3.4.78 \(\int x^m (a+b x)^2 (c+d x)^5 \, dx\) [378]
Optimal. Leaf size=231 \[ \frac {a^2 c^5 x^{1+m}}{1+m}+\frac {a c^4 (2 b c+5 a d) x^{2+m}}{2+m}+\frac {c^3 \left (b^2 c^2+10 a b c d+10 a^2 d^2\right ) x^{3+m}}{3+m}+\frac {5 c^2 d \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^{4+m}}{4+m}+\frac {5 c d^2 \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5+m}}{5+m}+\frac {d^3 \left (10 b^2 c^2+10 a b c d+a^2 d^2\right ) x^{6+m}}{6+m}+\frac {b d^4 (5 b c+2 a d) x^{7+m}}{7+m}+\frac {b^2 d^5 x^{8+m}}{8+m} \]
[Out]
a^2*c^5*x^(1+m)/(1+m)+a*c^4*(5*a*d+2*b*c)*x^(2+m)/(2+m)+c^3*(10*a^2*d^2+10*a*b*c*d+b^2*c^2)*x^(3+m)/(3+m)+5*c^
2*d*(2*a^2*d^2+4*a*b*c*d+b^2*c^2)*x^(4+m)/(4+m)+5*c*d^2*(a^2*d^2+4*a*b*c*d+2*b^2*c^2)*x^(5+m)/(5+m)+d^3*(a^2*d
^2+10*a*b*c*d+10*b^2*c^2)*x^(6+m)/(6+m)+b*d^4*(2*a*d+5*b*c)*x^(7+m)/(7+m)+b^2*d^5*x^(8+m)/(8+m)
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Rubi [A]
time = 0.09, antiderivative size = 231, 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 {5 c^2 d x^{m+4} \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )}{m+4}+\frac {5 c d^2 x^{m+5} \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )}{m+5}+\frac {d^3 x^{m+6} \left (a^2 d^2+10 a b c d+10 b^2 c^2\right )}{m+6}+\frac {c^3 x^{m+3} \left (10 a^2 d^2+10 a b c d+b^2 c^2\right )}{m+3}+\frac {a^2 c^5 x^{m+1}}{m+1}+\frac {a c^4 x^{m+2} (5 a d+2 b c)}{m+2}+\frac {b d^4 x^{m+7} (2 a d+5 b c)}{m+7}+\frac {b^2 d^5 x^{m+8}}{m+8} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^m*(a + b*x)^2*(c + d*x)^5,x]
[Out]
(a^2*c^5*x^(1 + m))/(1 + m) + (a*c^4*(2*b*c + 5*a*d)*x^(2 + m))/(2 + m) + (c^3*(b^2*c^2 + 10*a*b*c*d + 10*a^2*
d^2)*x^(3 + m))/(3 + m) + (5*c^2*d*(b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^(4 + m))/(4 + m) + (5*c*d^2*(2*b^2*c^2
+ 4*a*b*c*d + a^2*d^2)*x^(5 + m))/(5 + m) + (d^3*(10*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^(6 + m))/(6 + m) + (b*d
^4*(5*b*c + 2*a*d)*x^(7 + m))/(7 + m) + (b^2*d^5*x^(8 + m))/(8 + m)
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^m (a+b x)^2 (c+d x)^5 \, dx &=\int \left (a^2 c^5 x^m+a c^4 (2 b c+5 a d) x^{1+m}+c^3 \left (b^2 c^2+10 a b c d+10 a^2 d^2\right ) x^{2+m}+5 c^2 d \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^{3+m}+5 c d^2 \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^{4+m}+d^3 \left (10 b^2 c^2+10 a b c d+a^2 d^2\right ) x^{5+m}+b d^4 (5 b c+2 a d) x^{6+m}+b^2 d^5 x^{7+m}\right ) \, dx\\ &=\frac {a^2 c^5 x^{1+m}}{1+m}+\frac {a c^4 (2 b c+5 a d) x^{2+m}}{2+m}+\frac {c^3 \left (b^2 c^2+10 a b c d+10 a^2 d^2\right ) x^{3+m}}{3+m}+\frac {5 c^2 d \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^{4+m}}{4+m}+\frac {5 c d^2 \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5+m}}{5+m}+\frac {d^3 \left (10 b^2 c^2+10 a b c d+a^2 d^2\right ) x^{6+m}}{6+m}+\frac {b d^4 (5 b c+2 a d) x^{7+m}}{7+m}+\frac {b^2 d^5 x^{8+m}}{8+m}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 216, normalized size = 0.94 \begin {gather*} x^{1+m} \left (\frac {a^2 c^5}{1+m}+\frac {a c^4 (2 b c+5 a d) x}{2+m}+\frac {c^3 \left (b^2 c^2+10 a b c d+10 a^2 d^2\right ) x^2}{3+m}+\frac {5 c^2 d \left (b^2 c^2+4 a b c d+2 a^2 d^2\right ) x^3}{4+m}+\frac {5 c d^2 \left (2 b^2 c^2+4 a b c d+a^2 d^2\right ) x^4}{5+m}+\frac {d^3 \left (10 b^2 c^2+10 a b c d+a^2 d^2\right ) x^5}{6+m}+\frac {b d^4 (5 b c+2 a d) x^6}{7+m}+\frac {b^2 d^5 x^7}{8+m}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^m*(a + b*x)^2*(c + d*x)^5,x]
[Out]
x^(1 + m)*((a^2*c^5)/(1 + m) + (a*c^4*(2*b*c + 5*a*d)*x)/(2 + m) + (c^3*(b^2*c^2 + 10*a*b*c*d + 10*a^2*d^2)*x^
2)/(3 + m) + (5*c^2*d*(b^2*c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^3)/(4 + m) + (5*c*d^2*(2*b^2*c^2 + 4*a*b*c*d + a^2*d
^2)*x^4)/(5 + m) + (d^3*(10*b^2*c^2 + 10*a*b*c*d + a^2*d^2)*x^5)/(6 + m) + (b*d^4*(5*b*c + 2*a*d)*x^6)/(7 + m)
+ (b^2*d^5*x^7)/(8 + m))
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Maple [A]
time = 0.08, size = 254, normalized size = 1.10
| | |
| method |
result |
size |
| | |
| norman |
\(\frac {a^{2} c^{5} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {b^{2} d^{5} x^{8} {\mathrm e}^{m \ln \left (x \right )}}{8+m}+\frac {c^{3} \left (10 a^{2} d^{2}+10 a b c d +b^{2} c^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {d^{3} \left (a^{2} d^{2}+10 a b c d +10 b^{2} c^{2}\right ) x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}+\frac {a \,c^{4} \left (5 a d +2 b c \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {b \,d^{4} \left (2 a d +5 b c \right ) x^{7} {\mathrm e}^{m \ln \left (x \right )}}{7+m}+\frac {5 c \,d^{2} \left (a^{2} d^{2}+4 a b c d +2 b^{2} c^{2}\right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {5 c^{2} d \left (2 a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}\) |
\(254\) |
| risch |
\(\text {Expression too large to display}\) |
\(2057\) |
| gosper |
\(\text {Expression too large to display}\) |
\(2058\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(b*x+a)^2*(d*x+c)^5,x,method=_RETURNVERBOSE)
[Out]
a^2*c^5/(1+m)*x*exp(m*ln(x))+b^2*d^5/(8+m)*x^8*exp(m*ln(x))+c^3*(10*a^2*d^2+10*a*b*c*d+b^2*c^2)/(3+m)*x^3*exp(
m*ln(x))+d^3*(a^2*d^2+10*a*b*c*d+10*b^2*c^2)/(6+m)*x^6*exp(m*ln(x))+a*c^4*(5*a*d+2*b*c)/(2+m)*x^2*exp(m*ln(x))
+b*d^4*(2*a*d+5*b*c)/(7+m)*x^7*exp(m*ln(x))+5*c*d^2*(a^2*d^2+4*a*b*c*d+2*b^2*c^2)/(5+m)*x^5*exp(m*ln(x))+5*c^2
*d*(2*a^2*d^2+4*a*b*c*d+b^2*c^2)/(4+m)*x^4*exp(m*ln(x))
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Maxima [A]
time = 0.27, size = 339, normalized size = 1.47 \begin {gather*} \frac {b^{2} d^{5} x^{m + 8}}{m + 8} + \frac {5 \, b^{2} c d^{4} x^{m + 7}}{m + 7} + \frac {2 \, a b d^{5} x^{m + 7}}{m + 7} + \frac {10 \, b^{2} c^{2} d^{3} x^{m + 6}}{m + 6} + \frac {10 \, a b c d^{4} x^{m + 6}}{m + 6} + \frac {a^{2} d^{5} x^{m + 6}}{m + 6} + \frac {10 \, b^{2} c^{3} d^{2} x^{m + 5}}{m + 5} + \frac {20 \, a b c^{2} d^{3} x^{m + 5}}{m + 5} + \frac {5 \, a^{2} c d^{4} x^{m + 5}}{m + 5} + \frac {5 \, b^{2} c^{4} d x^{m + 4}}{m + 4} + \frac {20 \, a b c^{3} d^{2} x^{m + 4}}{m + 4} + \frac {10 \, a^{2} c^{2} d^{3} x^{m + 4}}{m + 4} + \frac {b^{2} c^{5} x^{m + 3}}{m + 3} + \frac {10 \, a b c^{4} d x^{m + 3}}{m + 3} + \frac {10 \, a^{2} c^{3} d^{2} x^{m + 3}}{m + 3} + \frac {2 \, a b c^{5} x^{m + 2}}{m + 2} + \frac {5 \, a^{2} c^{4} d x^{m + 2}}{m + 2} + \frac {a^{2} c^{5} x^{m + 1}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x+a)^2*(d*x+c)^5,x, algorithm="maxima")
[Out]
b^2*d^5*x^(m + 8)/(m + 8) + 5*b^2*c*d^4*x^(m + 7)/(m + 7) + 2*a*b*d^5*x^(m + 7)/(m + 7) + 10*b^2*c^2*d^3*x^(m
+ 6)/(m + 6) + 10*a*b*c*d^4*x^(m + 6)/(m + 6) + a^2*d^5*x^(m + 6)/(m + 6) + 10*b^2*c^3*d^2*x^(m + 5)/(m + 5) +
20*a*b*c^2*d^3*x^(m + 5)/(m + 5) + 5*a^2*c*d^4*x^(m + 5)/(m + 5) + 5*b^2*c^4*d*x^(m + 4)/(m + 4) + 20*a*b*c^3
*d^2*x^(m + 4)/(m + 4) + 10*a^2*c^2*d^3*x^(m + 4)/(m + 4) + b^2*c^5*x^(m + 3)/(m + 3) + 10*a*b*c^4*d*x^(m + 3)
/(m + 3) + 10*a^2*c^3*d^2*x^(m + 3)/(m + 3) + 2*a*b*c^5*x^(m + 2)/(m + 2) + 5*a^2*c^4*d*x^(m + 2)/(m + 2) + a^
2*c^5*x^(m + 1)/(m + 1)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1623 vs.
\(2 (231) = 462\).
time = 1.35, size = 1623, normalized size = 7.03 \begin {gather*} \frac {{\left ({\left (b^{2} d^{5} m^{7} + 28 \, b^{2} d^{5} m^{6} + 322 \, b^{2} d^{5} m^{5} + 1960 \, b^{2} d^{5} m^{4} + 6769 \, b^{2} d^{5} m^{3} + 13132 \, b^{2} d^{5} m^{2} + 13068 \, b^{2} d^{5} m + 5040 \, b^{2} d^{5}\right )} x^{8} + {\left ({\left (5 \, b^{2} c d^{4} + 2 \, a b d^{5}\right )} m^{7} + 28800 \, b^{2} c d^{4} + 11520 \, a b d^{5} + 29 \, {\left (5 \, b^{2} c d^{4} + 2 \, a b d^{5}\right )} m^{6} + 343 \, {\left (5 \, b^{2} c d^{4} + 2 \, a b d^{5}\right )} m^{5} + 2135 \, {\left (5 \, b^{2} c d^{4} + 2 \, a b d^{5}\right )} m^{4} + 7504 \, {\left (5 \, b^{2} c d^{4} + 2 \, a b d^{5}\right )} m^{3} + 14756 \, {\left (5 \, b^{2} c d^{4} + 2 \, a b d^{5}\right )} m^{2} + 14832 \, {\left (5 \, b^{2} c d^{4} + 2 \, a b d^{5}\right )} m\right )} x^{7} + {\left ({\left (10 \, b^{2} c^{2} d^{3} + 10 \, a b c d^{4} + a^{2} d^{5}\right )} m^{7} + 67200 \, b^{2} c^{2} d^{3} + 67200 \, a b c d^{4} + 6720 \, a^{2} d^{5} + 30 \, {\left (10 \, b^{2} c^{2} d^{3} + 10 \, a b c d^{4} + a^{2} d^{5}\right )} m^{6} + 366 \, {\left (10 \, b^{2} c^{2} d^{3} + 10 \, a b c d^{4} + a^{2} d^{5}\right )} m^{5} + 2340 \, {\left (10 \, b^{2} c^{2} d^{3} + 10 \, a b c d^{4} + a^{2} d^{5}\right )} m^{4} + 8409 \, {\left (10 \, b^{2} c^{2} d^{3} + 10 \, a b c d^{4} + a^{2} d^{5}\right )} m^{3} + 16830 \, {\left (10 \, b^{2} c^{2} d^{3} + 10 \, a b c d^{4} + a^{2} d^{5}\right )} m^{2} + 17144 \, {\left (10 \, b^{2} c^{2} d^{3} + 10 \, a b c d^{4} + a^{2} d^{5}\right )} m\right )} x^{6} + 5 \, {\left ({\left (2 \, b^{2} c^{3} d^{2} + 4 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} m^{7} + 16128 \, b^{2} c^{3} d^{2} + 32256 \, a b c^{2} d^{3} + 8064 \, a^{2} c d^{4} + 31 \, {\left (2 \, b^{2} c^{3} d^{2} + 4 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} m^{6} + 391 \, {\left (2 \, b^{2} c^{3} d^{2} + 4 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} m^{5} + 2581 \, {\left (2 \, b^{2} c^{3} d^{2} + 4 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} m^{4} + 9544 \, {\left (2 \, b^{2} c^{3} d^{2} + 4 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} m^{3} + 19564 \, {\left (2 \, b^{2} c^{3} d^{2} + 4 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} m^{2} + 20304 \, {\left (2 \, b^{2} c^{3} d^{2} + 4 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} m\right )} x^{5} + 5 \, {\left ({\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3}\right )} m^{7} + 10080 \, b^{2} c^{4} d + 40320 \, a b c^{3} d^{2} + 20160 \, a^{2} c^{2} d^{3} + 32 \, {\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3}\right )} m^{6} + 418 \, {\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3}\right )} m^{5} + 2864 \, {\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3}\right )} m^{4} + 10993 \, {\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3}\right )} m^{3} + 23312 \, {\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3}\right )} m^{2} + 24876 \, {\left (b^{2} c^{4} d + 4 \, a b c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3}\right )} m\right )} x^{4} + {\left ({\left (b^{2} c^{5} + 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} m^{7} + 13440 \, b^{2} c^{5} + 134400 \, a b c^{4} d + 134400 \, a^{2} c^{3} d^{2} + 33 \, {\left (b^{2} c^{5} + 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} m^{6} + 447 \, {\left (b^{2} c^{5} + 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} m^{5} + 3195 \, {\left (b^{2} c^{5} + 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} m^{4} + 12864 \, {\left (b^{2} c^{5} + 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} m^{3} + 28692 \, {\left (b^{2} c^{5} + 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} m^{2} + 32048 \, {\left (b^{2} c^{5} + 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} m\right )} x^{3} + {\left ({\left (2 \, a b c^{5} + 5 \, a^{2} c^{4} d\right )} m^{7} + 40320 \, a b c^{5} + 100800 \, a^{2} c^{4} d + 34 \, {\left (2 \, a b c^{5} + 5 \, a^{2} c^{4} d\right )} m^{6} + 478 \, {\left (2 \, a b c^{5} + 5 \, a^{2} c^{4} d\right )} m^{5} + 3580 \, {\left (2 \, a b c^{5} + 5 \, a^{2} c^{4} d\right )} m^{4} + 15289 \, {\left (2 \, a b c^{5} + 5 \, a^{2} c^{4} d\right )} m^{3} + 36706 \, {\left (2 \, a b c^{5} + 5 \, a^{2} c^{4} d\right )} m^{2} + 44712 \, {\left (2 \, a b c^{5} + 5 \, a^{2} c^{4} d\right )} m\right )} x^{2} + {\left (a^{2} c^{5} m^{7} + 35 \, a^{2} c^{5} m^{6} + 511 \, a^{2} c^{5} m^{5} + 4025 \, a^{2} c^{5} m^{4} + 18424 \, a^{2} c^{5} m^{3} + 48860 \, a^{2} c^{5} m^{2} + 69264 \, a^{2} c^{5} m + 40320 \, a^{2} c^{5}\right )} x\right )} x^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x+a)^2*(d*x+c)^5,x, algorithm="fricas")
[Out]
((b^2*d^5*m^7 + 28*b^2*d^5*m^6 + 322*b^2*d^5*m^5 + 1960*b^2*d^5*m^4 + 6769*b^2*d^5*m^3 + 13132*b^2*d^5*m^2 + 1
3068*b^2*d^5*m + 5040*b^2*d^5)*x^8 + ((5*b^2*c*d^4 + 2*a*b*d^5)*m^7 + 28800*b^2*c*d^4 + 11520*a*b*d^5 + 29*(5*
b^2*c*d^4 + 2*a*b*d^5)*m^6 + 343*(5*b^2*c*d^4 + 2*a*b*d^5)*m^5 + 2135*(5*b^2*c*d^4 + 2*a*b*d^5)*m^4 + 7504*(5*
b^2*c*d^4 + 2*a*b*d^5)*m^3 + 14756*(5*b^2*c*d^4 + 2*a*b*d^5)*m^2 + 14832*(5*b^2*c*d^4 + 2*a*b*d^5)*m)*x^7 + ((
10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^7 + 67200*b^2*c^2*d^3 + 67200*a*b*c*d^4 + 6720*a^2*d^5 + 30*(10*b^2
*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^6 + 366*(10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^5 + 2340*(10*b^2*c^2*
d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^4 + 8409*(10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m^3 + 16830*(10*b^2*c^2*d^3
+ 10*a*b*c*d^4 + a^2*d^5)*m^2 + 17144*(10*b^2*c^2*d^3 + 10*a*b*c*d^4 + a^2*d^5)*m)*x^6 + 5*((2*b^2*c^3*d^2 +
4*a*b*c^2*d^3 + a^2*c*d^4)*m^7 + 16128*b^2*c^3*d^2 + 32256*a*b*c^2*d^3 + 8064*a^2*c*d^4 + 31*(2*b^2*c^3*d^2 +
4*a*b*c^2*d^3 + a^2*c*d^4)*m^6 + 391*(2*b^2*c^3*d^2 + 4*a*b*c^2*d^3 + a^2*c*d^4)*m^5 + 2581*(2*b^2*c^3*d^2 + 4
*a*b*c^2*d^3 + a^2*c*d^4)*m^4 + 9544*(2*b^2*c^3*d^2 + 4*a*b*c^2*d^3 + a^2*c*d^4)*m^3 + 19564*(2*b^2*c^3*d^2 +
4*a*b*c^2*d^3 + a^2*c*d^4)*m^2 + 20304*(2*b^2*c^3*d^2 + 4*a*b*c^2*d^3 + a^2*c*d^4)*m)*x^5 + 5*((b^2*c^4*d + 4*
a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m^7 + 10080*b^2*c^4*d + 40320*a*b*c^3*d^2 + 20160*a^2*c^2*d^3 + 32*(b^2*c^4*d + 4
*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m^6 + 418*(b^2*c^4*d + 4*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m^5 + 2864*(b^2*c^4*d + 4*
a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m^4 + 10993*(b^2*c^4*d + 4*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m^3 + 23312*(b^2*c^4*d +
4*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m^2 + 24876*(b^2*c^4*d + 4*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*m)*x^4 + ((b^2*c^5 + 10
*a*b*c^4*d + 10*a^2*c^3*d^2)*m^7 + 13440*b^2*c^5 + 134400*a*b*c^4*d + 134400*a^2*c^3*d^2 + 33*(b^2*c^5 + 10*a*
b*c^4*d + 10*a^2*c^3*d^2)*m^6 + 447*(b^2*c^5 + 10*a*b*c^4*d + 10*a^2*c^3*d^2)*m^5 + 3195*(b^2*c^5 + 10*a*b*c^4
*d + 10*a^2*c^3*d^2)*m^4 + 12864*(b^2*c^5 + 10*a*b*c^4*d + 10*a^2*c^3*d^2)*m^3 + 28692*(b^2*c^5 + 10*a*b*c^4*d
+ 10*a^2*c^3*d^2)*m^2 + 32048*(b^2*c^5 + 10*a*b*c^4*d + 10*a^2*c^3*d^2)*m)*x^3 + ((2*a*b*c^5 + 5*a^2*c^4*d)*m
^7 + 40320*a*b*c^5 + 100800*a^2*c^4*d + 34*(2*a*b*c^5 + 5*a^2*c^4*d)*m^6 + 478*(2*a*b*c^5 + 5*a^2*c^4*d)*m^5 +
3580*(2*a*b*c^5 + 5*a^2*c^4*d)*m^4 + 15289*(2*a*b*c^5 + 5*a^2*c^4*d)*m^3 + 36706*(2*a*b*c^5 + 5*a^2*c^4*d)*m^
2 + 44712*(2*a*b*c^5 + 5*a^2*c^4*d)*m)*x^2 + (a^2*c^5*m^7 + 35*a^2*c^5*m^6 + 511*a^2*c^5*m^5 + 4025*a^2*c^5*m^
4 + 18424*a^2*c^5*m^3 + 48860*a^2*c^5*m^2 + 69264*a^2*c^5*m + 40320*a^2*c^5)*x)*x^m/(m^8 + 36*m^7 + 546*m^6 +
4536*m^5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 10401 vs.
\(2 (226) = 452\).
time = 1.06, size = 10401, normalized size = 45.03 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*(b*x+a)**2*(d*x+c)**5,x)
[Out]
Piecewise((-a**2*c**5/(7*x**7) - 5*a**2*c**4*d/(6*x**6) - 2*a**2*c**3*d**2/x**5 - 5*a**2*c**2*d**3/(2*x**4) -
5*a**2*c*d**4/(3*x**3) - a**2*d**5/(2*x**2) - a*b*c**5/(3*x**6) - 2*a*b*c**4*d/x**5 - 5*a*b*c**3*d**2/x**4 - 2
0*a*b*c**2*d**3/(3*x**3) - 5*a*b*c*d**4/x**2 - 2*a*b*d**5/x - b**2*c**5/(5*x**5) - 5*b**2*c**4*d/(4*x**4) - 10
*b**2*c**3*d**2/(3*x**3) - 5*b**2*c**2*d**3/x**2 - 5*b**2*c*d**4/x + b**2*d**5*log(x), Eq(m, -8)), (-a**2*c**5
/(6*x**6) - a**2*c**4*d/x**5 - 5*a**2*c**3*d**2/(2*x**4) - 10*a**2*c**2*d**3/(3*x**3) - 5*a**2*c*d**4/(2*x**2)
- a**2*d**5/x - 2*a*b*c**5/(5*x**5) - 5*a*b*c**4*d/(2*x**4) - 20*a*b*c**3*d**2/(3*x**3) - 10*a*b*c**2*d**3/x*
*2 - 10*a*b*c*d**4/x + 2*a*b*d**5*log(x) - b**2*c**5/(4*x**4) - 5*b**2*c**4*d/(3*x**3) - 5*b**2*c**3*d**2/x**2
- 10*b**2*c**2*d**3/x + 5*b**2*c*d**4*log(x) + b**2*d**5*x, Eq(m, -7)), (-a**2*c**5/(5*x**5) - 5*a**2*c**4*d/
(4*x**4) - 10*a**2*c**3*d**2/(3*x**3) - 5*a**2*c**2*d**3/x**2 - 5*a**2*c*d**4/x + a**2*d**5*log(x) - a*b*c**5/
(2*x**4) - 10*a*b*c**4*d/(3*x**3) - 10*a*b*c**3*d**2/x**2 - 20*a*b*c**2*d**3/x + 10*a*b*c*d**4*log(x) + 2*a*b*
d**5*x - b**2*c**5/(3*x**3) - 5*b**2*c**4*d/(2*x**2) - 10*b**2*c**3*d**2/x + 10*b**2*c**2*d**3*log(x) + 5*b**2
*c*d**4*x + b**2*d**5*x**2/2, Eq(m, -6)), (-a**2*c**5/(4*x**4) - 5*a**2*c**4*d/(3*x**3) - 5*a**2*c**3*d**2/x**
2 - 10*a**2*c**2*d**3/x + 5*a**2*c*d**4*log(x) + a**2*d**5*x - 2*a*b*c**5/(3*x**3) - 5*a*b*c**4*d/x**2 - 20*a*
b*c**3*d**2/x + 20*a*b*c**2*d**3*log(x) + 10*a*b*c*d**4*x + a*b*d**5*x**2 - b**2*c**5/(2*x**2) - 5*b**2*c**4*d
/x + 10*b**2*c**3*d**2*log(x) + 10*b**2*c**2*d**3*x + 5*b**2*c*d**4*x**2/2 + b**2*d**5*x**3/3, Eq(m, -5)), (-a
**2*c**5/(3*x**3) - 5*a**2*c**4*d/(2*x**2) - 10*a**2*c**3*d**2/x + 10*a**2*c**2*d**3*log(x) + 5*a**2*c*d**4*x
+ a**2*d**5*x**2/2 - a*b*c**5/x**2 - 10*a*b*c**4*d/x + 20*a*b*c**3*d**2*log(x) + 20*a*b*c**2*d**3*x + 5*a*b*c*
d**4*x**2 + 2*a*b*d**5*x**3/3 - b**2*c**5/x + 5*b**2*c**4*d*log(x) + 10*b**2*c**3*d**2*x + 5*b**2*c**2*d**3*x*
*2 + 5*b**2*c*d**4*x**3/3 + b**2*d**5*x**4/4, Eq(m, -4)), (-a**2*c**5/(2*x**2) - 5*a**2*c**4*d/x + 10*a**2*c**
3*d**2*log(x) + 10*a**2*c**2*d**3*x + 5*a**2*c*d**4*x**2/2 + a**2*d**5*x**3/3 - 2*a*b*c**5/x + 10*a*b*c**4*d*l
og(x) + 20*a*b*c**3*d**2*x + 10*a*b*c**2*d**3*x**2 + 10*a*b*c*d**4*x**3/3 + a*b*d**5*x**4/2 + b**2*c**5*log(x)
+ 5*b**2*c**4*d*x + 5*b**2*c**3*d**2*x**2 + 10*b**2*c**2*d**3*x**3/3 + 5*b**2*c*d**4*x**4/4 + b**2*d**5*x**5/
5, Eq(m, -3)), (-a**2*c**5/x + 5*a**2*c**4*d*log(x) + 10*a**2*c**3*d**2*x + 5*a**2*c**2*d**3*x**2 + 5*a**2*c*d
**4*x**3/3 + a**2*d**5*x**4/4 + 2*a*b*c**5*log(x) + 10*a*b*c**4*d*x + 10*a*b*c**3*d**2*x**2 + 20*a*b*c**2*d**3
*x**3/3 + 5*a*b*c*d**4*x**4/2 + 2*a*b*d**5*x**5/5 + b**2*c**5*x + 5*b**2*c**4*d*x**2/2 + 10*b**2*c**3*d**2*x**
3/3 + 5*b**2*c**2*d**3*x**4/2 + b**2*c*d**4*x**5 + b**2*d**5*x**6/6, Eq(m, -2)), (a**2*c**5*log(x) + 5*a**2*c*
*4*d*x + 5*a**2*c**3*d**2*x**2 + 10*a**2*c**2*d**3*x**3/3 + 5*a**2*c*d**4*x**4/4 + a**2*d**5*x**5/5 + 2*a*b*c*
*5*x + 5*a*b*c**4*d*x**2 + 20*a*b*c**3*d**2*x**3/3 + 5*a*b*c**2*d**3*x**4 + 2*a*b*c*d**4*x**5 + a*b*d**5*x**6/
3 + b**2*c**5*x**2/2 + 5*b**2*c**4*d*x**3/3 + 5*b**2*c**3*d**2*x**4/2 + 2*b**2*c**2*d**3*x**5 + 5*b**2*c*d**4*
x**6/6 + b**2*d**5*x**7/7, Eq(m, -1)), (a**2*c**5*m**7*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m
**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 35*a**2*c**5*m**6*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 453
6*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 511*a**2*c**5*m**5*x*x**m/(m**8 + 36*m**7
+ 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 4025*a**2*c**5*m**4*x*x*
*m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 18424*
a**2*c**5*m**3*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*
m + 40320) + 48860*a**2*c**5*m**2*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 11
8124*m**2 + 109584*m + 40320) + 69264*a**2*c**5*m*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 +
67284*m**3 + 118124*m**2 + 109584*m + 40320) + 40320*a**2*c**5*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5
+ 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 5*a**2*c**4*d*m**7*x**2*x**m/(m**8 + 36*m**7 + 5
46*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 170*a**2*c**4*d*m**6*x**2*x*
*m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 2390*a
**2*c**4*d*m**5*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109
584*m + 40320) + 17900*a**2*c**4*d*m**4*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*
m**3 + 118124*m**2 + 109584*m + 40320) + 76445*a**2*c**4*d*m**3*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m*
*5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 183530*a**2*c**4*d*m**2*x**2*x**m/(m**8 + 36*
m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 6728...
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2524 vs.
\(2 (231) = 462\).
time = 1.19, size = 2524, normalized size = 10.93 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x+a)^2*(d*x+c)^5,x, algorithm="giac")
[Out]
(b^2*d^5*m^7*x^8*x^m + 5*b^2*c*d^4*m^7*x^7*x^m + 2*a*b*d^5*m^7*x^7*x^m + 28*b^2*d^5*m^6*x^8*x^m + 10*b^2*c^2*d
^3*m^7*x^6*x^m + 10*a*b*c*d^4*m^7*x^6*x^m + a^2*d^5*m^7*x^6*x^m + 145*b^2*c*d^4*m^6*x^7*x^m + 58*a*b*d^5*m^6*x
^7*x^m + 322*b^2*d^5*m^5*x^8*x^m + 10*b^2*c^3*d^2*m^7*x^5*x^m + 20*a*b*c^2*d^3*m^7*x^5*x^m + 5*a^2*c*d^4*m^7*x
^5*x^m + 300*b^2*c^2*d^3*m^6*x^6*x^m + 300*a*b*c*d^4*m^6*x^6*x^m + 30*a^2*d^5*m^6*x^6*x^m + 1715*b^2*c*d^4*m^5
*x^7*x^m + 686*a*b*d^5*m^5*x^7*x^m + 1960*b^2*d^5*m^4*x^8*x^m + 5*b^2*c^4*d*m^7*x^4*x^m + 20*a*b*c^3*d^2*m^7*x
^4*x^m + 10*a^2*c^2*d^3*m^7*x^4*x^m + 310*b^2*c^3*d^2*m^6*x^5*x^m + 620*a*b*c^2*d^3*m^6*x^5*x^m + 155*a^2*c*d^
4*m^6*x^5*x^m + 3660*b^2*c^2*d^3*m^5*x^6*x^m + 3660*a*b*c*d^4*m^5*x^6*x^m + 366*a^2*d^5*m^5*x^6*x^m + 10675*b^
2*c*d^4*m^4*x^7*x^m + 4270*a*b*d^5*m^4*x^7*x^m + 6769*b^2*d^5*m^3*x^8*x^m + b^2*c^5*m^7*x^3*x^m + 10*a*b*c^4*d
*m^7*x^3*x^m + 10*a^2*c^3*d^2*m^7*x^3*x^m + 160*b^2*c^4*d*m^6*x^4*x^m + 640*a*b*c^3*d^2*m^6*x^4*x^m + 320*a^2*
c^2*d^3*m^6*x^4*x^m + 3910*b^2*c^3*d^2*m^5*x^5*x^m + 7820*a*b*c^2*d^3*m^5*x^5*x^m + 1955*a^2*c*d^4*m^5*x^5*x^m
+ 23400*b^2*c^2*d^3*m^4*x^6*x^m + 23400*a*b*c*d^4*m^4*x^6*x^m + 2340*a^2*d^5*m^4*x^6*x^m + 37520*b^2*c*d^4*m^
3*x^7*x^m + 15008*a*b*d^5*m^3*x^7*x^m + 13132*b^2*d^5*m^2*x^8*x^m + 2*a*b*c^5*m^7*x^2*x^m + 5*a^2*c^4*d*m^7*x^
2*x^m + 33*b^2*c^5*m^6*x^3*x^m + 330*a*b*c^4*d*m^6*x^3*x^m + 330*a^2*c^3*d^2*m^6*x^3*x^m + 2090*b^2*c^4*d*m^5*
x^4*x^m + 8360*a*b*c^3*d^2*m^5*x^4*x^m + 4180*a^2*c^2*d^3*m^5*x^4*x^m + 25810*b^2*c^3*d^2*m^4*x^5*x^m + 51620*
a*b*c^2*d^3*m^4*x^5*x^m + 12905*a^2*c*d^4*m^4*x^5*x^m + 84090*b^2*c^2*d^3*m^3*x^6*x^m + 84090*a*b*c*d^4*m^3*x^
6*x^m + 8409*a^2*d^5*m^3*x^6*x^m + 73780*b^2*c*d^4*m^2*x^7*x^m + 29512*a*b*d^5*m^2*x^7*x^m + 13068*b^2*d^5*m*x
^8*x^m + a^2*c^5*m^7*x*x^m + 68*a*b*c^5*m^6*x^2*x^m + 170*a^2*c^4*d*m^6*x^2*x^m + 447*b^2*c^5*m^5*x^3*x^m + 44
70*a*b*c^4*d*m^5*x^3*x^m + 4470*a^2*c^3*d^2*m^5*x^3*x^m + 14320*b^2*c^4*d*m^4*x^4*x^m + 57280*a*b*c^3*d^2*m^4*
x^4*x^m + 28640*a^2*c^2*d^3*m^4*x^4*x^m + 95440*b^2*c^3*d^2*m^3*x^5*x^m + 190880*a*b*c^2*d^3*m^3*x^5*x^m + 477
20*a^2*c*d^4*m^3*x^5*x^m + 168300*b^2*c^2*d^3*m^2*x^6*x^m + 168300*a*b*c*d^4*m^2*x^6*x^m + 16830*a^2*d^5*m^2*x
^6*x^m + 74160*b^2*c*d^4*m*x^7*x^m + 29664*a*b*d^5*m*x^7*x^m + 5040*b^2*d^5*x^8*x^m + 35*a^2*c^5*m^6*x*x^m + 9
56*a*b*c^5*m^5*x^2*x^m + 2390*a^2*c^4*d*m^5*x^2*x^m + 3195*b^2*c^5*m^4*x^3*x^m + 31950*a*b*c^4*d*m^4*x^3*x^m +
31950*a^2*c^3*d^2*m^4*x^3*x^m + 54965*b^2*c^4*d*m^3*x^4*x^m + 219860*a*b*c^3*d^2*m^3*x^4*x^m + 109930*a^2*c^2
*d^3*m^3*x^4*x^m + 195640*b^2*c^3*d^2*m^2*x^5*x^m + 391280*a*b*c^2*d^3*m^2*x^5*x^m + 97820*a^2*c*d^4*m^2*x^5*x
^m + 171440*b^2*c^2*d^3*m*x^6*x^m + 171440*a*b*c*d^4*m*x^6*x^m + 17144*a^2*d^5*m*x^6*x^m + 28800*b^2*c*d^4*x^7
*x^m + 11520*a*b*d^5*x^7*x^m + 511*a^2*c^5*m^5*x*x^m + 7160*a*b*c^5*m^4*x^2*x^m + 17900*a^2*c^4*d*m^4*x^2*x^m
+ 12864*b^2*c^5*m^3*x^3*x^m + 128640*a*b*c^4*d*m^3*x^3*x^m + 128640*a^2*c^3*d^2*m^3*x^3*x^m + 116560*b^2*c^4*d
*m^2*x^4*x^m + 466240*a*b*c^3*d^2*m^2*x^4*x^m + 233120*a^2*c^2*d^3*m^2*x^4*x^m + 203040*b^2*c^3*d^2*m*x^5*x^m
+ 406080*a*b*c^2*d^3*m*x^5*x^m + 101520*a^2*c*d^4*m*x^5*x^m + 67200*b^2*c^2*d^3*x^6*x^m + 67200*a*b*c*d^4*x^6*
x^m + 6720*a^2*d^5*x^6*x^m + 4025*a^2*c^5*m^4*x*x^m + 30578*a*b*c^5*m^3*x^2*x^m + 76445*a^2*c^4*d*m^3*x^2*x^m
+ 28692*b^2*c^5*m^2*x^3*x^m + 286920*a*b*c^4*d*m^2*x^3*x^m + 286920*a^2*c^3*d^2*m^2*x^3*x^m + 124380*b^2*c^4*d
*m*x^4*x^m + 497520*a*b*c^3*d^2*m*x^4*x^m + 248760*a^2*c^2*d^3*m*x^4*x^m + 80640*b^2*c^3*d^2*x^5*x^m + 161280*
a*b*c^2*d^3*x^5*x^m + 40320*a^2*c*d^4*x^5*x^m + 18424*a^2*c^5*m^3*x*x^m + 73412*a*b*c^5*m^2*x^2*x^m + 183530*a
^2*c^4*d*m^2*x^2*x^m + 32048*b^2*c^5*m*x^3*x^m + 320480*a*b*c^4*d*m*x^3*x^m + 320480*a^2*c^3*d^2*m*x^3*x^m + 5
0400*b^2*c^4*d*x^4*x^m + 201600*a*b*c^3*d^2*x^4*x^m + 100800*a^2*c^2*d^3*x^4*x^m + 48860*a^2*c^5*m^2*x*x^m + 8
9424*a*b*c^5*m*x^2*x^m + 223560*a^2*c^4*d*m*x^2*x^m + 13440*b^2*c^5*x^3*x^m + 134400*a*b*c^4*d*x^3*x^m + 13440
0*a^2*c^3*d^2*x^3*x^m + 69264*a^2*c^5*m*x*x^m + 40320*a*b*c^5*x^2*x^m + 100800*a^2*c^4*d*x^2*x^m + 40320*a^2*c
^5*x*x^m)/(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)
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Mupad [B]
time = 1.21, size = 781, normalized size = 3.38 \begin {gather*} \frac {b^2\,d^5\,x^m\,x^8\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {c^3\,x^m\,x^3\,\left (10\,a^2\,d^2+10\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {d^3\,x^m\,x^6\,\left (a^2\,d^2+10\,a\,b\,c\,d+10\,b^2\,c^2\right )\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a^2\,c^5\,x\,x^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a\,c^4\,x^m\,x^2\,\left (5\,a\,d+2\,b\,c\right )\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {b\,d^4\,x^m\,x^7\,\left (2\,a\,d+5\,b\,c\right )\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {5\,c\,d^2\,x^m\,x^5\,\left (a^2\,d^2+4\,a\,b\,c\,d+2\,b^2\,c^2\right )\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {5\,c^2\,d\,x^m\,x^4\,\left (2\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(a + b*x)^2*(c + d*x)^5,x)
[Out]
(b^2*d^5*x^m*x^8*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))/(109584*m + 1181
24*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (c^3*x^m*x^3*(10*a^2*d^2 + b^2*c
^2 + 10*a*b*c*d)*(32048*m + 28692*m^2 + 12864*m^3 + 3195*m^4 + 447*m^5 + 33*m^6 + m^7 + 13440))/(109584*m + 11
8124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (d^3*x^m*x^6*(a^2*d^2 + 10*b^2
*c^2 + 10*a*b*c*d)*(17144*m + 16830*m^2 + 8409*m^3 + 2340*m^4 + 366*m^5 + 30*m^6 + m^7 + 6720))/(109584*m + 11
8124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (a^2*c^5*x*x^m*(69264*m + 4886
0*m^2 + 18424*m^3 + 4025*m^4 + 511*m^5 + 35*m^6 + m^7 + 40320))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4
+ 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (a*c^4*x^m*x^2*(5*a*d + 2*b*c)*(44712*m + 36706*m^2 + 15289*m^
3 + 3580*m^4 + 478*m^5 + 34*m^6 + m^7 + 20160))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 54
6*m^6 + 36*m^7 + m^8 + 40320) + (b*d^4*x^m*x^7*(2*a*d + 5*b*c)*(14832*m + 14756*m^2 + 7504*m^3 + 2135*m^4 + 34
3*m^5 + 29*m^6 + m^7 + 5760))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m
^8 + 40320) + (5*c*d^2*x^m*x^5*(a^2*d^2 + 2*b^2*c^2 + 4*a*b*c*d)*(20304*m + 19564*m^2 + 9544*m^3 + 2581*m^4 +
391*m^5 + 31*m^6 + m^7 + 8064))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 +
m^8 + 40320) + (5*c^2*d*x^m*x^4*(2*a^2*d^2 + b^2*c^2 + 4*a*b*c*d)*(24876*m + 23312*m^2 + 10993*m^3 + 2864*m^4
+ 418*m^5 + 32*m^6 + m^7 + 10080))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m
^7 + m^8 + 40320)
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