**5) - 36*a**4*b*c*d**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b*
*9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a**4*b*d**3*x*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x +
600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 625*a**4*b*d**3*x/(60*a**5*b**6
+ 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 9*a**3*b**2
*c**2*d/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**1
1*x**5) - 180*a**3*b**2*c*d**2*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 3
00*a*b**10*x**4 + 60*b**11*x**5) + 600*a**3*b**2*d**3*x**2*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*
a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 1100*a**3*b**2*d**3*x**2/(60*a**5*b*
*6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 2*a**2*b*
*3*c**3/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**1
1*x**5) - 45*a**2*b**3*c**2*d*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 30
0*a*b**10*x**4 + 60*b**11*x**5) - 360*a**2*b**3*c*d**2*x**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x*
*2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 600*a**2*b**3*d**3*x**3*log(a/b + x)/(60*a**5*b*
*6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 900*a**2*
b**3*d**3*x**3/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 +
60*b**11*x**5) - 10*a*b**4*c**3*x/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 +
300*a*b**10*x**4 + 60*b**11*x**5) - 90*a*b**4*c**2*d*x**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2
+ 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 360*a*b**4*c*d**2*x**3/(60*a**5*b**6 + 300*a**4*b*
*7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 300*a*b**4*d**3*x**4*log(
a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**
11*x**5) + 300*a*b**4*d**3*x**4/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 30
0*a*b**10*x**4 + 60*b**11*x**5) - 20*b**5*c**3*x**2/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600
*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 90*b**5*c**2*d*x**3/(60*a**5*b**6 + 300*a**4*b**7*x + 60
0*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) - 180*b**5*c*d**2*x**4/(60*a**5*b**6
+ 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) + 60*b**5*d**
3*x**5*log(a/b + x)/(60*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x*
*4 + 60*b**11*x**5), Eq(n, -6)), (-60*a**5*d**3*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**
2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 125*a**5*d**3/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*
b**9*x**3 + 12*b**10*x**4) + 36*a**4*b*c*d**2*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2
+ 48*a*b**9*x**3 + 12*b**10*x**4) + 75*a**4*b*c*d**2/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a
*b**9*x**3 + 12*b**10*x**4) - 240*a**4*b*d**3*x*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**
2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 440*a**4*b*d**3*x/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 4
8*a*b**9*x**3 + 12*b**10*x**4) - 9*a**3*b**2*c**2*d/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*
b**9*x**3 + 12*b**10*x**4) + 144*a**3*b**2*c*d**2*x*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8
*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) + 264*a**3*b**2*c*d**2*x/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8
*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 360*a**3*b**2*d**3*x**2*log(a/b + x)/(12*a**4*b**6 + 48*a**3*b**7*x
+ 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 540*a**3*b**2*d**3*x**2/(12*a**4*b**6 + 48*a**3*b**7*x
+ 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - a**2*b**3*c**3/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a*
*2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 36*a**2*b**3*c**2*d*x/(12*a**4*b**6 + 48*a**3*b**7*x + 72*a**
2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) + 216*a**2*b**3*c*d**2*x**2*log(a/b + x)/(12*a**4*b**6 + 48*a**3
*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) + 324*a**2*b**3*c*d**2*x**2/(12*a**4*b**6 + 48*a
**3*b**7*x + 72*a**2*b**8*x**2 + 48*a*b**9*x**3 + 12*b**10*x**4) - 240*a**2*b**3*d**3*x**3*log(a/b + x)/(12*a*
*4*b**6 + 48*a**3*b**7*x + 72*a**2*b**8*x**2 + ...
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1873 vs.
\(2 (212) = 424\).
time = 1.04, size = 1873, normalized size = 8.83 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*(b*x+a)^n*(d*x+c)^3,x, algorithm="giac")
[Out]
((b*x + a)^n*b^6*d^3*n^5*x^6 + 3*(b*x + a)^n*b^6*c*d^2*n^5*x^5 + (b*x + a)^n*a*b^5*d^3*n^5*x^5 + 15*(b*x + a)^
n*b^6*d^3*n^4*x^6 + 3*(b*x + a)^n*b^6*c^2*d*n^5*x^4 + 3*(b*x + a)^n*a*b^5*c*d^2*n^5*x^4 + 48*(b*x + a)^n*b^6*c
*d^2*n^4*x^5 + 10*(b*x + a)^n*a*b^5*d^3*n^4*x^5 + 85*(b*x + a)^n*b^6*d^3*n^3*x^6 + (b*x + a)^n*b^6*c^3*n^5*x^3
+ 3*(b*x + a)^n*a*b^5*c^2*d*n^5*x^3 + 51*(b*x + a)^n*b^6*c^2*d*n^4*x^4 + 36*(b*x + a)^n*a*b^5*c*d^2*n^4*x^4 -
5*(b*x + a)^n*a^2*b^4*d^3*n^4*x^4 + 285*(b*x + a)^n*b^6*c*d^2*n^3*x^5 + 35*(b*x + a)^n*a*b^5*d^3*n^3*x^5 + 22
5*(b*x + a)^n*b^6*d^3*n^2*x^6 + (b*x + a)^n*a*b^5*c^3*n^5*x^2 + 18*(b*x + a)^n*b^6*c^3*n^4*x^3 + 42*(b*x + a)^
n*a*b^5*c^2*d*n^4*x^3 - 12*(b*x + a)^n*a^2*b^4*c*d^2*n^4*x^3 + 321*(b*x + a)^n*b^6*c^2*d*n^3*x^4 + 141*(b*x +
a)^n*a*b^5*c*d^2*n^3*x^4 - 30*(b*x + a)^n*a^2*b^4*d^3*n^3*x^4 + 780*(b*x + a)^n*b^6*c*d^2*n^2*x^5 + 50*(b*x +
a)^n*a*b^5*d^3*n^2*x^5 + 274*(b*x + a)^n*b^6*d^3*n*x^6 + 16*(b*x + a)^n*a*b^5*c^3*n^4*x^2 - 9*(b*x + a)^n*a^2*
b^4*c^2*d*n^4*x^2 + 121*(b*x + a)^n*b^6*c^3*n^3*x^3 + 195*(b*x + a)^n*a*b^5*c^2*d*n^3*x^3 - 108*(b*x + a)^n*a^
2*b^4*c*d^2*n^3*x^3 + 20*(b*x + a)^n*a^3*b^3*d^3*n^3*x^3 + 921*(b*x + a)^n*b^6*c^2*d*n^2*x^4 + 216*(b*x + a)^n
*a*b^5*c*d^2*n^2*x^4 - 55*(b*x + a)^n*a^2*b^4*d^3*n^2*x^4 + 972*(b*x + a)^n*b^6*c*d^2*n*x^5 + 24*(b*x + a)^n*a
*b^5*d^3*n*x^5 + 120*(b*x + a)^n*b^6*d^3*x^6 - 2*(b*x + a)^n*a^2*b^4*c^3*n^4*x + 89*(b*x + a)^n*a*b^5*c^3*n^3*
x^2 - 108*(b*x + a)^n*a^2*b^4*c^2*d*n^3*x^2 + 36*(b*x + a)^n*a^3*b^3*c*d^2*n^3*x^2 + 372*(b*x + a)^n*b^6*c^3*n
^2*x^3 + 336*(b*x + a)^n*a*b^5*c^2*d*n^2*x^3 - 240*(b*x + a)^n*a^2*b^4*c*d^2*n^2*x^3 + 60*(b*x + a)^n*a^3*b^3*
d^3*n^2*x^3 + 1188*(b*x + a)^n*b^6*c^2*d*n*x^4 + 108*(b*x + a)^n*a*b^5*c*d^2*n*x^4 - 30*(b*x + a)^n*a^2*b^4*d^
3*n*x^4 + 432*(b*x + a)^n*b^6*c*d^2*x^5 - 30*(b*x + a)^n*a^2*b^4*c^3*n^3*x + 18*(b*x + a)^n*a^3*b^3*c^2*d*n^3*
x + 194*(b*x + a)^n*a*b^5*c^3*n^2*x^2 - 369*(b*x + a)^n*a^2*b^4*c^2*d*n^2*x^2 + 252*(b*x + a)^n*a^3*b^3*c*d^2*
n^2*x^2 - 60*(b*x + a)^n*a^4*b^2*d^3*n^2*x^2 + 508*(b*x + a)^n*b^6*c^3*n*x^3 + 180*(b*x + a)^n*a*b^5*c^2*d*n*x
^3 - 144*(b*x + a)^n*a^2*b^4*c*d^2*n*x^3 + 40*(b*x + a)^n*a^3*b^3*d^3*n*x^3 + 540*(b*x + a)^n*b^6*c^2*d*x^4 +
2*(b*x + a)^n*a^3*b^3*c^3*n^3 - 148*(b*x + a)^n*a^2*b^4*c^3*n^2*x + 198*(b*x + a)^n*a^3*b^3*c^2*d*n^2*x - 72*(
b*x + a)^n*a^4*b^2*c*d^2*n^2*x + 120*(b*x + a)^n*a*b^5*c^3*n*x^2 - 270*(b*x + a)^n*a^2*b^4*c^2*d*n*x^2 + 216*(
b*x + a)^n*a^3*b^3*c*d^2*n*x^2 - 60*(b*x + a)^n*a^4*b^2*d^3*n*x^2 + 240*(b*x + a)^n*b^6*c^3*x^3 + 30*(b*x + a)
^n*a^3*b^3*c^3*n^2 - 18*(b*x + a)^n*a^4*b^2*c^2*d*n^2 - 240*(b*x + a)^n*a^2*b^4*c^3*n*x + 540*(b*x + a)^n*a^3*
b^3*c^2*d*n*x - 432*(b*x + a)^n*a^4*b^2*c*d^2*n*x + 120*(b*x + a)^n*a^5*b*d^3*n*x + 148*(b*x + a)^n*a^3*b^3*c^
3*n - 198*(b*x + a)^n*a^4*b^2*c^2*d*n + 72*(b*x + a)^n*a^5*b*c*d^2*n + 240*(b*x + a)^n*a^3*b^3*c^3 - 540*(b*x
+ a)^n*a^4*b^2*c^2*d + 432*(b*x + a)^n*a^5*b*c*d^2 - 120*(b*x + a)^n*a^6*d^3)/(b^6*n^6 + 21*b^6*n^5 + 175*b^6*
n^4 + 735*b^6*n^3 + 1624*b^6*n^2 + 1764*b^6*n + 720*b^6)
________________________________________________________________________________________
Mupad [B]
time = 1.54, size = 867, normalized size = 4.09 \begin {gather*} \frac {d^3\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}+\frac {2\,a^3\,{\left (a+b\,x\right )}^n\,\left (-60\,a^3\,d^3+36\,a^2\,b\,c\,d^2\,n+216\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d\,n^2-99\,a\,b^2\,c^2\,d\,n-270\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+15\,b^3\,c^3\,n^2+74\,b^3\,c^3\,n+120\,b^3\,c^3\right )}{b^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (20\,a^3\,d^3\,n-12\,a^2\,b\,c\,d^2\,n^2-72\,a^2\,b\,c\,d^2\,n+3\,a\,b^2\,c^2\,d\,n^3+33\,a\,b^2\,c^2\,d\,n^2+90\,a\,b^2\,c^2\,d\,n+b^3\,c^3\,n^3+15\,b^3\,c^3\,n^2+74\,b^3\,c^3\,n+120\,b^3\,c^3\right )}{b^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {2\,a^2\,n\,x\,{\left (a+b\,x\right )}^n\,\left (-60\,a^3\,d^3+36\,a^2\,b\,c\,d^2\,n+216\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d\,n^2-99\,a\,b^2\,c^2\,d\,n-270\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+15\,b^3\,c^3\,n^2+74\,b^3\,c^3\,n+120\,b^3\,c^3\right )}{b^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {d^2\,x^5\,{\left (a+b\,x\right )}^n\,\left (18\,b\,c+a\,d\,n+3\,b\,c\,n\right )\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {d\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (-5\,a^2\,d^2\,n+3\,a\,b\,c\,d\,n^2+18\,a\,b\,c\,d\,n+3\,b^2\,c^2\,n^2+33\,b^2\,c^2\,n+90\,b^2\,c^2\right )}{b^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-60\,a^3\,d^3+36\,a^2\,b\,c\,d^2\,n+216\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d\,n^2-99\,a\,b^2\,c^2\,d\,n-270\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+15\,b^3\,c^3\,n^2+74\,b^3\,c^3\,n+120\,b^3\,c^3\right )}{b^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*(a + b*x)^n*(c + d*x)^3,x)
[Out]
(d^3*x^6*(a + b*x)^n*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 +
21*n^5 + n^6 + 720) + (2*a^3*(a + b*x)^n*(120*b^3*c^3 - 60*a^3*d^3 + 74*b^3*c^3*n + 15*b^3*c^3*n^2 + b^3*c^3*
n^3 - 270*a*b^2*c^2*d + 216*a^2*b*c*d^2 - 99*a*b^2*c^2*d*n + 36*a^2*b*c*d^2*n - 9*a*b^2*c^2*d*n^2))/(b^6*(1764
*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (x^3*(a + b*x)^n*(3*n + n^2 + 2)*(120*b^3*c^3 + 20*
a^3*d^3*n + 74*b^3*c^3*n + 15*b^3*c^3*n^2 + b^3*c^3*n^3 + 90*a*b^2*c^2*d*n - 72*a^2*b*c*d^2*n + 33*a*b^2*c^2*d
*n^2 - 12*a^2*b*c*d^2*n^2 + 3*a*b^2*c^2*d*n^3))/(b^3*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 7
20)) - (2*a^2*n*x*(a + b*x)^n*(120*b^3*c^3 - 60*a^3*d^3 + 74*b^3*c^3*n + 15*b^3*c^3*n^2 + b^3*c^3*n^3 - 270*a*
b^2*c^2*d + 216*a^2*b*c*d^2 - 99*a*b^2*c^2*d*n + 36*a^2*b*c*d^2*n - 9*a*b^2*c^2*d*n^2))/(b^5*(1764*n + 1624*n^
2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (d^2*x^5*(a + b*x)^n*(18*b*c + a*d*n + 3*b*c*n)*(50*n + 35*n^2
+ 10*n^3 + n^4 + 24))/(b*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (d*x^4*(a + b*x)^n*(1
1*n + 6*n^2 + n^3 + 6)*(90*b^2*c^2 - 5*a^2*d^2*n + 33*b^2*c^2*n + 3*b^2*c^2*n^2 + 18*a*b*c*d*n + 3*a*b*c*d*n^2
))/(b^2*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (a*n*x^2*(n + 1)*(a + b*x)^n*(120*b^3*
c^3 - 60*a^3*d^3 + 74*b^3*c^3*n + 15*b^3*c^3*n^2 + b^3*c^3*n^3 - 270*a*b^2*c^2*d + 216*a^2*b*c*d^2 - 99*a*b^2*
c^2*d*n + 36*a^2*b*c*d^2*n - 9*a*b^2*c^2*d*n^2))/(b^4*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 +
720))
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