3.15.23 \(\int (b d+2 c d x)^m (a+b x+c x^2)^3 \, dx\) [1423]

Optimal. Leaf size=141 \[ -\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{1+m}}{128 c^4 d (1+m)}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3+m}}{128 c^4 d^3 (3+m)}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5+m}}{128 c^4 d^5 (5+m)}+\frac {(b d+2 c d x)^{7+m}}{128 c^4 d^7 (7+m)} \]

[Out]

-1/128*(-4*a*c+b^2)^3*(2*c*d*x+b*d)^(1+m)/c^4/d/(1+m)+3/128*(-4*a*c+b^2)^2*(2*c*d*x+b*d)^(3+m)/c^4/d^3/(3+m)-3
/128*(-4*a*c+b^2)*(2*c*d*x+b*d)^(5+m)/c^4/d^5/(5+m)+1/128*(2*c*d*x+b*d)^(7+m)/c^4/d^7/(7+m)

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Rubi [A]
time = 0.06, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \begin {gather*} -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac {(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3,x]

[Out]

-1/128*((b^2 - 4*a*c)^3*(b*d + 2*c*d*x)^(1 + m))/(c^4*d*(1 + m)) + (3*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(3 + m))
/(128*c^4*d^3*(3 + m)) - (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5 + m))/(128*c^4*d^5*(5 + m)) + (b*d + 2*c*d*x)^(7
+ m)/(128*c^4*d^7*(7 + m))

Rule 697

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {align*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3 (b d+2 c d x)^m}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{2+m}}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{4+m}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{6+m}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{1+m}}{128 c^4 d (1+m)}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3+m}}{128 c^4 d^3 (3+m)}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5+m}}{128 c^4 d^5 (5+m)}+\frac {(b d+2 c d x)^{7+m}}{128 c^4 d^7 (7+m)}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 103, normalized size = 0.73 \begin {gather*} \frac {(b+2 c x) (d (b+2 c x))^m \left (-\frac {\left (b^2-4 a c\right )^3}{1+m}+\frac {3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{3+m}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^4}{5+m}+\frac {(b+2 c x)^6}{7+m}\right )}{128 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3,x]

[Out]

((b + 2*c*x)*(d*(b + 2*c*x))^m*(-((b^2 - 4*a*c)^3/(1 + m)) + (3*(b^2 - 4*a*c)^2*(b + 2*c*x)^2)/(3 + m) - (3*(b
^2 - 4*a*c)*(b + 2*c*x)^4)/(5 + m) + (b + 2*c*x)^6/(7 + m)))/(128*c^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs. \(2(133)=266\).
time = 0.72, size = 626, normalized size = 4.44

method	result	size
norman	\(\frac {c^{3} x^{7} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{7+m}+\frac {7 b \,c^{2} x^{6} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 \left (7+m \right )}+\frac {5 b \left (6 a c m +2 b^{2} m +42 a c +7 b^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{4 \left (m^{2}+12 m +35\right )}+\frac {b \left (4 a^{3} c^{3} m^{3}+60 a^{3} c^{3} m^{2}-6 a^{2} b^{2} c^{2} m^{2}+284 a^{3} c^{3} m -72 a^{2} b^{2} c^{2} m +6 a \,b^{4} c m +420 a^{3} c^{3}-210 a^{2} b^{2} c^{2}+42 a \,b^{4} c -3 b^{6}\right ) {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{8 c^{4} \left (m^{4}+16 m^{3}+86 m^{2}+176 m +105\right )}+\frac {\left (6 a^{2} c^{2} m^{2}+12 a \,b^{2} c \,m^{2}+b^{4} m^{2}+72 a^{2} c^{2} m +114 a \,b^{2} c m +2 b^{4} m +210 a^{2} c^{2}+210 a c \,b^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 c \left (m^{3}+15 m^{2}+71 m +105\right )}+\frac {3 c \left (2 a c m +3 b^{2} m +14 a c +14 b^{2}\right ) x^{5} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{2 \left (m^{2}+12 m +35\right )}+\frac {\left (4 a^{3} c^{3} m^{3}+6 a^{2} b^{2} c^{2} m^{3}+60 a^{3} c^{3} m^{2}+72 a^{2} b^{2} c^{2} m^{2}-6 a \,b^{4} c \,m^{2}+284 a^{3} c^{3} m +210 a^{2} b^{2} c^{2} m -42 a \,b^{4} c m +3 b^{6} m +420 a^{3} c^{3}\right ) x \,{\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{4 c^{3} \left (m^{4}+16 m^{3}+86 m^{2}+176 m +105\right )}+\frac {3 \left (6 a^{2} c^{2} m^{2}+2 a \,b^{2} c \,m^{2}+72 a^{2} c^{2} m +14 a \,b^{2} c m -b^{4} m +210 a^{2} c^{2}\right ) b \,x^{2} {\mathrm e}^{m \ln \left (2 c d x +b d \right )}}{4 c^{2} \left (m^{3}+15 m^{2}+71 m +105\right )}\)	\(626\)
gosper	\(\frac {\left (2 c d x +b d \right )^{m} \left (4 c^{6} m^{3} x^{6}+12 b \,c^{5} m^{3} x^{5}+36 c^{6} m^{2} x^{6}+12 a \,c^{5} m^{3} x^{4}+12 b^{2} c^{4} m^{3} x^{4}+108 b \,c^{5} m^{2} x^{5}+92 c^{6} m \,x^{6}+24 a b \,c^{4} m^{3} x^{3}+132 a \,c^{5} m^{2} x^{4}+4 b^{3} c^{3} m^{3} x^{3}+102 b^{2} c^{4} m^{2} x^{4}+276 b \,c^{5} m \,x^{5}+60 c^{6} x^{6}+12 a^{2} c^{4} m^{3} x^{2}+12 a \,b^{2} c^{3} m^{3} x^{2}+264 a b \,c^{4} m^{2} x^{3}+372 a \,c^{5} m \,x^{4}+24 b^{3} c^{3} m^{2} x^{3}+252 b^{2} c^{4} m \,x^{4}+180 b \,c^{5} x^{5}+12 a^{2} b \,c^{3} m^{3} x +156 a^{2} c^{4} m^{2} x^{2}+120 a \,b^{2} c^{3} m^{2} x^{2}+744 a b \,c^{4} m \,x^{3}+252 a \,c^{5} x^{4}-6 b^{4} c^{2} m^{2} x^{2}+44 b^{3} c^{3} m \,x^{3}+162 b^{2} c^{4} x^{4}+4 a^{3} c^{3} m^{3}+156 a^{2} b \,c^{3} m^{2} x +564 a^{2} c^{4} m \,x^{2}-12 a \,b^{3} c^{2} m^{2} x +276 a \,b^{2} c^{3} m \,x^{2}+504 a b \,c^{4} x^{3}-18 b^{4} c^{2} m \,x^{2}+24 b^{3} x^{3} c^{3}+60 a^{3} c^{3} m^{2}-6 a^{2} b^{2} c^{2} m^{2}+564 a^{2} b \,c^{3} m x +420 a^{2} c^{4} x^{2}-96 a \,b^{3} c^{2} m x +168 a \,b^{2} c^{3} x^{2}+6 b^{5} c m x -12 b^{4} c^{2} x^{2}+284 a^{3} c^{3} m -72 a^{2} b^{2} c^{2} m +420 a^{2} b \,c^{3} x +6 a \,b^{4} c m -84 a \,b^{3} c^{2} x +6 b^{5} c x +420 a^{3} c^{3}-210 a^{2} b^{2} c^{2}+42 a \,b^{4} c -3 b^{6}\right ) \left (2 c x +b \right )}{8 c^{4} \left (m^{4}+16 m^{3}+86 m^{2}+176 m +105\right )}\)	\(653\)
risch	\(\frac {\left (8 c^{7} m^{3} x^{7}+28 b \,c^{6} m^{3} x^{6}+72 c^{7} m^{2} x^{7}+24 a \,c^{6} m^{3} x^{5}+36 b^{2} c^{5} m^{3} x^{5}+252 b \,c^{6} m^{2} x^{6}+184 c^{7} m \,x^{7}+60 a b \,c^{5} m^{3} x^{4}+264 a \,c^{6} m^{2} x^{5}+20 b^{3} c^{4} m^{3} x^{4}+312 b^{2} c^{5} m^{2} x^{5}+644 b \,c^{6} m \,x^{6}+120 c^{7} x^{7}+24 a^{2} c^{5} m^{3} x^{3}+48 a \,b^{2} c^{4} m^{3} x^{3}+660 a b \,c^{5} m^{2} x^{4}+744 a \,c^{6} m \,x^{5}+4 b^{4} c^{3} m^{3} x^{3}+150 b^{3} c^{4} m^{2} x^{4}+780 b^{2} c^{5} m \,x^{5}+420 b \,c^{6} x^{6}+36 a^{2} b \,c^{4} m^{3} x^{2}+312 a^{2} c^{5} m^{2} x^{3}+12 a \,b^{3} c^{3} m^{3} x^{2}+504 a \,b^{2} c^{4} m^{2} x^{3}+1860 a b \,c^{5} m \,x^{4}+504 a \,c^{6} x^{5}+12 b^{4} c^{3} m^{2} x^{3}+340 b^{3} c^{4} m \,x^{4}+504 b^{2} c^{5} x^{5}+8 a^{3} c^{4} m^{3} x +12 a^{2} b^{2} c^{3} m^{3} x +468 a^{2} b \,c^{4} m^{2} x^{2}+1128 a^{2} c^{5} m \,x^{3}+96 a \,b^{3} c^{3} m^{2} x^{2}+1296 a \,b^{2} c^{4} m \,x^{3}+1260 a b \,c^{5} x^{4}-6 b^{5} c^{2} m^{2} x^{2}+8 b^{4} c^{3} m \,x^{3}+210 b^{3} c^{4} x^{4}+4 a^{3} b \,c^{3} m^{3}+120 a^{3} c^{4} m^{2} x +144 a^{2} b^{2} c^{3} m^{2} x +1692 a^{2} b \,c^{4} m \,x^{2}+840 a^{2} c^{5} x^{3}-12 a \,b^{4} c^{2} m^{2} x +84 a \,b^{3} c^{3} m \,x^{2}+840 a \,b^{2} c^{4} x^{3}-6 b^{5} c^{2} m \,x^{2}+60 a^{3} b \,c^{3} m^{2}+568 a^{3} c^{4} m x -6 a^{2} b^{3} c^{2} m^{2}+420 a^{2} b^{2} c^{3} m x +1260 a^{2} b \,c^{4} x^{2}-84 a \,b^{4} c^{2} m x +6 b^{6} c m x +284 a^{3} b \,c^{3} m +840 a^{3} c^{4} x -72 a^{2} b^{3} c^{2} m +6 a \,b^{5} c m +420 a^{3} b \,c^{3}-210 a^{2} b^{3} c^{2}+42 a \,b^{5} c -3 b^{7}\right ) \left (d \left (2 c x +b \right )\right )^{m}}{8 \left (5+m \right ) \left (7+m \right ) \left (3+m \right ) \left (1+m \right ) c^{4}}\)	\(799\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

c^3/(7+m)*x^7*exp(m*ln(2*c*d*x+b*d))+7/2*b*c^2/(7+m)*x^6*exp(m*ln(2*c*d*x+b*d))+5/4*b*(6*a*c*m+2*b^2*m+42*a*c+
7*b^2)/(m^2+12*m+35)*x^4*exp(m*ln(2*c*d*x+b*d))+1/8*b*(4*a^3*c^3*m^3+60*a^3*c^3*m^2-6*a^2*b^2*c^2*m^2+284*a^3*
c^3*m-72*a^2*b^2*c^2*m+6*a*b^4*c*m+420*a^3*c^3-210*a^2*b^2*c^2+42*a*b^4*c-3*b^6)/c^4/(m^4+16*m^3+86*m^2+176*m+
105)*exp(m*ln(2*c*d*x+b*d))+1/2/c*(6*a^2*c^2*m^2+12*a*b^2*c*m^2+b^4*m^2+72*a^2*c^2*m+114*a*b^2*c*m+2*b^4*m+210
*a^2*c^2+210*a*b^2*c)/(m^3+15*m^2+71*m+105)*x^3*exp(m*ln(2*c*d*x+b*d))+3/2*c*(2*a*c*m+3*b^2*m+14*a*c+14*b^2)/(
m^2+12*m+35)*x^5*exp(m*ln(2*c*d*x+b*d))+1/4*(4*a^3*c^3*m^3+6*a^2*b^2*c^2*m^3+60*a^3*c^3*m^2+72*a^2*b^2*c^2*m^2
-6*a*b^4*c*m^2+284*a^3*c^3*m+210*a^2*b^2*c^2*m-42*a*b^4*c*m+3*b^6*m+420*a^3*c^3)/c^3/(m^4+16*m^3+86*m^2+176*m+
105)*x*exp(m*ln(2*c*d*x+b*d))+3/4*(6*a^2*c^2*m^2+2*a*b^2*c*m^2+72*a^2*c^2*m+14*a*b^2*c*m-b^4*m+210*a^2*c^2)*b/
c^2/(m^3+15*m^2+71*m+105)*x^2*exp(m*ln(2*c*d*x+b*d))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (133) = 266\).
time = 0.35, size = 1299, normalized size = 9.21 \begin {gather*} \frac {3 \, {\left (4 \, c^{2} d^{m} {\left (m + 1\right )} x^{2} + 2 \, b c d^{m} m x - b^{2} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a^{2} b}{4 \, {\left (m^{2} + 3 \, m + 2\right )} c^{2}} + \frac {3 \, {\left (4 \, {\left (m^{2} + 3 \, m + 2\right )} c^{3} d^{m} x^{3} + 2 \, {\left (m^{2} + m\right )} b c^{2} d^{m} x^{2} - 2 \, b^{2} c d^{m} m x + b^{3} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a b^{2}}{4 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{3}} + \frac {3 \, {\left (4 \, {\left (m^{2} + 3 \, m + 2\right )} c^{3} d^{m} x^{3} + 2 \, {\left (m^{2} + m\right )} b c^{2} d^{m} x^{2} - 2 \, b^{2} c d^{m} m x + b^{3} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a^{2}}{4 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{2}} + \frac {{\left (2 \, c d x + b d\right )}^{m + 1} a^{3}}{2 \, c d {\left (m + 1\right )}} + \frac {{\left (8 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{4} d^{m} x^{4} + 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b c^{3} d^{m} x^{3} - 6 \, {\left (m^{2} + m\right )} b^{2} c^{2} d^{m} x^{2} + 6 \, b^{3} c d^{m} m x - 3 \, b^{4} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} b^{3}}{8 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{4}} + \frac {3 \, {\left (8 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} c^{4} d^{m} x^{4} + 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b c^{3} d^{m} x^{3} - 6 \, {\left (m^{2} + m\right )} b^{2} c^{2} d^{m} x^{2} + 6 \, b^{3} c d^{m} m x - 3 \, b^{4} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a b}{4 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{3}} + \frac {3 \, {\left (4 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{5} d^{m} x^{5} + 2 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b c^{4} d^{m} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{2} c^{3} d^{m} x^{3} + 6 \, {\left (m^{2} + m\right )} b^{3} c^{2} d^{m} x^{2} - 6 \, b^{4} c d^{m} m x + 3 \, b^{5} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} b^{2}}{4 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} c^{4}} + \frac {3 \, {\left (4 \, {\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} c^{5} d^{m} x^{5} + 2 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b c^{4} d^{m} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{2} c^{3} d^{m} x^{3} + 6 \, {\left (m^{2} + m\right )} b^{3} c^{2} d^{m} x^{2} - 6 \, b^{4} c d^{m} m x + 3 \, b^{5} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} a}{4 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} c^{3}} + \frac {3 \, {\left (8 \, {\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} c^{6} d^{m} x^{6} + 4 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} b c^{5} d^{m} x^{5} - 10 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b^{2} c^{4} d^{m} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{3} c^{3} d^{m} x^{3} - 30 \, {\left (m^{2} + m\right )} b^{4} c^{2} d^{m} x^{2} + 30 \, b^{5} c d^{m} m x - 15 \, b^{6} d^{m}\right )} {\left (2 \, c x + b\right )}^{m} b}{8 \, {\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} c^{4}} + \frac {{\left (8 \, {\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} c^{7} d^{m} x^{7} + 4 \, {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} b c^{6} d^{m} x^{6} - 12 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} b^{2} c^{5} d^{m} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b^{3} c^{4} d^{m} x^{4} - 60 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{4} c^{3} d^{m} x^{3} + 90 \, {\left (m^{2} + m\right )} b^{5} c^{2} d^{m} x^{2} - 90 \, b^{6} c d^{m} m x + 45 \, b^{7} d^{m}\right )} {\left (2 \, c x + b\right )}^{m}}{8 \, {\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

3/4*(4*c^2*d^m*(m + 1)*x^2 + 2*b*c*d^m*m*x - b^2*d^m)*(2*c*x + b)^m*a^2*b/((m^2 + 3*m + 2)*c^2) + 3/4*(4*(m^2
+ 3*m + 2)*c^3*d^m*x^3 + 2*(m^2 + m)*b*c^2*d^m*x^2 - 2*b^2*c*d^m*m*x + b^3*d^m)*(2*c*x + b)^m*a*b^2/((m^3 + 6*
m^2 + 11*m + 6)*c^3) + 3/4*(4*(m^2 + 3*m + 2)*c^3*d^m*x^3 + 2*(m^2 + m)*b*c^2*d^m*x^2 - 2*b^2*c*d^m*m*x + b^3*
d^m)*(2*c*x + b)^m*a^2/((m^3 + 6*m^2 + 11*m + 6)*c^2) + 1/2*(2*c*d*x + b*d)^(m + 1)*a^3/(c*d*(m + 1)) + 1/8*(8
*(m^3 + 6*m^2 + 11*m + 6)*c^4*d^m*x^4 + 4*(m^3 + 3*m^2 + 2*m)*b*c^3*d^m*x^3 - 6*(m^2 + m)*b^2*c^2*d^m*x^2 + 6*
b^3*c*d^m*m*x - 3*b^4*d^m)*(2*c*x + b)^m*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*c^4) + 3/4*(8*(m^3 + 6*m^2 +
11*m + 6)*c^4*d^m*x^4 + 4*(m^3 + 3*m^2 + 2*m)*b*c^3*d^m*x^3 - 6*(m^2 + m)*b^2*c^2*d^m*x^2 + 6*b^3*c*d^m*m*x -
3*b^4*d^m)*(2*c*x + b)^m*a*b/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*c^3) + 3/4*(4*(m^4 + 10*m^3 + 35*m^2 + 50*m
+ 24)*c^5*d^m*x^5 + 2*(m^4 + 6*m^3 + 11*m^2 + 6*m)*b*c^4*d^m*x^4 - 4*(m^3 + 3*m^2 + 2*m)*b^2*c^3*d^m*x^3 + 6*
(m^2 + m)*b^3*c^2*d^m*x^2 - 6*b^4*c*d^m*m*x + 3*b^5*d^m)*(2*c*x + b)^m*b^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 +
274*m + 120)*c^4) + 3/4*(4*(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*c^5*d^m*x^5 + 2*(m^4 + 6*m^3 + 11*m^2 + 6*m)*b
*c^4*d^m*x^4 - 4*(m^3 + 3*m^2 + 2*m)*b^2*c^3*d^m*x^3 + 6*(m^2 + m)*b^3*c^2*d^m*x^2 - 6*b^4*c*d^m*m*x + 3*b^5*d
^m)*(2*c*x + b)^m*a/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*c^3) + 3/8*(8*(m^5 + 15*m^4 + 85*m^3 + 22
5*m^2 + 274*m + 120)*c^6*d^m*x^6 + 4*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*b*c^5*d^m*x^5 - 10*(m^4 + 6*m^3 +
11*m^2 + 6*m)*b^2*c^4*d^m*x^4 + 20*(m^3 + 3*m^2 + 2*m)*b^3*c^3*d^m*x^3 - 30*(m^2 + m)*b^4*c^2*d^m*x^2 + 30*b^
5*c*d^m*m*x - 15*b^6*d^m)*(2*c*x + b)^m*b/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*c^4) +
1/8*(8*(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*c^7*d^m*x^7 + 4*(m^6 + 15*m^5 + 85*m^4 +
225*m^3 + 274*m^2 + 120*m)*b*c^6*d^m*x^6 - 12*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*b^2*c^5*d^m*x^5 + 30*(m^
4 + 6*m^3 + 11*m^2 + 6*m)*b^3*c^4*d^m*x^4 - 60*(m^3 + 3*m^2 + 2*m)*b^4*c^3*d^m*x^3 + 90*(m^2 + m)*b^5*c^2*d^m*
x^2 - 90*b^6*c*d^m*m*x + 45*b^7*d^m)*(2*c*x + b)^m/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2
+ 13068*m + 5040)*c^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 691 vs. \(2 (133) = 266\).
time = 1.74, size = 691, normalized size = 4.90 \begin {gather*} \frac {{\left (4 \, a^{3} b c^{3} m^{3} + 8 \, {\left (c^{7} m^{3} + 9 \, c^{7} m^{2} + 23 \, c^{7} m + 15 \, c^{7}\right )} x^{7} - 3 \, b^{7} + 42 \, a b^{5} c - 210 \, a^{2} b^{3} c^{2} + 420 \, a^{3} b c^{3} + 28 \, {\left (b c^{6} m^{3} + 9 \, b c^{6} m^{2} + 23 \, b c^{6} m + 15 \, b c^{6}\right )} x^{6} + 12 \, {\left (42 \, b^{2} c^{5} + 42 \, a c^{6} + {\left (3 \, b^{2} c^{5} + 2 \, a c^{6}\right )} m^{3} + 2 \, {\left (13 \, b^{2} c^{5} + 11 \, a c^{6}\right )} m^{2} + {\left (65 \, b^{2} c^{5} + 62 \, a c^{6}\right )} m\right )} x^{5} + 10 \, {\left (21 \, b^{3} c^{4} + 126 \, a b c^{5} + 2 \, {\left (b^{3} c^{4} + 3 \, a b c^{5}\right )} m^{3} + 3 \, {\left (5 \, b^{3} c^{4} + 22 \, a b c^{5}\right )} m^{2} + 2 \, {\left (17 \, b^{3} c^{4} + 93 \, a b c^{5}\right )} m\right )} x^{4} + 4 \, {\left (210 \, a b^{2} c^{4} + 210 \, a^{2} c^{5} + {\left (b^{4} c^{3} + 12 \, a b^{2} c^{4} + 6 \, a^{2} c^{5}\right )} m^{3} + 3 \, {\left (b^{4} c^{3} + 42 \, a b^{2} c^{4} + 26 \, a^{2} c^{5}\right )} m^{2} + 2 \, {\left (b^{4} c^{3} + 162 \, a b^{2} c^{4} + 141 \, a^{2} c^{5}\right )} m\right )} x^{3} - 6 \, {\left (a^{2} b^{3} c^{2} - 10 \, a^{3} b c^{3}\right )} m^{2} + 6 \, {\left (210 \, a^{2} b c^{4} + 2 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b c^{4}\right )} m^{3} - {\left (b^{5} c^{2} - 16 \, a b^{3} c^{3} - 78 \, a^{2} b c^{4}\right )} m^{2} - {\left (b^{5} c^{2} - 14 \, a b^{3} c^{3} - 282 \, a^{2} b c^{4}\right )} m\right )} x^{2} + 2 \, {\left (3 \, a b^{5} c - 36 \, a^{2} b^{3} c^{2} + 142 \, a^{3} b c^{3}\right )} m + 2 \, {\left (420 \, a^{3} c^{4} + 2 \, {\left (3 \, a^{2} b^{2} c^{3} + 2 \, a^{3} c^{4}\right )} m^{3} - 6 \, {\left (a b^{4} c^{2} - 12 \, a^{2} b^{2} c^{3} - 10 \, a^{3} c^{4}\right )} m^{2} + {\left (3 \, b^{6} c - 42 \, a b^{4} c^{2} + 210 \, a^{2} b^{2} c^{3} + 284 \, a^{3} c^{4}\right )} m\right )} x\right )} {\left (2 \, c d x + b d\right )}^{m}}{8 \, {\left (c^{4} m^{4} + 16 \, c^{4} m^{3} + 86 \, c^{4} m^{2} + 176 \, c^{4} m + 105 \, c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/8*(4*a^3*b*c^3*m^3 + 8*(c^7*m^3 + 9*c^7*m^2 + 23*c^7*m + 15*c^7)*x^7 - 3*b^7 + 42*a*b^5*c - 210*a^2*b^3*c^2
+ 420*a^3*b*c^3 + 28*(b*c^6*m^3 + 9*b*c^6*m^2 + 23*b*c^6*m + 15*b*c^6)*x^6 + 12*(42*b^2*c^5 + 42*a*c^6 + (3*b^
2*c^5 + 2*a*c^6)*m^3 + 2*(13*b^2*c^5 + 11*a*c^6)*m^2 + (65*b^2*c^5 + 62*a*c^6)*m)*x^5 + 10*(21*b^3*c^4 + 126*a
*b*c^5 + 2*(b^3*c^4 + 3*a*b*c^5)*m^3 + 3*(5*b^3*c^4 + 22*a*b*c^5)*m^2 + 2*(17*b^3*c^4 + 93*a*b*c^5)*m)*x^4 + 4
*(210*a*b^2*c^4 + 210*a^2*c^5 + (b^4*c^3 + 12*a*b^2*c^4 + 6*a^2*c^5)*m^3 + 3*(b^4*c^3 + 42*a*b^2*c^4 + 26*a^2*
c^5)*m^2 + 2*(b^4*c^3 + 162*a*b^2*c^4 + 141*a^2*c^5)*m)*x^3 - 6*(a^2*b^3*c^2 - 10*a^3*b*c^3)*m^2 + 6*(210*a^2*
b*c^4 + 2*(a*b^3*c^3 + 3*a^2*b*c^4)*m^3 - (b^5*c^2 - 16*a*b^3*c^3 - 78*a^2*b*c^4)*m^2 - (b^5*c^2 - 14*a*b^3*c^
3 - 282*a^2*b*c^4)*m)*x^2 + 2*(3*a*b^5*c - 36*a^2*b^3*c^2 + 142*a^3*b*c^3)*m + 2*(420*a^3*c^4 + 2*(3*a^2*b^2*c
^3 + 2*a^3*c^4)*m^3 - 6*(a*b^4*c^2 - 12*a^2*b^2*c^3 - 10*a^3*c^4)*m^2 + (3*b^6*c - 42*a*b^4*c^2 + 210*a^2*b^2*
c^3 + 284*a^3*c^4)*m)*x)*(2*c*d*x + b*d)^m/(c^4*m^4 + 16*c^4*m^3 + 86*c^4*m^2 + 176*c^4*m + 105*c^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 9491 vs. \(2 (133) = 266\).
time = 2.09, size = 9491, normalized size = 67.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**m*(c*x**2+b*x+a)**3,x)

[Out]

Piecewise(((b*d)**m*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), Eq(c, 0)), (-128*a**3*c**3/(1536*b
**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*
c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 48*a**2*b**2*c**2/(1536*b**6*c**4*d**7 + 1
8432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 +
294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 576*a**2*b*c**3*x/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d*
*7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**
7*x**5 + 98304*c**10*d**7*x**6) - 576*a**2*c**4*x**2/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**
4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c
**10*d**7*x**6) - 24*a*b**4*c/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 2457
60*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 288*a
*b**3*c**2*x/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7
*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 1440*a*b**2*c**3*x**2/
(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 36864
0*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 2304*a*b*c**4*x**3/(1536*b**6*c**4*
d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7
*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 1152*a*c**5*x**4/(1536*b**6*c**4*d**7 + 18432*b**5*
c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c
**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 12*b**6*log(b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*
x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x
**5 + 98304*c**10*d**7*x**6) + 11*b**6/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x*
*2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6
) + 144*b**5*c*x*log(b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 +
245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 1
08*b**5*c*x/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*
x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 720*b**4*c**2*x**2*log(
b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7
*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 396*b**4*c**2*x**2/(15
36*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b
**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 1920*b**3*c**3*x**3*log(b/(2*c) + x)/(
1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640
*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 576*b**3*c**3*x**3/(1536*b**6*c**4*d
**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*
x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 2880*b**2*c**4*x**4*log(b/(2*c) + x)/(1536*b**6*c**4
*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**
7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 288*b**2*c**4*x**4/(1536*b**6*c**4*d**7 + 18432*b*
*5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*
b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 2304*b*c**5*x**5*log(b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**
5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b
*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 768*c**6*x**6*log(b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**5*c*
*5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**
9*d**7*x**5 + 98304*c**10*d**7*x**6), Eq(m, -7)), (-64*a**3*c**3/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x +
12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 48*a**2*b**2*c**2/(512*b**4*c**4*
d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 384
*a**2*b*c**3*x/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**
3 + 8192*c**8*d**5*x**4) - 384*a**2*c**4*x**2/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d*
*5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1506 vs. \(2 (133) = 266\).
time = 1.36, size = 1506, normalized size = 10.68 \begin {gather*} \frac {8 \, {\left (2 \, c d x + b d\right )}^{m} c^{7} m^{3} x^{7} + 28 \, {\left (2 \, c d x + b d\right )}^{m} b c^{6} m^{3} x^{6} + 72 \, {\left (2 \, c d x + b d\right )}^{m} c^{7} m^{2} x^{7} + 36 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{5} m^{3} x^{5} + 24 \, {\left (2 \, c d x + b d\right )}^{m} a c^{6} m^{3} x^{5} + 252 \, {\left (2 \, c d x + b d\right )}^{m} b c^{6} m^{2} x^{6} + 184 \, {\left (2 \, c d x + b d\right )}^{m} c^{7} m x^{7} + 20 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{4} m^{3} x^{4} + 60 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{5} m^{3} x^{4} + 312 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{5} m^{2} x^{5} + 264 \, {\left (2 \, c d x + b d\right )}^{m} a c^{6} m^{2} x^{5} + 644 \, {\left (2 \, c d x + b d\right )}^{m} b c^{6} m x^{6} + 120 \, {\left (2 \, c d x + b d\right )}^{m} c^{7} x^{7} + 4 \, {\left (2 \, c d x + b d\right )}^{m} b^{4} c^{3} m^{3} x^{3} + 48 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{4} m^{3} x^{3} + 24 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{5} m^{3} x^{3} + 150 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{4} m^{2} x^{4} + 660 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{5} m^{2} x^{4} + 780 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{5} m x^{5} + 744 \, {\left (2 \, c d x + b d\right )}^{m} a c^{6} m x^{5} + 420 \, {\left (2 \, c d x + b d\right )}^{m} b c^{6} x^{6} + 12 \, {\left (2 \, c d x + b d\right )}^{m} a b^{3} c^{3} m^{3} x^{2} + 36 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{4} m^{3} x^{2} + 12 \, {\left (2 \, c d x + b d\right )}^{m} b^{4} c^{3} m^{2} x^{3} + 504 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{4} m^{2} x^{3} + 312 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{5} m^{2} x^{3} + 340 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{4} m x^{4} + 1860 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{5} m x^{4} + 504 \, {\left (2 \, c d x + b d\right )}^{m} b^{2} c^{5} x^{5} + 504 \, {\left (2 \, c d x + b d\right )}^{m} a c^{6} x^{5} + 12 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{2} c^{3} m^{3} x + 8 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} c^{4} m^{3} x - 6 \, {\left (2 \, c d x + b d\right )}^{m} b^{5} c^{2} m^{2} x^{2} + 96 \, {\left (2 \, c d x + b d\right )}^{m} a b^{3} c^{3} m^{2} x^{2} + 468 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{4} m^{2} x^{2} + 8 \, {\left (2 \, c d x + b d\right )}^{m} b^{4} c^{3} m x^{3} + 1296 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{4} m x^{3} + 1128 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{5} m x^{3} + 210 \, {\left (2 \, c d x + b d\right )}^{m} b^{3} c^{4} x^{4} + 1260 \, {\left (2 \, c d x + b d\right )}^{m} a b c^{5} x^{4} + 4 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} b c^{3} m^{3} - 12 \, {\left (2 \, c d x + b d\right )}^{m} a b^{4} c^{2} m^{2} x + 144 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{2} c^{3} m^{2} x + 120 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} c^{4} m^{2} x - 6 \, {\left (2 \, c d x + b d\right )}^{m} b^{5} c^{2} m x^{2} + 84 \, {\left (2 \, c d x + b d\right )}^{m} a b^{3} c^{3} m x^{2} + 1692 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{4} m x^{2} + 840 \, {\left (2 \, c d x + b d\right )}^{m} a b^{2} c^{4} x^{3} + 840 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} c^{5} x^{3} - 6 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{3} c^{2} m^{2} + 60 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} b c^{3} m^{2} + 6 \, {\left (2 \, c d x + b d\right )}^{m} b^{6} c m x - 84 \, {\left (2 \, c d x + b d\right )}^{m} a b^{4} c^{2} m x + 420 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{2} c^{3} m x + 568 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} c^{4} m x + 1260 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b c^{4} x^{2} + 6 \, {\left (2 \, c d x + b d\right )}^{m} a b^{5} c m - 72 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{3} c^{2} m + 284 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} b c^{3} m + 840 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} c^{4} x - 3 \, {\left (2 \, c d x + b d\right )}^{m} b^{7} + 42 \, {\left (2 \, c d x + b d\right )}^{m} a b^{5} c - 210 \, {\left (2 \, c d x + b d\right )}^{m} a^{2} b^{3} c^{2} + 420 \, {\left (2 \, c d x + b d\right )}^{m} a^{3} b c^{3}}{8 \, {\left (c^{4} m^{4} + 16 \, c^{4} m^{3} + 86 \, c^{4} m^{2} + 176 \, c^{4} m + 105 \, c^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/8*(8*(2*c*d*x + b*d)^m*c^7*m^3*x^7 + 28*(2*c*d*x + b*d)^m*b*c^6*m^3*x^6 + 72*(2*c*d*x + b*d)^m*c^7*m^2*x^7 +
36*(2*c*d*x + b*d)^m*b^2*c^5*m^3*x^5 + 24*(2*c*d*x + b*d)^m*a*c^6*m^3*x^5 + 252*(2*c*d*x + b*d)^m*b*c^6*m^2*x
^6 + 184*(2*c*d*x + b*d)^m*c^7*m*x^7 + 20*(2*c*d*x + b*d)^m*b^3*c^4*m^3*x^4 + 60*(2*c*d*x + b*d)^m*a*b*c^5*m^3
*x^4 + 312*(2*c*d*x + b*d)^m*b^2*c^5*m^2*x^5 + 264*(2*c*d*x + b*d)^m*a*c^6*m^2*x^5 + 644*(2*c*d*x + b*d)^m*b*c
^6*m*x^6 + 120*(2*c*d*x + b*d)^m*c^7*x^7 + 4*(2*c*d*x + b*d)^m*b^4*c^3*m^3*x^3 + 48*(2*c*d*x + b*d)^m*a*b^2*c^
4*m^3*x^3 + 24*(2*c*d*x + b*d)^m*a^2*c^5*m^3*x^3 + 150*(2*c*d*x + b*d)^m*b^3*c^4*m^2*x^4 + 660*(2*c*d*x + b*d)
^m*a*b*c^5*m^2*x^4 + 780*(2*c*d*x + b*d)^m*b^2*c^5*m*x^5 + 744*(2*c*d*x + b*d)^m*a*c^6*m*x^5 + 420*(2*c*d*x +
b*d)^m*b*c^6*x^6 + 12*(2*c*d*x + b*d)^m*a*b^3*c^3*m^3*x^2 + 36*(2*c*d*x + b*d)^m*a^2*b*c^4*m^3*x^2 + 12*(2*c*d
*x + b*d)^m*b^4*c^3*m^2*x^3 + 504*(2*c*d*x + b*d)^m*a*b^2*c^4*m^2*x^3 + 312*(2*c*d*x + b*d)^m*a^2*c^5*m^2*x^3
+ 340*(2*c*d*x + b*d)^m*b^3*c^4*m*x^4 + 1860*(2*c*d*x + b*d)^m*a*b*c^5*m*x^4 + 504*(2*c*d*x + b*d)^m*b^2*c^5*x
^5 + 504*(2*c*d*x + b*d)^m*a*c^6*x^5 + 12*(2*c*d*x + b*d)^m*a^2*b^2*c^3*m^3*x + 8*(2*c*d*x + b*d)^m*a^3*c^4*m^
3*x - 6*(2*c*d*x + b*d)^m*b^5*c^2*m^2*x^2 + 96*(2*c*d*x + b*d)^m*a*b^3*c^3*m^2*x^2 + 468*(2*c*d*x + b*d)^m*a^2
*b*c^4*m^2*x^2 + 8*(2*c*d*x + b*d)^m*b^4*c^3*m*x^3 + 1296*(2*c*d*x + b*d)^m*a*b^2*c^4*m*x^3 + 1128*(2*c*d*x +
b*d)^m*a^2*c^5*m*x^3 + 210*(2*c*d*x + b*d)^m*b^3*c^4*x^4 + 1260*(2*c*d*x + b*d)^m*a*b*c^5*x^4 + 4*(2*c*d*x + b
*d)^m*a^3*b*c^3*m^3 - 12*(2*c*d*x + b*d)^m*a*b^4*c^2*m^2*x + 144*(2*c*d*x + b*d)^m*a^2*b^2*c^3*m^2*x + 120*(2*
c*d*x + b*d)^m*a^3*c^4*m^2*x - 6*(2*c*d*x + b*d)^m*b^5*c^2*m*x^2 + 84*(2*c*d*x + b*d)^m*a*b^3*c^3*m*x^2 + 1692
*(2*c*d*x + b*d)^m*a^2*b*c^4*m*x^2 + 840*(2*c*d*x + b*d)^m*a*b^2*c^4*x^3 + 840*(2*c*d*x + b*d)^m*a^2*c^5*x^3 -
6*(2*c*d*x + b*d)^m*a^2*b^3*c^2*m^2 + 60*(2*c*d*x + b*d)^m*a^3*b*c^3*m^2 + 6*(2*c*d*x + b*d)^m*b^6*c*m*x - 84
*(2*c*d*x + b*d)^m*a*b^4*c^2*m*x + 420*(2*c*d*x + b*d)^m*a^2*b^2*c^3*m*x + 568*(2*c*d*x + b*d)^m*a^3*c^4*m*x +
1260*(2*c*d*x + b*d)^m*a^2*b*c^4*x^2 + 6*(2*c*d*x + b*d)^m*a*b^5*c*m - 72*(2*c*d*x + b*d)^m*a^2*b^3*c^2*m + 2
84*(2*c*d*x + b*d)^m*a^3*b*c^3*m + 840*(2*c*d*x + b*d)^m*a^3*c^4*x - 3*(2*c*d*x + b*d)^m*b^7 + 42*(2*c*d*x + b
*d)^m*a*b^5*c - 210*(2*c*d*x + b*d)^m*a^2*b^3*c^2 + 420*(2*c*d*x + b*d)^m*a^3*b*c^3)/(c^4*m^4 + 16*c^4*m^3 + 8
6*c^4*m^2 + 176*c^4*m + 105*c^4)

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Mupad [B]
time = 1.11, size = 655, normalized size = 4.65 \begin {gather*} {\left (b\,d+2\,c\,d\,x\right )}^m\,\left (\frac {4\,a^3\,b\,c^3\,m^3+60\,a^3\,b\,c^3\,m^2+284\,a^3\,b\,c^3\,m+420\,a^3\,b\,c^3-6\,a^2\,b^3\,c^2\,m^2-72\,a^2\,b^3\,c^2\,m-210\,a^2\,b^3\,c^2+6\,a\,b^5\,c\,m+42\,a\,b^5\,c-3\,b^7}{8\,c^4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {c^3\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {x\,\left (8\,a^3\,c^4\,m^3+120\,a^3\,c^4\,m^2+568\,a^3\,c^4\,m+840\,a^3\,c^4+12\,a^2\,b^2\,c^3\,m^3+144\,a^2\,b^2\,c^3\,m^2+420\,a^2\,b^2\,c^3\,m-12\,a\,b^4\,c^2\,m^2-84\,a\,b^4\,c^2\,m+6\,b^6\,c\,m\right )}{8\,c^4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {5\,b\,x^4\,\left (m^2+4\,m+3\right )\,\left (42\,a\,c+2\,b^2\,m+7\,b^2+6\,a\,c\,m\right )}{4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {3\,c\,x^5\,\left (m^2+4\,m+3\right )\,\left (14\,a\,c+3\,b^2\,m+14\,b^2+2\,a\,c\,m\right )}{2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {x^3\,\left (m+1\right )\,\left (6\,a^2\,c^2\,m^2+72\,a^2\,c^2\,m+210\,a^2\,c^2+12\,a\,b^2\,c\,m^2+114\,a\,b^2\,c\,m+210\,a\,b^2\,c+b^4\,m^2+2\,b^4\,m\right )}{2\,c\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {7\,b\,c^2\,x^6\,\left (m^3+9\,m^2+23\,m+15\right )}{2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {3\,b\,x^2\,\left (m+1\right )\,\left (6\,a^2\,c^2\,m^2+72\,a^2\,c^2\,m+210\,a^2\,c^2+2\,a\,b^2\,c\,m^2+14\,a\,b^2\,c\,m-b^4\,m\right )}{4\,c^2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3,x)

[Out]

(b*d + 2*c*d*x)^m*((420*a^3*b*c^3 - 3*b^7 - 210*a^2*b^3*c^2 + 42*a*b^5*c - 72*a^2*b^3*c^2*m + 60*a^3*b*c^3*m^2
+ 4*a^3*b*c^3*m^3 + 6*a*b^5*c*m - 6*a^2*b^3*c^2*m^2 + 284*a^3*b*c^3*m)/(8*c^4*(176*m + 86*m^2 + 16*m^3 + m^4
+ 105)) + (c^3*x^7*(23*m + 9*m^2 + m^3 + 15))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (x*(840*a^3*c^4 + 568*a^
3*c^4*m + 120*a^3*c^4*m^2 + 8*a^3*c^4*m^3 + 6*b^6*c*m + 420*a^2*b^2*c^3*m - 12*a*b^4*c^2*m^2 + 144*a^2*b^2*c^3
*m^2 + 12*a^2*b^2*c^3*m^3 - 84*a*b^4*c^2*m))/(8*c^4*(176*m + 86*m^2 + 16*m^3 + m^4 + 105)) + (5*b*x^4*(4*m + m
^2 + 3)*(42*a*c + 2*b^2*m + 7*b^2 + 6*a*c*m))/(4*(176*m + 86*m^2 + 16*m^3 + m^4 + 105)) + (3*c*x^5*(4*m + m^2
+ 3)*(14*a*c + 3*b^2*m + 14*b^2 + 2*a*c*m))/(2*(176*m + 86*m^2 + 16*m^3 + m^4 + 105)) + (x^3*(m + 1)*(2*b^4*m
+ 210*a^2*c^2 + b^4*m^2 + 72*a^2*c^2*m + 6*a^2*c^2*m^2 + 210*a*b^2*c + 114*a*b^2*c*m + 12*a*b^2*c*m^2))/(2*c*(
176*m + 86*m^2 + 16*m^3 + m^4 + 105)) + (7*b*c^2*x^6*(23*m + 9*m^2 + m^3 + 15))/(2*(176*m + 86*m^2 + 16*m^3 +
m^4 + 105)) + (3*b*x^2*(m + 1)*(210*a^2*c^2 - b^4*m + 72*a^2*c^2*m + 6*a^2*c^2*m^2 + 14*a*b^2*c*m + 2*a*b^2*c*
m^2))/(4*c^2*(176*m + 86*m^2 + 16*m^3 + m^4 + 105)))

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