3.1.37 \(\int (d x)^m (A+B x+C x^2) (a+b x^2+c x^4)^3 \, dx\) [37]

Optimal. Leaf size=399 \[ \frac {a^3 A (d x)^{1+m}}{d (1+m)}+\frac {a^3 B (d x)^{2+m}}{d^2 (2+m)}+\frac {a^2 (3 A b+a C) (d x)^{3+m}}{d^3 (3+m)}+\frac {3 a^2 b B (d x)^{4+m}}{d^4 (4+m)}+\frac {3 a \left (A \left (b^2+a c\right )+a b C\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {3 a B \left (b^2+a c\right ) (d x)^{6+m}}{d^6 (6+m)}+\frac {\left (A \left (b^3+6 a b c\right )+3 a \left (b^2+a c\right ) C\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {b B \left (b^2+6 a c\right ) (d x)^{8+m}}{d^8 (8+m)}+\frac {\left (3 A c \left (b^2+a c\right )+b \left (b^2+6 a c\right ) C\right ) (d x)^{9+m}}{d^9 (9+m)}+\frac {3 B c \left (b^2+a c\right ) (d x)^{10+m}}{d^{10} (10+m)}+\frac {3 c \left (A b c+\left (b^2+a c\right ) C\right ) (d x)^{11+m}}{d^{11} (11+m)}+\frac {3 b B c^2 (d x)^{12+m}}{d^{12} (12+m)}+\frac {c^2 (A c+3 b C) (d x)^{13+m}}{d^{13} (13+m)}+\frac {B c^3 (d x)^{14+m}}{d^{14} (14+m)}+\frac {c^3 C (d x)^{15+m}}{d^{15} (15+m)} \]

[Out]

a^3*A*(d*x)^(1+m)/d/(1+m)+a^3*B*(d*x)^(2+m)/d^2/(2+m)+a^2*(3*A*b+C*a)*(d*x)^(3+m)/d^3/(3+m)+3*a^2*b*B*(d*x)^(4
+m)/d^4/(4+m)+3*a*(A*(a*c+b^2)+a*b*C)*(d*x)^(5+m)/d^5/(5+m)+3*a*B*(a*c+b^2)*(d*x)^(6+m)/d^6/(6+m)+(A*(6*a*b*c+
b^3)+3*a*(a*c+b^2)*C)*(d*x)^(7+m)/d^7/(7+m)+b*B*(6*a*c+b^2)*(d*x)^(8+m)/d^8/(8+m)+(3*A*c*(a*c+b^2)+b*(6*a*c+b^
2)*C)*(d*x)^(9+m)/d^9/(9+m)+3*B*c*(a*c+b^2)*(d*x)^(10+m)/d^10/(10+m)+3*c*(A*b*c+(a*c+b^2)*C)*(d*x)^(11+m)/d^11
/(11+m)+3*b*B*c^2*(d*x)^(12+m)/d^12/(12+m)+c^2*(A*c+3*C*b)*(d*x)^(13+m)/d^13/(13+m)+B*c^3*(d*x)^(14+m)/d^14/(1
4+m)+c^3*C*(d*x)^(15+m)/d^15/(15+m)

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Rubi [A]
time = 0.28, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1642} \begin {gather*} \frac {a^3 A (d x)^{m+1}}{d (m+1)}+\frac {a^3 B (d x)^{m+2}}{d^2 (m+2)}+\frac {a^2 (d x)^{m+3} (a C+3 A b)}{d^3 (m+3)}+\frac {3 a^2 b B (d x)^{m+4}}{d^4 (m+4)}+\frac {3 c (d x)^{m+11} \left (C \left (a c+b^2\right )+A b c\right )}{d^{11} (m+11)}+\frac {(d x)^{m+9} \left (3 A c \left (a c+b^2\right )+b C \left (6 a c+b^2\right )\right )}{d^9 (m+9)}+\frac {3 a (d x)^{m+5} \left (A \left (a c+b^2\right )+a b C\right )}{d^5 (m+5)}+\frac {(d x)^{m+7} \left (A \left (6 a b c+b^3\right )+3 a C \left (a c+b^2\right )\right )}{d^7 (m+7)}+\frac {3 B c \left (a c+b^2\right ) (d x)^{m+10}}{d^{10} (m+10)}+\frac {b B \left (6 a c+b^2\right ) (d x)^{m+8}}{d^8 (m+8)}+\frac {3 a B \left (a c+b^2\right ) (d x)^{m+6}}{d^6 (m+6)}+\frac {c^2 (d x)^{m+13} (A c+3 b C)}{d^{13} (m+13)}+\frac {3 b B c^2 (d x)^{m+12}}{d^{12} (m+12)}+\frac {B c^3 (d x)^{m+14}}{d^{14} (m+14)}+\frac {c^3 C (d x)^{m+15}}{d^{15} (m+15)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^3,x]

[Out]

(a^3*A*(d*x)^(1 + m))/(d*(1 + m)) + (a^3*B*(d*x)^(2 + m))/(d^2*(2 + m)) + (a^2*(3*A*b + a*C)*(d*x)^(3 + m))/(d
^3*(3 + m)) + (3*a^2*b*B*(d*x)^(4 + m))/(d^4*(4 + m)) + (3*a*(A*(b^2 + a*c) + a*b*C)*(d*x)^(5 + m))/(d^5*(5 +
m)) + (3*a*B*(b^2 + a*c)*(d*x)^(6 + m))/(d^6*(6 + m)) + ((A*(b^3 + 6*a*b*c) + 3*a*(b^2 + a*c)*C)*(d*x)^(7 + m)
)/(d^7*(7 + m)) + (b*B*(b^2 + 6*a*c)*(d*x)^(8 + m))/(d^8*(8 + m)) + ((3*A*c*(b^2 + a*c) + b*(b^2 + 6*a*c)*C)*(
d*x)^(9 + m))/(d^9*(9 + m)) + (3*B*c*(b^2 + a*c)*(d*x)^(10 + m))/(d^10*(10 + m)) + (3*c*(A*b*c + (b^2 + a*c)*C
)*(d*x)^(11 + m))/(d^11*(11 + m)) + (3*b*B*c^2*(d*x)^(12 + m))/(d^12*(12 + m)) + (c^2*(A*c + 3*b*C)*(d*x)^(13
+ m))/(d^13*(13 + m)) + (B*c^3*(d*x)^(14 + m))/(d^14*(14 + m)) + (c^3*C*(d*x)^(15 + m))/(d^15*(15 + m))

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right )^3 \, dx &=\int \left (a^3 A (d x)^m+\frac {a^3 B (d x)^{1+m}}{d}+\frac {a^2 (3 A b+a C) (d x)^{2+m}}{d^2}+\frac {3 a^2 b B (d x)^{3+m}}{d^3}+\frac {3 a \left (A \left (b^2+a c\right )+a b C\right ) (d x)^{4+m}}{d^4}+\frac {3 a B \left (b^2+a c\right ) (d x)^{5+m}}{d^5}+\frac {\left (A \left (b^3+6 a b c\right )+3 a \left (b^2+a c\right ) C\right ) (d x)^{6+m}}{d^6}+\frac {b B \left (b^2+6 a c\right ) (d x)^{7+m}}{d^7}+\frac {\left (3 A c \left (b^2+a c\right )+b \left (b^2+6 a c\right ) C\right ) (d x)^{8+m}}{d^8}+\frac {3 B c \left (b^2+a c\right ) (d x)^{9+m}}{d^9}+\frac {3 c \left (A b c+\left (b^2+a c\right ) C\right ) (d x)^{10+m}}{d^{10}}+\frac {3 b B c^2 (d x)^{11+m}}{d^{11}}+\frac {c^2 (A c+3 b C) (d x)^{12+m}}{d^{12}}+\frac {B c^3 (d x)^{13+m}}{d^{13}}+\frac {c^3 C (d x)^{14+m}}{d^{14}}\right ) \, dx\\ &=\frac {a^3 A (d x)^{1+m}}{d (1+m)}+\frac {a^3 B (d x)^{2+m}}{d^2 (2+m)}+\frac {a^2 (3 A b+a C) (d x)^{3+m}}{d^3 (3+m)}+\frac {3 a^2 b B (d x)^{4+m}}{d^4 (4+m)}+\frac {3 a \left (A \left (b^2+a c\right )+a b C\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {3 a B \left (b^2+a c\right ) (d x)^{6+m}}{d^6 (6+m)}+\frac {\left (A \left (b^3+6 a b c\right )+3 a \left (b^2+a c\right ) C\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {b B \left (b^2+6 a c\right ) (d x)^{8+m}}{d^8 (8+m)}+\frac {\left (3 A c \left (b^2+a c\right )+b \left (b^2+6 a c\right ) C\right ) (d x)^{9+m}}{d^9 (9+m)}+\frac {3 B c \left (b^2+a c\right ) (d x)^{10+m}}{d^{10} (10+m)}+\frac {3 c \left (A b c+\left (b^2+a c\right ) C\right ) (d x)^{11+m}}{d^{11} (11+m)}+\frac {3 b B c^2 (d x)^{12+m}}{d^{12} (12+m)}+\frac {c^2 (A c+3 b C) (d x)^{13+m}}{d^{13} (13+m)}+\frac {B c^3 (d x)^{14+m}}{d^{14} (14+m)}+\frac {c^3 C (d x)^{15+m}}{d^{15} (15+m)}\\ \end {align*}

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Mathematica [A]
time = 2.11, size = 296, normalized size = 0.74 \begin {gather*} x (d x)^m \left (\frac {a^3 A}{1+m}+\frac {a^3 B x}{2+m}+\frac {a^2 (3 A b+a C) x^2}{3+m}+\frac {3 a^2 b B x^3}{4+m}+\frac {3 a \left (A \left (b^2+a c\right )+a b C\right ) x^4}{5+m}+\frac {3 a B \left (b^2+a c\right ) x^5}{6+m}+\frac {\left (A \left (b^3+6 a b c\right )+3 a \left (b^2+a c\right ) C\right ) x^6}{7+m}+\frac {b B \left (b^2+6 a c\right ) x^7}{8+m}+\frac {\left (3 A c \left (b^2+a c\right )+b \left (b^2+6 a c\right ) C\right ) x^8}{9+m}+\frac {3 B c \left (b^2+a c\right ) x^9}{10+m}+\frac {3 c \left (A b c+\left (b^2+a c\right ) C\right ) x^{10}}{11+m}+\frac {3 b B c^2 x^{11}}{12+m}+\frac {c^2 (A c+3 b C) x^{12}}{13+m}+\frac {B c^3 x^{13}}{14+m}+\frac {c^3 C x^{14}}{15+m}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^3,x]

[Out]

x*(d*x)^m*((a^3*A)/(1 + m) + (a^3*B*x)/(2 + m) + (a^2*(3*A*b + a*C)*x^2)/(3 + m) + (3*a^2*b*B*x^3)/(4 + m) + (
3*a*(A*(b^2 + a*c) + a*b*C)*x^4)/(5 + m) + (3*a*B*(b^2 + a*c)*x^5)/(6 + m) + ((A*(b^3 + 6*a*b*c) + 3*a*(b^2 +
a*c)*C)*x^6)/(7 + m) + (b*B*(b^2 + 6*a*c)*x^7)/(8 + m) + ((3*A*c*(b^2 + a*c) + b*(b^2 + 6*a*c)*C)*x^8)/(9 + m)
+ (3*B*c*(b^2 + a*c)*x^9)/(10 + m) + (3*c*(A*b*c + (b^2 + a*c)*C)*x^10)/(11 + m) + (3*b*B*c^2*x^11)/(12 + m)
+ (c^2*(A*c + 3*b*C)*x^12)/(13 + m) + (B*c^3*x^13)/(14 + m) + (c^3*C*x^14)/(15 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5519\) vs. \(2(399)=798\).
time = 0.04, size = 5520, normalized size = 13.83

method	result	size
gosper	\(\text {Expression too large to display}\)	\(5520\)
risch	\(\text {Expression too large to display}\)	\(5520\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [A]
time = 0.43, size = 611, normalized size = 1.53 \begin {gather*} \frac {C c^{3} d^{m} x^{15} x^{m}}{m + 15} + \frac {B c^{3} d^{m} x^{14} x^{m}}{m + 14} + \frac {3 \, C b c^{2} d^{m} x^{13} x^{m}}{m + 13} + \frac {A c^{3} d^{m} x^{13} x^{m}}{m + 13} + \frac {3 \, B b c^{2} d^{m} x^{12} x^{m}}{m + 12} + \frac {3 \, C b^{2} c d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, C a c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, A b c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {3 \, B b^{2} c d^{m} x^{10} x^{m}}{m + 10} + \frac {3 \, B a c^{2} d^{m} x^{10} x^{m}}{m + 10} + \frac {C b^{3} d^{m} x^{9} x^{m}}{m + 9} + \frac {6 \, C a b c d^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, A b^{2} c d^{m} x^{9} x^{m}}{m + 9} + \frac {3 \, A a c^{2} d^{m} x^{9} x^{m}}{m + 9} + \frac {B b^{3} d^{m} x^{8} x^{m}}{m + 8} + \frac {6 \, B a b c d^{m} x^{8} x^{m}}{m + 8} + \frac {3 \, C a b^{2} d^{m} x^{7} x^{m}}{m + 7} + \frac {A b^{3} d^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, C a^{2} c d^{m} x^{7} x^{m}}{m + 7} + \frac {6 \, A a b c d^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a b^{2} d^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, B a^{2} c d^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, C a^{2} b d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, A a^{2} c d^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, B a^{2} b d^{m} x^{4} x^{m}}{m + 4} + \frac {C a^{3} d^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, A a^{2} b d^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{3} d^{m} x^{2} x^{m}}{m + 2} + \frac {\left (d x\right )^{m + 1} A a^{3}}{d {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^3,x, algorithm="maxima")

[Out]

C*c^3*d^m*x^15*x^m/(m + 15) + B*c^3*d^m*x^14*x^m/(m + 14) + 3*C*b*c^2*d^m*x^13*x^m/(m + 13) + A*c^3*d^m*x^13*x
^m/(m + 13) + 3*B*b*c^2*d^m*x^12*x^m/(m + 12) + 3*C*b^2*c*d^m*x^11*x^m/(m + 11) + 3*C*a*c^2*d^m*x^11*x^m/(m +
11) + 3*A*b*c^2*d^m*x^11*x^m/(m + 11) + 3*B*b^2*c*d^m*x^10*x^m/(m + 10) + 3*B*a*c^2*d^m*x^10*x^m/(m + 10) + C*
b^3*d^m*x^9*x^m/(m + 9) + 6*C*a*b*c*d^m*x^9*x^m/(m + 9) + 3*A*b^2*c*d^m*x^9*x^m/(m + 9) + 3*A*a*c^2*d^m*x^9*x^
m/(m + 9) + B*b^3*d^m*x^8*x^m/(m + 8) + 6*B*a*b*c*d^m*x^8*x^m/(m + 8) + 3*C*a*b^2*d^m*x^7*x^m/(m + 7) + A*b^3*
d^m*x^7*x^m/(m + 7) + 3*C*a^2*c*d^m*x^7*x^m/(m + 7) + 6*A*a*b*c*d^m*x^7*x^m/(m + 7) + 3*B*a*b^2*d^m*x^6*x^m/(m
+ 6) + 3*B*a^2*c*d^m*x^6*x^m/(m + 6) + 3*C*a^2*b*d^m*x^5*x^m/(m + 5) + 3*A*a*b^2*d^m*x^5*x^m/(m + 5) + 3*A*a^
2*c*d^m*x^5*x^m/(m + 5) + 3*B*a^2*b*d^m*x^4*x^m/(m + 4) + C*a^3*d^m*x^3*x^m/(m + 3) + 3*A*a^2*b*d^m*x^3*x^m/(m
+ 3) + B*a^3*d^m*x^2*x^m/(m + 2) + (d*x)^(m + 1)*A*a^3/(d*(m + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3898 vs. \(2 (399) = 798\).
time = 0.43, size = 3898, normalized size = 9.77 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

((C*c^3*m^14 + 105*C*c^3*m^13 + 5005*C*c^3*m^12 + 143325*C*c^3*m^11 + 2749747*C*c^3*m^10 + 37312275*C*c^3*m^9
+ 368411615*C*c^3*m^8 + 2681453775*C*c^3*m^7 + 14409322928*C*c^3*m^6 + 56663366760*C*c^3*m^5 + 159721605680*C*
c^3*m^4 + 310989260400*C*c^3*m^3 + 392156797824*C*c^3*m^2 + 283465647360*C*c^3*m + 87178291200*C*c^3)*x^15 + (
B*c^3*m^14 + 106*B*c^3*m^13 + 5096*B*c^3*m^12 + 147056*B*c^3*m^11 + 2840838*B*c^3*m^10 + 38786748*B*c^3*m^9 +
385081268*B*c^3*m^8 + 2816490248*B*c^3*m^7 + 15200266081*B*c^3*m^6 + 59999485546*B*c^3*m^5 + 169679309436*B*c^
3*m^4 + 331303013496*B*c^3*m^3 + 418753514880*B*c^3*m^2 + 303268406400*B*c^3*m + 93405312000*B*c^3)*x^14 + ((3
*C*b*c^2 + A*c^3)*m^14 + 107*(3*C*b*c^2 + A*c^3)*m^13 + 5189*(3*C*b*c^2 + A*c^3)*m^12 + 150943*(3*C*b*c^2 + A*
c^3)*m^11 + 2937363*(3*C*b*c^2 + A*c^3)*m^10 + 40372761*(3*C*b*c^2 + A*c^3)*m^9 + 403249847*(3*C*b*c^2 + A*c^3
)*m^8 + 2965379989*(3*C*b*c^2 + A*c^3)*m^7 + 16081189696*(3*C*b*c^2 + A*c^3)*m^6 + 63747744632*(3*C*b*c^2 + A*
c^3)*m^5 + 180951426864*(3*C*b*c^2 + A*c^3)*m^4 + 301771008000*C*b*c^2 + 100590336000*A*c^3 + 354444796368*(3*
C*b*c^2 + A*c^3)*m^3 + 449213351040*(3*C*b*c^2 + A*c^3)*m^2 + 326044051200*(3*C*b*c^2 + A*c^3)*m)*x^13 + 3*(B*
b*c^2*m^14 + 108*B*b*c^2*m^13 + 5284*B*b*c^2*m^12 + 154992*B*b*c^2*m^11 + 3039718*B*b*c^2*m^10 + 42081864*B*b*
c^2*m^9 + 423113372*B*b*c^2*m^8 + 3130267536*B*b*c^2*m^7 + 17067919121*B*b*c^2*m^6 + 67988181228*B*b*c^2*m^5 +
193813932344*B*b*c^2*m^4 + 381046157472*B*b*c^2*m^3 + 484441814160*B*b*c^2*m^2 + 352515844800*B*b*c^2*m + 108
972864000*B*b*c^2)*x^12 + 3*((C*b^2*c + (C*a + A*b)*c^2)*m^14 + 109*(C*b^2*c + (C*a + A*b)*c^2)*m^13 + 5381*(C
*b^2*c + (C*a + A*b)*c^2)*m^12 + 159209*(C*b^2*c + (C*a + A*b)*c^2)*m^11 + 3148323*(C*b^2*c + (C*a + A*b)*c^2)
*m^10 + 43926927*(C*b^2*c + (C*a + A*b)*c^2)*m^9 + 444899543*(C*b^2*c + (C*a + A*b)*c^2)*m^8 + 3313733027*(C*b
^2*c + (C*a + A*b)*c^2)*m^7 + 18180066256*(C*b^2*c + (C*a + A*b)*c^2)*m^6 + 72822481864*(C*b^2*c + (C*a + A*b)
*c^2)*m^5 + 208624806576*(C*b^2*c + (C*a + A*b)*c^2)*m^4 + 118879488000*C*b^2*c + 411940473264*(C*b^2*c + (C*a
+ A*b)*c^2)*m^3 + 118879488000*(C*a + A*b)*c^2 + 525650497920*(C*b^2*c + (C*a + A*b)*c^2)*m^2 + 383662137600*
(C*b^2*c + (C*a + A*b)*c^2)*m)*x^11 + 3*((B*b^2*c + B*a*c^2)*m^14 + 110*(B*b^2*c + B*a*c^2)*m^13 + 5480*(B*b^2
*c + B*a*c^2)*m^12 + 163600*(B*b^2*c + B*a*c^2)*m^11 + 3263622*(B*b^2*c + B*a*c^2)*m^10 + 45922260*(B*b^2*c +
B*a*c^2)*m^9 + 468873140*(B*b^2*c + B*a*c^2)*m^8 + 3518896600*(B*b^2*c + B*a*c^2)*m^7 + 19442163553*(B*b^2*c +
B*a*c^2)*m^6 + 78381575150*(B*b^2*c + B*a*c^2)*m^5 + 225856355580*(B*b^2*c + B*a*c^2)*m^4 + 130767436800*B*b^
2*c + 130767436800*B*a*c^2 + 448249789800*(B*b^2*c + B*a*c^2)*m^3 + 574497805824*(B*b^2*c + B*a*c^2)*m^2 + 420
839556480*(B*b^2*c + B*a*c^2)*m)*x^10 + ((C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^14 + 111*(C*b^3 + 3*A*a
*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^13 + 5581*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^12 + 168171*(C*b^3 + 3
*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^11 + 3386083*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^10 + 48083733*(
C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^9 + 495342143*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^8 + 3
749548713*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^7 + 20885191136*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^
2)*c)*m^6 + 84836490456*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^5 + 246143692976*(C*b^3 + 3*A*a*c^2 + 3*
(2*C*a*b + A*b^2)*c)*m^4 + 145297152000*C*b^3 + 435891456000*A*a*c^2 + 491520108816*(C*b^3 + 3*A*a*c^2 + 3*(2*
C*a*b + A*b^2)*c)*m^3 + 633314724480*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m^2 + 435891456000*(2*C*a*b +
A*b^2)*c + 465985094400*(C*b^3 + 3*A*a*c^2 + 3*(2*C*a*b + A*b^2)*c)*m)*x^9 + ((B*b^3 + 6*B*a*b*c)*m^14 + 112*
(B*b^3 + 6*B*a*b*c)*m^13 + 5684*(B*b^3 + 6*B*a*b*c)*m^12 + 172928*(B*b^3 + 6*B*a*b*c)*m^11 + 3516198*(B*b^3 +
6*B*a*b*c)*m^10 + 50428896*(B*b^3 + 6*B*a*b*c)*m^9 + 524664572*(B*b^3 + 6*B*a*b*c)*m^8 + 4010311424*(B*b^3 + 6
*B*a*b*c)*m^7 + 22548638161*(B*b^3 + 6*B*a*b*c)*m^6 + 92414105392*(B*b^3 + 6*B*a*b*c)*m^5 + 270359263944*(B*b^
3 + 6*B*a*b*c)*m^4 + 163459296000*B*b^3 + 980755776000*B*a*b*c + 543939234048*(B*b^3 + 6*B*a*b*c)*m^3 + 705481
831440*(B*b^3 + 6*B*a*b*c)*m^2 + 521962963200*(B*b^3 + 6*B*a*b*c)*m)*x^8 + ((3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*
A*a*b)*c)*m^14 + 113*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^13 + 5789*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2
*A*a*b)*c)*m^12 + 177877*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^11 + 3654483*(3*C*a*b^2 + A*b^3 + 3*(C*
a^2 + 2*A*a*b)*c)*m^10 + 52977099*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^9 + 557256047*(3*C*a*b^2 + A*b
^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^8 + 4306835671*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^7 + 24483279856*(3*
C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^6 + 101420251688*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^5 +
299730345264*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a*b)*c)*m^4 + 560431872000*C*a*b^2 + 186810624000*A*b^3 + 608
700928752*(3*C*a*b^2 + A*b^3 + 3*(C*a^2 + 2*A*a...

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 47658 vs. \(2 (379) = 758\).
time = 3.36, size = 47658, normalized size = 119.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(C*x**2+B*x+A)*(c*x**4+b*x**2+a)**3,x)

[Out]

Piecewise(((-A*a**3/(14*x**14) - A*a**2*b/(4*x**12) - 3*A*a**2*c/(10*x**10) - 3*A*a*b**2/(10*x**10) - 3*A*a*b*
c/(4*x**8) - A*a*c**2/(2*x**6) - A*b**3/(8*x**8) - A*b**2*c/(2*x**6) - 3*A*b*c**2/(4*x**4) - A*c**3/(2*x**2) -
B*a**3/(13*x**13) - 3*B*a**2*b/(11*x**11) - B*a**2*c/(3*x**9) - B*a*b**2/(3*x**9) - 6*B*a*b*c/(7*x**7) - 3*B*
a*c**2/(5*x**5) - B*b**3/(7*x**7) - 3*B*b**2*c/(5*x**5) - B*b*c**2/x**3 - B*c**3/x - C*a**3/(12*x**12) - 3*C*a
**2*b/(10*x**10) - 3*C*a**2*c/(8*x**8) - 3*C*a*b**2/(8*x**8) - C*a*b*c/x**6 - 3*C*a*c**2/(4*x**4) - C*b**3/(6*
x**6) - 3*C*b**2*c/(4*x**4) - 3*C*b*c**2/(2*x**2) + C*c**3*log(x))/d**15, Eq(m, -15)), ((-A*a**3/(13*x**13) -
3*A*a**2*b/(11*x**11) - A*a**2*c/(3*x**9) - A*a*b**2/(3*x**9) - 6*A*a*b*c/(7*x**7) - 3*A*a*c**2/(5*x**5) - A*b
**3/(7*x**7) - 3*A*b**2*c/(5*x**5) - A*b*c**2/x**3 - A*c**3/x - B*a**3/(12*x**12) - 3*B*a**2*b/(10*x**10) - 3*
B*a**2*c/(8*x**8) - 3*B*a*b**2/(8*x**8) - B*a*b*c/x**6 - 3*B*a*c**2/(4*x**4) - B*b**3/(6*x**6) - 3*B*b**2*c/(4
*x**4) - 3*B*b*c**2/(2*x**2) + B*c**3*log(x) - C*a**3/(11*x**11) - C*a**2*b/(3*x**9) - 3*C*a**2*c/(7*x**7) - 3
*C*a*b**2/(7*x**7) - 6*C*a*b*c/(5*x**5) - C*a*c**2/x**3 - C*b**3/(5*x**5) - C*b**2*c/x**3 - 3*C*b*c**2/x + C*c
**3*x)/d**14, Eq(m, -14)), ((-A*a**3/(12*x**12) - 3*A*a**2*b/(10*x**10) - 3*A*a**2*c/(8*x**8) - 3*A*a*b**2/(8*
x**8) - A*a*b*c/x**6 - 3*A*a*c**2/(4*x**4) - A*b**3/(6*x**6) - 3*A*b**2*c/(4*x**4) - 3*A*b*c**2/(2*x**2) + A*c
**3*log(x) - B*a**3/(11*x**11) - B*a**2*b/(3*x**9) - 3*B*a**2*c/(7*x**7) - 3*B*a*b**2/(7*x**7) - 6*B*a*b*c/(5*
x**5) - B*a*c**2/x**3 - B*b**3/(5*x**5) - B*b**2*c/x**3 - 3*B*b*c**2/x + B*c**3*x - C*a**3/(10*x**10) - 3*C*a*
*2*b/(8*x**8) - C*a**2*c/(2*x**6) - C*a*b**2/(2*x**6) - 3*C*a*b*c/(2*x**4) - 3*C*a*c**2/(2*x**2) - C*b**3/(4*x
**4) - 3*C*b**2*c/(2*x**2) + 3*C*b*c**2*log(x) + C*c**3*x**2/2)/d**13, Eq(m, -13)), ((-A*a**3/(11*x**11) - A*a
**2*b/(3*x**9) - 3*A*a**2*c/(7*x**7) - 3*A*a*b**2/(7*x**7) - 6*A*a*b*c/(5*x**5) - A*a*c**2/x**3 - A*b**3/(5*x*
*5) - A*b**2*c/x**3 - 3*A*b*c**2/x + A*c**3*x - B*a**3/(10*x**10) - 3*B*a**2*b/(8*x**8) - B*a**2*c/(2*x**6) -
B*a*b**2/(2*x**6) - 3*B*a*b*c/(2*x**4) - 3*B*a*c**2/(2*x**2) - B*b**3/(4*x**4) - 3*B*b**2*c/(2*x**2) + 3*B*b*c
**2*log(x) + B*c**3*x**2/2 - C*a**3/(9*x**9) - 3*C*a**2*b/(7*x**7) - 3*C*a**2*c/(5*x**5) - 3*C*a*b**2/(5*x**5)
- 2*C*a*b*c/x**3 - 3*C*a*c**2/x - C*b**3/(3*x**3) - 3*C*b**2*c/x + 3*C*b*c**2*x + C*c**3*x**3/3)/d**12, Eq(m,
-12)), ((-A*a**3/(10*x**10) - 3*A*a**2*b/(8*x**8) - A*a**2*c/(2*x**6) - A*a*b**2/(2*x**6) - 3*A*a*b*c/(2*x**4
) - 3*A*a*c**2/(2*x**2) - A*b**3/(4*x**4) - 3*A*b**2*c/(2*x**2) + 3*A*b*c**2*log(x) + A*c**3*x**2/2 - B*a**3/(
9*x**9) - 3*B*a**2*b/(7*x**7) - 3*B*a**2*c/(5*x**5) - 3*B*a*b**2/(5*x**5) - 2*B*a*b*c/x**3 - 3*B*a*c**2/x - B*
b**3/(3*x**3) - 3*B*b**2*c/x + 3*B*b*c**2*x + B*c**3*x**3/3 - C*a**3/(8*x**8) - C*a**2*b/(2*x**6) - 3*C*a**2*c
/(4*x**4) - 3*C*a*b**2/(4*x**4) - 3*C*a*b*c/x**2 + 3*C*a*c**2*log(x) - C*b**3/(2*x**2) + 3*C*b**2*c*log(x) + 3
*C*b*c**2*x**2/2 + C*c**3*x**4/4)/d**11, Eq(m, -11)), ((-A*a**3/(9*x**9) - 3*A*a**2*b/(7*x**7) - 3*A*a**2*c/(5
*x**5) - 3*A*a*b**2/(5*x**5) - 2*A*a*b*c/x**3 - 3*A*a*c**2/x - A*b**3/(3*x**3) - 3*A*b**2*c/x + 3*A*b*c**2*x +
A*c**3*x**3/3 - B*a**3/(8*x**8) - B*a**2*b/(2*x**6) - 3*B*a**2*c/(4*x**4) - 3*B*a*b**2/(4*x**4) - 3*B*a*b*c/x
**2 + 3*B*a*c**2*log(x) - B*b**3/(2*x**2) + 3*B*b**2*c*log(x) + 3*B*b*c**2*x**2/2 + B*c**3*x**4/4 - C*a**3/(7*
x**7) - 3*C*a**2*b/(5*x**5) - C*a**2*c/x**3 - C*a*b**2/x**3 - 6*C*a*b*c/x + 3*C*a*c**2*x - C*b**3/x + 3*C*b**2
*c*x + C*b*c**2*x**3 + C*c**3*x**5/5)/d**10, Eq(m, -10)), ((-A*a**3/(8*x**8) - A*a**2*b/(2*x**6) - 3*A*a**2*c/
(4*x**4) - 3*A*a*b**2/(4*x**4) - 3*A*a*b*c/x**2 + 3*A*a*c**2*log(x) - A*b**3/(2*x**2) + 3*A*b**2*c*log(x) + 3*
A*b*c**2*x**2/2 + A*c**3*x**4/4 - B*a**3/(7*x**7) - 3*B*a**2*b/(5*x**5) - B*a**2*c/x**3 - B*a*b**2/x**3 - 6*B*
a*b*c/x + 3*B*a*c**2*x - B*b**3/x + 3*B*b**2*c*x + B*b*c**2*x**3 + B*c**3*x**5/5 - C*a**3/(6*x**6) - 3*C*a**2*
b/(4*x**4) - 3*C*a**2*c/(2*x**2) - 3*C*a*b**2/(2*x**2) + 6*C*a*b*c*log(x) + 3*C*a*c**2*x**2/2 + C*b**3*log(x)
+ 3*C*b**2*c*x**2/2 + 3*C*b*c**2*x**4/4 + C*c**3*x**6/6)/d**9, Eq(m, -9)), ((-A*a**3/(7*x**7) - 3*A*a**2*b/(5*
x**5) - A*a**2*c/x**3 - A*a*b**2/x**3 - 6*A*a*b*c/x + 3*A*a*c**2*x - A*b**3/x + 3*A*b**2*c*x + A*b*c**2*x**3 +
A*c**3*x**5/5 - B*a**3/(6*x**6) - 3*B*a**2*b/(4*x**4) - 3*B*a**2*c/(2*x**2) - 3*B*a*b**2/(2*x**2) + 6*B*a*b*c
*log(x) + 3*B*a*c**2*x**2/2 + B*b**3*log(x) + 3*B*b**2*c*x**2/2 + 3*B*b*c**2*x**4/4 + B*c**3*x**6/6 - C*a**3/(
5*x**5) - C*a**2*b/x**3 - 3*C*a**2*c/x - 3*C*a*b**2/x + 6*C*a*b*c*x + C*a*c**2*x**3 + C*b**3*x + C*b**2*c*x**3
+ 3*C*b*c**2*x**5/5 + C*c**3*x**7/7)/d**8, Eq(m, -8)), ((-A*a**3/(6*x**6) - 3*A*a**2*b/(4*x**4) - 3*A*a**2*c/
(2*x**2) - 3*A*a*b**2/(2*x**2) + 6*A*a*b*c*log(x) + 3*A*a*c**2*x**2/2 + A*b**3*log(x) + 3*A*b**2*c*x**2/2 + 3*
A*b*c**2*x**4/4 + A*c**3*x**6/6 - B*a**3/(5*x**5) - B*a**2*b/x**3 - 3*B*a**2*c/x - 3*B*a*b**2/x + 6*B*a*b*c*x
+ B*a*c**2*x**3 + B*b**3*x + B*b**2*c*x**3 + 3*...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 7808 vs. \(2 (399) = 798\).
time = 3.72, size = 7808, normalized size = 19.57 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(C*x^2+B*x+A)*(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

((d*x)^m*C*c^3*m^14*x^15 + (d*x)^m*B*c^3*m^14*x^14 + 105*(d*x)^m*C*c^3*m^13*x^15 + 3*(d*x)^m*C*b*c^2*m^14*x^13
+ (d*x)^m*A*c^3*m^14*x^13 + 106*(d*x)^m*B*c^3*m^13*x^14 + 5005*(d*x)^m*C*c^3*m^12*x^15 + 3*(d*x)^m*B*b*c^2*m^
14*x^12 + 321*(d*x)^m*C*b*c^2*m^13*x^13 + 107*(d*x)^m*A*c^3*m^13*x^13 + 5096*(d*x)^m*B*c^3*m^12*x^14 + 143325*
(d*x)^m*C*c^3*m^11*x^15 + 3*(d*x)^m*C*b^2*c*m^14*x^11 + 3*(d*x)^m*C*a*c^2*m^14*x^11 + 3*(d*x)^m*A*b*c^2*m^14*x
^11 + 324*(d*x)^m*B*b*c^2*m^13*x^12 + 15567*(d*x)^m*C*b*c^2*m^12*x^13 + 5189*(d*x)^m*A*c^3*m^12*x^13 + 147056*
(d*x)^m*B*c^3*m^11*x^14 + 2749747*(d*x)^m*C*c^3*m^10*x^15 + 3*(d*x)^m*B*b^2*c*m^14*x^10 + 3*(d*x)^m*B*a*c^2*m^
14*x^10 + 327*(d*x)^m*C*b^2*c*m^13*x^11 + 327*(d*x)^m*C*a*c^2*m^13*x^11 + 327*(d*x)^m*A*b*c^2*m^13*x^11 + 1585
2*(d*x)^m*B*b*c^2*m^12*x^12 + 452829*(d*x)^m*C*b*c^2*m^11*x^13 + 150943*(d*x)^m*A*c^3*m^11*x^13 + 2840838*(d*x
)^m*B*c^3*m^10*x^14 + 37312275*(d*x)^m*C*c^3*m^9*x^15 + (d*x)^m*C*b^3*m^14*x^9 + 6*(d*x)^m*C*a*b*c*m^14*x^9 +
3*(d*x)^m*A*b^2*c*m^14*x^9 + 3*(d*x)^m*A*a*c^2*m^14*x^9 + 330*(d*x)^m*B*b^2*c*m^13*x^10 + 330*(d*x)^m*B*a*c^2*
m^13*x^10 + 16143*(d*x)^m*C*b^2*c*m^12*x^11 + 16143*(d*x)^m*C*a*c^2*m^12*x^11 + 16143*(d*x)^m*A*b*c^2*m^12*x^1
1 + 464976*(d*x)^m*B*b*c^2*m^11*x^12 + 8812089*(d*x)^m*C*b*c^2*m^10*x^13 + 2937363*(d*x)^m*A*c^3*m^10*x^13 + 3
8786748*(d*x)^m*B*c^3*m^9*x^14 + 368411615*(d*x)^m*C*c^3*m^8*x^15 + (d*x)^m*B*b^3*m^14*x^8 + 6*(d*x)^m*B*a*b*c
*m^14*x^8 + 111*(d*x)^m*C*b^3*m^13*x^9 + 666*(d*x)^m*C*a*b*c*m^13*x^9 + 333*(d*x)^m*A*b^2*c*m^13*x^9 + 333*(d*
x)^m*A*a*c^2*m^13*x^9 + 16440*(d*x)^m*B*b^2*c*m^12*x^10 + 16440*(d*x)^m*B*a*c^2*m^12*x^10 + 477627*(d*x)^m*C*b
^2*c*m^11*x^11 + 477627*(d*x)^m*C*a*c^2*m^11*x^11 + 477627*(d*x)^m*A*b*c^2*m^11*x^11 + 9119154*(d*x)^m*B*b*c^2
*m^10*x^12 + 121118283*(d*x)^m*C*b*c^2*m^9*x^13 + 40372761*(d*x)^m*A*c^3*m^9*x^13 + 385081268*(d*x)^m*B*c^3*m^
8*x^14 + 2681453775*(d*x)^m*C*c^3*m^7*x^15 + 3*(d*x)^m*C*a*b^2*m^14*x^7 + (d*x)^m*A*b^3*m^14*x^7 + 3*(d*x)^m*C
*a^2*c*m^14*x^7 + 6*(d*x)^m*A*a*b*c*m^14*x^7 + 112*(d*x)^m*B*b^3*m^13*x^8 + 672*(d*x)^m*B*a*b*c*m^13*x^8 + 558
1*(d*x)^m*C*b^3*m^12*x^9 + 33486*(d*x)^m*C*a*b*c*m^12*x^9 + 16743*(d*x)^m*A*b^2*c*m^12*x^9 + 16743*(d*x)^m*A*a
*c^2*m^12*x^9 + 490800*(d*x)^m*B*b^2*c*m^11*x^10 + 490800*(d*x)^m*B*a*c^2*m^11*x^10 + 9444969*(d*x)^m*C*b^2*c*
m^10*x^11 + 9444969*(d*x)^m*C*a*c^2*m^10*x^11 + 9444969*(d*x)^m*A*b*c^2*m^10*x^11 + 126245592*(d*x)^m*B*b*c^2*
m^9*x^12 + 1209749541*(d*x)^m*C*b*c^2*m^8*x^13 + 403249847*(d*x)^m*A*c^3*m^8*x^13 + 2816490248*(d*x)^m*B*c^3*m
^7*x^14 + 14409322928*(d*x)^m*C*c^3*m^6*x^15 + 3*(d*x)^m*B*a*b^2*m^14*x^6 + 3*(d*x)^m*B*a^2*c*m^14*x^6 + 339*(
d*x)^m*C*a*b^2*m^13*x^7 + 113*(d*x)^m*A*b^3*m^13*x^7 + 339*(d*x)^m*C*a^2*c*m^13*x^7 + 678*(d*x)^m*A*a*b*c*m^13
*x^7 + 5684*(d*x)^m*B*b^3*m^12*x^8 + 34104*(d*x)^m*B*a*b*c*m^12*x^8 + 168171*(d*x)^m*C*b^3*m^11*x^9 + 1009026*
(d*x)^m*C*a*b*c*m^11*x^9 + 504513*(d*x)^m*A*b^2*c*m^11*x^9 + 504513*(d*x)^m*A*a*c^2*m^11*x^9 + 9790866*(d*x)^m
*B*b^2*c*m^10*x^10 + 9790866*(d*x)^m*B*a*c^2*m^10*x^10 + 131780781*(d*x)^m*C*b^2*c*m^9*x^11 + 131780781*(d*x)^
m*C*a*c^2*m^9*x^11 + 131780781*(d*x)^m*A*b*c^2*m^9*x^11 + 1269340116*(d*x)^m*B*b*c^2*m^8*x^12 + 8896139967*(d*
x)^m*C*b*c^2*m^7*x^13 + 2965379989*(d*x)^m*A*c^3*m^7*x^13 + 15200266081*(d*x)^m*B*c^3*m^6*x^14 + 56663366760*(
d*x)^m*C*c^3*m^5*x^15 + 3*(d*x)^m*C*a^2*b*m^14*x^5 + 3*(d*x)^m*A*a*b^2*m^14*x^5 + 3*(d*x)^m*A*a^2*c*m^14*x^5 +
342*(d*x)^m*B*a*b^2*m^13*x^6 + 342*(d*x)^m*B*a^2*c*m^13*x^6 + 17367*(d*x)^m*C*a*b^2*m^12*x^7 + 5789*(d*x)^m*A
*b^3*m^12*x^7 + 17367*(d*x)^m*C*a^2*c*m^12*x^7 + 34734*(d*x)^m*A*a*b*c*m^12*x^7 + 172928*(d*x)^m*B*b^3*m^11*x^
8 + 1037568*(d*x)^m*B*a*b*c*m^11*x^8 + 3386083*(d*x)^m*C*b^3*m^10*x^9 + 20316498*(d*x)^m*C*a*b*c*m^10*x^9 + 10
158249*(d*x)^m*A*b^2*c*m^10*x^9 + 10158249*(d*x)^m*A*a*c^2*m^10*x^9 + 137766780*(d*x)^m*B*b^2*c*m^9*x^10 + 137
766780*(d*x)^m*B*a*c^2*m^9*x^10 + 1334698629*(d*x)^m*C*b^2*c*m^8*x^11 + 1334698629*(d*x)^m*C*a*c^2*m^8*x^11 +
1334698629*(d*x)^m*A*b*c^2*m^8*x^11 + 9390802608*(d*x)^m*B*b*c^2*m^7*x^12 + 48243569088*(d*x)^m*C*b*c^2*m^6*x^
13 + 16081189696*(d*x)^m*A*c^3*m^6*x^13 + 59999485546*(d*x)^m*B*c^3*m^5*x^14 + 159721605680*(d*x)^m*C*c^3*m^4*
x^15 + 3*(d*x)^m*B*a^2*b*m^14*x^4 + 345*(d*x)^m*C*a^2*b*m^13*x^5 + 345*(d*x)^m*A*a*b^2*m^13*x^5 + 345*(d*x)^m*
A*a^2*c*m^13*x^5 + 17688*(d*x)^m*B*a*b^2*m^12*x^6 + 17688*(d*x)^m*B*a^2*c*m^12*x^6 + 533631*(d*x)^m*C*a*b^2*m^
11*x^7 + 177877*(d*x)^m*A*b^3*m^11*x^7 + 533631*(d*x)^m*C*a^2*c*m^11*x^7 + 1067262*(d*x)^m*A*a*b*c*m^11*x^7 +
3516198*(d*x)^m*B*b^3*m^10*x^8 + 21097188*(d*x)^m*B*a*b*c*m^10*x^8 + 48083733*(d*x)^m*C*b^3*m^9*x^9 + 28850239
8*(d*x)^m*C*a*b*c*m^9*x^9 + 144251199*(d*x)^m*A*b^2*c*m^9*x^9 + 144251199*(d*x)^m*A*a*c^2*m^9*x^9 + 1406619420
*(d*x)^m*B*b^2*c*m^8*x^10 + 1406619420*(d*x)^m*B*a*c^2*m^8*x^10 + 9941199081*(d*x)^m*C*b^2*c*m^7*x^11 + 994119
9081*(d*x)^m*C*a*c^2*m^7*x^11 + 9941199081*(d*x)^m*A*b*c^2*m^7*x^11 + 51203757363*(d*x)^m*B*b*c^2*m^6*x^12 + 1
91243233896*(d*x)^m*C*b*c^2*m^5*x^13 + 63747744...

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Mupad [B]
time = 3.28, size = 2500, normalized size = 6.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(A + B*x + C*x^2)*(a + b*x^2 + c*x^4)^3,x)

[Out]

(x^7*(d*x)^m*(A*b^3 + 3*C*a*b^2 + 3*C*a^2*c + 6*A*a*b*c)*(593193196800*m + 796089202560*m^2 + 608700928752*m^3
+ 299730345264*m^4 + 101420251688*m^5 + 24483279856*m^6 + 4306835671*m^7 + 557256047*m^8 + 52977099*m^9 + 365
4483*m^10 + 177877*m^11 + 5789*m^12 + 113*m^13 + m^14 + 186810624000))/(4339163001600*m + 6165817614720*m^2 +
5056995703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 54631129553*m^7 + 8207628000*m^
8 + 928095740*m^9 + 78558480*m^10 + 4899622*m^11 + 218400*m^12 + 6580*m^13 + 120*m^14 + m^15 + 1307674368000)
+ (x^9*(d*x)^m*(C*b^3 + 3*A*a*c^2 + 3*A*b^2*c + 6*C*a*b*c)*(465985094400*m + 633314724480*m^2 + 491520108816*m
^3 + 246143692976*m^4 + 84836490456*m^5 + 20885191136*m^6 + 3749548713*m^7 + 495342143*m^8 + 48083733*m^9 + 33
86083*m^10 + 168171*m^11 + 5581*m^12 + 111*m^13 + m^14 + 145297152000))/(4339163001600*m + 6165817614720*m^2 +
5056995703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 54631129553*m^7 + 8207628000*m
^8 + 928095740*m^9 + 78558480*m^10 + 4899622*m^11 + 218400*m^12 + 6580*m^13 + 120*m^14 + m^15 + 1307674368000)
+ (B*c^3*x^14*(d*x)^m*(303268406400*m + 418753514880*m^2 + 331303013496*m^3 + 169679309436*m^4 + 59999485546*
m^5 + 15200266081*m^6 + 2816490248*m^7 + 385081268*m^8 + 38786748*m^9 + 2840838*m^10 + 147056*m^11 + 5096*m^12
+ 106*m^13 + m^14 + 93405312000))/(4339163001600*m + 6165817614720*m^2 + 5056995703824*m^3 + 2706813345600*m^
4 + 1009672107080*m^5 + 272803210680*m^6 + 54631129553*m^7 + 8207628000*m^8 + 928095740*m^9 + 78558480*m^10 +
4899622*m^11 + 218400*m^12 + 6580*m^13 + 120*m^14 + m^15 + 1307674368000) + (B*a^3*x^2*(d*x)^m*(1842662908800*
m + 2161577352960*m^2 + 1447709175432*m^3 + 629552085084*m^4 + 190060010998*m^5 + 41371599841*m^6 + 6629764856
*m^7 + 788931572*m^8 + 69582084*m^9 + 4488198*m^10 + 205712*m^11 + 6344*m^12 + 118*m^13 + m^14 + 653837184000)
)/(4339163001600*m + 6165817614720*m^2 + 5056995703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210
680*m^6 + 54631129553*m^7 + 8207628000*m^8 + 928095740*m^9 + 78558480*m^10 + 4899622*m^11 + 218400*m^12 + 6580
*m^13 + 120*m^14 + m^15 + 1307674368000) + (3*a*x^5*(d*x)^m*(A*b^2 + A*a*c + C*a*b)*(815525625600*m + 10700583
97824*m^2 + 797387461200*m^3 + 381885176880*m^4 + 125557386040*m^5 + 29449164928*m^6 + 5036392925*m^7 + 634247
015*m^8 + 58769745*m^9 + 3957747*m^10 + 188375*m^11 + 6005*m^12 + 115*m^13 + m^14 + 261534873600))/(4339163001
600*m + 6165817614720*m^2 + 5056995703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 546
31129553*m^7 + 8207628000*m^8 + 928095740*m^9 + 78558480*m^10 + 4899622*m^11 + 218400*m^12 + 6580*m^13 + 120*m
^14 + m^15 + 1307674368000) + (3*c*x^11*(d*x)^m*(C*b^2 + A*b*c + C*a*c)*(383662137600*m + 525650497920*m^2 + 4
11940473264*m^3 + 208624806576*m^4 + 72822481864*m^5 + 18180066256*m^6 + 3313733027*m^7 + 444899543*m^8 + 4392
6927*m^9 + 3148323*m^10 + 159209*m^11 + 5381*m^12 + 109*m^13 + m^14 + 118879488000))/(4339163001600*m + 616581
7614720*m^2 + 5056995703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 54631129553*m^7 +
8207628000*m^8 + 928095740*m^9 + 78558480*m^10 + 4899622*m^11 + 218400*m^12 + 6580*m^13 + 120*m^14 + m^15 + 1
307674368000) + (a^2*x^3*(d*x)^m*(3*A*b + C*a)*(1301090515200*m + 1621575699840*m^2 + 1145140001328*m^3 + 5205
57781424*m^4 + 163038108552*m^5 + 36588367376*m^6 + 6014254059*m^7 + 731124647*m^8 + 65657031*m^9 + 4300483*m^
10 + 199713*m^11 + 6229*m^12 + 117*m^13 + m^14 + 435891456000))/(4339163001600*m + 6165817614720*m^2 + 5056995
703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 54631129553*m^7 + 8207628000*m^8 + 928
095740*m^9 + 78558480*m^10 + 4899622*m^11 + 218400*m^12 + 6580*m^13 + 120*m^14 + m^15 + 1307674368000) + (c^2*
x^13*(d*x)^m*(A*c + 3*C*b)*(326044051200*m + 449213351040*m^2 + 354444796368*m^3 + 180951426864*m^4 + 63747744
632*m^5 + 16081189696*m^6 + 2965379989*m^7 + 403249847*m^8 + 40372761*m^9 + 2937363*m^10 + 150943*m^11 + 5189*
m^12 + 107*m^13 + m^14 + 100590336000))/(4339163001600*m + 6165817614720*m^2 + 5056995703824*m^3 + 27068133456
00*m^4 + 1009672107080*m^5 + 272803210680*m^6 + 54631129553*m^7 + 8207628000*m^8 + 928095740*m^9 + 78558480*m^
10 + 4899622*m^11 + 218400*m^12 + 6580*m^13 + 120*m^14 + m^15 + 1307674368000) + (A*a^3*x*(d*x)^m*(30314886336
00*m + 3134328981120*m^2 + 1922666722704*m^3 + 784146622896*m^4 + 225525484184*m^5 + 47277726496*m^6 + 7353403
057*m^7 + 854224943*m^8 + 73870797*m^9 + 4687683*m^10 + 211939*m^11 + 6461*m^12 + 119*m^13 + m^14 + 1307674368
000))/(4339163001600*m + 6165817614720*m^2 + 5056995703824*m^3 + 2706813345600*m^4 + 1009672107080*m^5 + 27280
3210680*m^6 + 54631129553*m^7 + 8207628000*m^8 + 928095740*m^9 + 78558480*m^10 + 4899622*m^11 + 218400*m^12 +
6580*m^13 + 120*m^14 + m^15 + 1307674368000) + (C*c^3*x^15*(d*x)^m*(283465647360*m + 392156797824*m^2 + 310989
260400*m^3 + 159721605680*m^4 + 56663366760*m^5 + 14409322928*m^6 + 2681453775*m^7 + 368411615*m^8 + 37312275*
m^9 + 2749747*m^10 + 143325*m^11 + 5005*m^12 + ...

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