3.1.38 \(\int (d x)^m (A+B x+C x^2) (a+b x^2+c x^4)^2 \, dx\) [38]

Optimal. Leaf size=260 \[ \frac {a^2 A (d x)^{1+m}}{d (1+m)}+\frac {a^2 B (d x)^{2+m}}{d^2 (2+m)}+\frac {a (2 A b+a C) (d x)^{3+m}}{d^3 (3+m)}+\frac {2 a b B (d x)^{4+m}}{d^4 (4+m)}+\frac {\left (A \left (b^2+2 a c\right )+2 a b C\right ) (d x)^{5+m}}{d^5 (5+m)}+\frac {B \left (b^2+2 a c\right ) (d x)^{6+m}}{d^6 (6+m)}+\frac {\left (2 A b c+\left (b^2+2 a c\right ) C\right ) (d x)^{7+m}}{d^7 (7+m)}+\frac {2 b B c (d x)^{8+m}}{d^8 (8+m)}+\frac {c (A c+2 b C) (d x)^{9+m}}{d^9 (9+m)}+\frac {B c^2 (d x)^{10+m}}{d^{10} (10+m)}+\frac {c^2 C (d x)^{11+m}}{d^{11} (11+m)} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 260, 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^2 A (d x)^{m+1}}{d (m+1)}+\frac {a^2 B (d x)^{m+2}}{d^2 (m+2)}+\frac {(d x)^{m+7} \left (C \left (2 a c+b^2\right )+2 A b c\right )}{d^7 (m+7)}+\frac {(d x)^{m+5} \left (A \left (2 a c+b^2\right )+2 a b C\right )}{d^5 (m+5)}+\frac {a (d x)^{m+3} (a C+2 A b)}{d^3 (m+3)}+\frac {B \left (2 a c+b^2\right ) (d x)^{m+6}}{d^6 (m+6)}+\frac {2 a b B (d x)^{m+4}}{d^4 (m+4)}+\frac {c (d x)^{m+9} (A c+2 b C)}{d^9 (m+9)}+\frac {2 b B c (d x)^{m+8}}{d^8 (m+8)}+\frac {B c^2 (d x)^{m+10}}{d^{10} (m+10)}+\frac {c^2 C (d x)^{m+11}}{d^{11} (m+11)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.90, size = 187, normalized size = 0.72 \begin {gather*} (d x)^m \left (\frac {a^2 A x}{1+m}+\frac {a^2 B x^2}{2+m}+\frac {a (2 A b+a C) x^3}{3+m}+\frac {2 a b B x^4}{4+m}+\frac {\left (A b^2+2 a A c+2 a b C\right ) x^5}{5+m}+\frac {B \left (b^2+2 a c\right ) x^6}{6+m}+\frac {\left (2 A b c+b^2 C+2 a c C\right ) x^7}{7+m}+\frac {2 b B c x^8}{8+m}+\frac {c (A c+2 b C) x^9}{9+m}+\frac {B c^2 x^{10}}{10+m}+\frac {c^2 C x^{11}}{11+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)^2,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2186\) vs. \(2(260)=520\).
time = 0.02, size = 2187, normalized size = 8.41

method	result	size
gosper	\(\text {Expression too large to display}\)	\(2187\)
risch	\(\text {Expression too large to display}\)	\(2187\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

x*(C*c^2*m^10*x^10+B*c^2*m^10*x^9+55*C*c^2*m^9*x^10+A*c^2*m^10*x^8+56*B*c^2*m^9*x^9+2*C*b*c*m^10*x^8+1320*C*c^
2*m^8*x^10+57*A*c^2*m^9*x^8+2*B*b*c*m^10*x^7+1365*B*c^2*m^8*x^9+114*C*b*c*m^9*x^8+18150*C*c^2*m^7*x^10+2*A*b*c
*m^10*x^6+1412*A*c^2*m^8*x^8+116*B*b*c*m^9*x^7+19020*B*c^2*m^7*x^9+2*C*a*c*m^10*x^6+C*b^2*m^10*x^6+2824*C*b*c*
m^8*x^8+157773*C*c^2*m^6*x^10+118*A*b*c*m^9*x^6+19962*A*c^2*m^7*x^8+2*B*a*c*m^10*x^5+B*b^2*m^10*x^5+2922*B*b*c
*m^8*x^7+167223*B*c^2*m^6*x^9+118*C*a*c*m^9*x^6+59*C*b^2*m^9*x^6+39924*C*b*c*m^7*x^8+902055*C*c^2*m^5*x^10+2*A
*a*c*m^10*x^4+A*b^2*m^10*x^4+3024*A*b*c*m^8*x^6+177765*A*c^2*m^6*x^8+120*B*a*c*m^9*x^5+60*B*b^2*m^9*x^5+41964*
B*b*c*m^7*x^7+965328*B*c^2*m^5*x^9+2*C*a*b*m^10*x^4+3024*C*a*c*m^8*x^6+1512*C*b^2*m^8*x^6+355530*C*b*c*m^6*x^8
+3416930*C*c^2*m^4*x^10+122*A*a*c*m^9*x^4+61*A*b^2*m^9*x^4+44172*A*b*c*m^7*x^6+1037673*A*c^2*m^5*x^8+2*B*a*b*m
^10*x^3+3130*B*a*c*m^8*x^5+1565*B*b^2*m^8*x^5+379134*B*b*c*m^6*x^7+3686255*B*c^2*m^4*x^9+122*C*a*b*m^9*x^4+441
72*C*a*c*m^7*x^6+22086*C*b^2*m^7*x^6+2075346*C*b*c*m^5*x^8+8409500*C*c^2*m^3*x^10+2*A*a*b*m^10*x^2+3240*A*a*c*
m^8*x^4+1620*A*b^2*m^8*x^4+405642*A*b*c*m^6*x^6+4000478*A*c^2*m^4*x^8+124*B*a*b*m^9*x^3+46560*B*a*c*m^7*x^5+23
280*B*b^2*m^7*x^5+2242044*B*b*c*m^5*x^7+9133180*B*c^2*m^3*x^9+C*a^2*m^10*x^2+3240*C*a*b*m^8*x^4+405642*C*a*c*m
^6*x^6+202821*C*b^2*m^6*x^6+8000956*C*b*c*m^4*x^8+12753576*C*c^2*m^2*x^10+126*A*a*b*m^9*x^2+49140*A*a*c*m^7*x^
4+24570*A*b^2*m^7*x^4+2435622*A*b*c*m^5*x^6+9991428*A*c^2*m^3*x^8+B*a^2*m^10*x+3354*B*a*b*m^8*x^3+435486*B*a*c
*m^6*x^5+217743*B*b^2*m^6*x^5+8742718*B*b*c*m^4*x^7+13926276*B*c^2*m^2*x^9+63*C*a^2*m^9*x^2+49140*C*a*b*m^7*x^
4+2435622*C*a*c*m^5*x^6+1217811*C*b^2*m^5*x^6+19982856*C*b*c*m^3*x^8+10628640*C*c^2*m*x^10+A*a^2*m^10+3472*A*a
*b*m^8*x^2+469146*A*a*c*m^6*x^4+234573*A*b^2*m^6*x^4+9629716*A*b*c*m^4*x^6+15335224*A*c^2*m^2*x^8+64*B*a^2*m^9
*x+51924*B*a*b*m^7*x^3+2662200*B*a*c*m^5*x^5+1331100*B*b^2*m^5*x^5+22049716*B*b*c*m^3*x^7+11655216*B*c^2*m*x^9
+1736*C*a^2*m^8*x^2+469146*C*a*b*m^6*x^4+9629716*C*a*c*m^4*x^6+4814858*C*b^2*m^4*x^6+30670448*C*b*c*m^2*x^8+36
28800*C*c^2*x^10+65*A*a^2*m^9+54924*A*a*b*m^7*x^2+2929386*A*a*c*m^5*x^4+1464693*A*b^2*m^5*x^4+24583448*A*b*c*m
^3*x^6+12900960*A*c^2*m*x^8+1797*B*a^2*m^8*x+507150*B*a*b*m^6*x^3+10705870*B*a*c*m^4*x^5+5352935*B*b^2*m^4*x^5
[phone]*B*b*c*m^2*x^7+3991680*B*c^2*x^9+27462*C*a^2*m^7*x^2+2929386*C*a*b*m^5*x^4+24583448*C*a*c*m^3*x^6+122
91724*C*b^2*m^3*x^6+25801920*C*b*c*m*x^8+1860*A*a^2*m^8+550074*A*a*b*m^6*x^2+12032140*A*a*c*m^4*x^4+6016070*A*
b^2*m^4*x^4+38432016*A*b*c*m^2*x^6+4435200*A*c^2*x^8+29076*B*a^2*m^7*x+3246516*B*a*b*m^5*x^3+27756240*B*a*c*m^
3*x^5+13878120*B*b^2*m^3*x^5+28888560*B*b*c*m*x^7+275037*C*a^2*m^6*x^2+12032140*C*a*b*m^4*x^4+38432016*C*a*c*m
^2*x^6+19216008*C*b^2*m^2*x^6+8870400*C*b*c*x^8+30810*A*a^2*m^7+3624894*A*a*b*m^5*x^2+31830760*A*a*c*m^3*x^4+1
5915380*A*b^2*m^3*x^4+32811840*A*b*c*m*x^6+299271*B*a^2*m^6*x+13693006*B*a*b*m^4*x^3+43978712*B*a*c*m^2*x^5+21
989356*B*b^2*m^2*x^5+9979200*B*b*c*x^7+1812447*C*a^2*m^5*x^2+31830760*C*a*b*m^3*x^4+32811840*C*a*c*m*x^6+16405
920*C*b^2*m*x^6+326613*A*a^2*m^6+15804388*A*a*b*m^4*x^2+51362352*A*a*c*m^2*x^4+25681176*A*b^2*m^2*x^4+11404800
*A*b*c*x^6+2039016*B*a^2*m^5*x+37219436*B*a*b*m^3*x^3+37963680*B*a*c*m*x^5+18981840*B*b^2*m*x^5+7902194*C*a^2*
m^4*x^2+51362352*C*a*b*m^2*x^4+11404800*C*a*c*x^6+5702400*C*b^2*x^6+2310945*A*a^2*m^5+44578296*A*a*b*m^3*x^2+4
5024192*A*a*c*m*x^4+22512096*A*b^2*m*x^4+9261503*B*a^2*m^4*x+61638408*B*a*b*m^2*x^3+13305600*B*a*c*x^5+6652800
*B*b^2*x^5+22289148*C*a^2*m^3*x^2+45024192*C*a*b*m*x^4+11028590*A*a^2*m^4+76781264*A*a*b*m^2*x^2+15966720*A*a*
c*x^4+7983360*A*b^2*x^4+27472724*B*a^2*m^3*x+55282320*B*a*b*m*x^3+38390632*C*a^2*m^2*x^2+15966720*C*a*b*x^4+34
967140*A*a^2*m^3+71492160*A*a*b*m*x^2+50312628*B*a^2*m^2*x+19958400*B*a*b*x^3+35746080*C*a^2*m*x^2+70290936*A*
a^2*m^2+26611200*A*a*b*x^2+50292720*B*a^2*m*x+13305600*C*a^2*x^2+80627040*A*a^2*m+19958400*B*a^2*x+39916800*A*
a^2)*(d*x)^m/(11+m)/(10+m)/(9+m)/(8+m)/(7+m)/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)

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Maxima [A]
time = 0.34, size = 344, normalized size = 1.32 \begin {gather*} \frac {C c^{2} d^{m} x^{11} x^{m}}{m + 11} + \frac {B c^{2} d^{m} x^{10} x^{m}}{m + 10} + \frac {2 \, C b c d^{m} x^{9} x^{m}}{m + 9} + \frac {A c^{2} d^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, B b c d^{m} x^{8} x^{m}}{m + 8} + \frac {C b^{2} d^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, C a c d^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, A b c d^{m} x^{7} x^{m}}{m + 7} + \frac {B b^{2} d^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, B a c d^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, C a b d^{m} x^{5} x^{m}}{m + 5} + \frac {A b^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, A a c d^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, B a b d^{m} x^{4} x^{m}}{m + 4} + \frac {C a^{2} d^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a b d^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{2} d^{m} x^{2} x^{m}}{m + 2} + \frac {\left (d x\right )^{m + 1} A a^{2}}{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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1603 vs. \(2 (260) = 520\).
time = 0.41, size = 1603, normalized size = 6.17 \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)^2,x, algorithm="fricas")

[Out]

((C*c^2*m^10 + 55*C*c^2*m^9 + 1320*C*c^2*m^8 + 18150*C*c^2*m^7 + 157773*C*c^2*m^6 + 902055*C*c^2*m^5 + 3416930
*C*c^2*m^4 + 8409500*C*c^2*m^3 + 12753576*C*c^2*m^2 + 10628640*C*c^2*m + 3628800*C*c^2)*x^11 + (B*c^2*m^10 + 5
6*B*c^2*m^9 + 1365*B*c^2*m^8 + 19020*B*c^2*m^7 + 167223*B*c^2*m^6 + 965328*B*c^2*m^5 + 3686255*B*c^2*m^4 + 913
3180*B*c^2*m^3 + 13926276*B*c^2*m^2 + 11655216*B*c^2*m + 3991680*B*c^2)*x^10 + ((2*C*b*c + A*c^2)*m^10 + 57*(2
*C*b*c + A*c^2)*m^9 + 1412*(2*C*b*c + A*c^2)*m^8 + 19962*(2*C*b*c + A*c^2)*m^7 + 177765*(2*C*b*c + A*c^2)*m^6
+ 1037673*(2*C*b*c + A*c^2)*m^5 + 4000478*(2*C*b*c + A*c^2)*m^4 + 9991428*(2*C*b*c + A*c^2)*m^3 + 8870400*C*b*
c + 4435200*A*c^2 + 15335224*(2*C*b*c + A*c^2)*m^2 + 12900960*(2*C*b*c + A*c^2)*m)*x^9 + 2*(B*b*c*m^10 + 58*B*
b*c*m^9 + 1461*B*b*c*m^8 + 20982*B*b*c*m^7 + 189567*B*b*c*m^6 + 1121022*B*b*c*m^5 + 4371359*B*b*c*m^4 + 110248
58*B*b*c*m^3 + 17059212*B*b*c*m^2 + 14444280*B*b*c*m + 4989600*B*b*c)*x^8 + ((C*b^2 + 2*(C*a + A*b)*c)*m^10 +
59*(C*b^2 + 2*(C*a + A*b)*c)*m^9 + 1512*(C*b^2 + 2*(C*a + A*b)*c)*m^8 + 22086*(C*b^2 + 2*(C*a + A*b)*c)*m^7 +
202821*(C*b^2 + 2*(C*a + A*b)*c)*m^6 + 1217811*(C*b^2 + 2*(C*a + A*b)*c)*m^5 + 4814858*(C*b^2 + 2*(C*a + A*b)*
c)*m^4 + 12291724*(C*b^2 + 2*(C*a + A*b)*c)*m^3 + 5702400*C*b^2 + 19216008*(C*b^2 + 2*(C*a + A*b)*c)*m^2 + 114
04800*(C*a + A*b)*c + 16405920*(C*b^2 + 2*(C*a + A*b)*c)*m)*x^7 + ((B*b^2 + 2*B*a*c)*m^10 + 60*(B*b^2 + 2*B*a*
c)*m^9 + 1565*(B*b^2 + 2*B*a*c)*m^8 + 23280*(B*b^2 + 2*B*a*c)*m^7 + 217743*(B*b^2 + 2*B*a*c)*m^6 + 1331100*(B*
b^2 + 2*B*a*c)*m^5 + 5352935*(B*b^2 + 2*B*a*c)*m^4 + 13878120*(B*b^2 + 2*B*a*c)*m^3 + 6652800*B*b^2 + 13305600
*B*a*c + 21989356*(B*b^2 + 2*B*a*c)*m^2 + 18981840*(B*b^2 + 2*B*a*c)*m)*x^6 + ((2*C*a*b + A*b^2 + 2*A*a*c)*m^1
0 + 61*(2*C*a*b + A*b^2 + 2*A*a*c)*m^9 + 1620*(2*C*a*b + A*b^2 + 2*A*a*c)*m^8 + 24570*(2*C*a*b + A*b^2 + 2*A*a
*c)*m^7 + 234573*(2*C*a*b + A*b^2 + 2*A*a*c)*m^6 + 1464693*(2*C*a*b + A*b^2 + 2*A*a*c)*m^5 + 6016070*(2*C*a*b
+ A*b^2 + 2*A*a*c)*m^4 + 15915380*(2*C*a*b + A*b^2 + 2*A*a*c)*m^3 + 15966720*C*a*b + 7983360*A*b^2 + 15966720*
A*a*c + 25681176*(2*C*a*b + A*b^2 + 2*A*a*c)*m^2 + 22512096*(2*C*a*b + A*b^2 + 2*A*a*c)*m)*x^5 + 2*(B*a*b*m^10
+ 62*B*a*b*m^9 + 1677*B*a*b*m^8 + 25962*B*a*b*m^7 + 253575*B*a*b*m^6 + 1623258*B*a*b*m^5 + 6846503*B*a*b*m^4
+ 18609718*B*a*b*m^3 + 30819204*B*a*b*m^2 + 27641160*B*a*b*m + 9979200*B*a*b)*x^4 + ((C*a^2 + 2*A*a*b)*m^10 +
63*(C*a^2 + 2*A*a*b)*m^9 + 1736*(C*a^2 + 2*A*a*b)*m^8 + 27462*(C*a^2 + 2*A*a*b)*m^7 + 275037*(C*a^2 + 2*A*a*b)
*m^6 + 1812447*(C*a^2 + 2*A*a*b)*m^5 + 7902194*(C*a^2 + 2*A*a*b)*m^4 + 22289148*(C*a^2 + 2*A*a*b)*m^3 + 133056
00*C*a^2 + 26611200*A*a*b + 38390632*(C*a^2 + 2*A*a*b)*m^2 + 35746080*(C*a^2 + 2*A*a*b)*m)*x^3 + (B*a^2*m^10 +
64*B*a^2*m^9 + 1797*B*a^2*m^8 + 29076*B*a^2*m^7 + 299271*B*a^2*m^6 + 2039016*B*a^2*m^5 + 9261503*B*a^2*m^4 +
27472724*B*a^2*m^3 + 50312628*B*a^2*m^2 + 50292720*B*a^2*m + 19958400*B*a^2)*x^2 + (A*a^2*m^10 + 65*A*a^2*m^9
+ 1860*A*a^2*m^8 + 30810*A*a^2*m^7 + 326613*A*a^2*m^6 + 2310945*A*a^2*m^5 + 11028590*A*a^2*m^4 + 34967140*A*a^
2*m^3 + 70290936*A*a^2*m^2 + 80627040*A*a^2*m + 39916800*A*a^2)*x)*(d*x)^m/(m^11 + 66*m^10 + 1925*m^9 + 32670*
m^8 + 357423*m^7 + 2637558*m^6 + 13339535*m^5 + 45995730*m^4 + 105258076*m^3 + 150917976*m^2 + 120543840*m + 3
9916800)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 16323 vs. \(2 (245) = 490\).
time = 1.47, size = 16323, normalized size = 62.78 \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)**2,x)

[Out]

Piecewise(((-A*a**2/(10*x**10) - A*a*b/(4*x**8) - A*a*c/(3*x**6) - A*b**2/(6*x**6) - A*b*c/(2*x**4) - A*c**2/(
2*x**2) - B*a**2/(9*x**9) - 2*B*a*b/(7*x**7) - 2*B*a*c/(5*x**5) - B*b**2/(5*x**5) - 2*B*b*c/(3*x**3) - B*c**2/
x - C*a**2/(8*x**8) - C*a*b/(3*x**6) - C*a*c/(2*x**4) - C*b**2/(4*x**4) - C*b*c/x**2 + C*c**2*log(x))/d**11, E
q(m, -11)), ((-A*a**2/(9*x**9) - 2*A*a*b/(7*x**7) - 2*A*a*c/(5*x**5) - A*b**2/(5*x**5) - 2*A*b*c/(3*x**3) - A*
c**2/x - B*a**2/(8*x**8) - B*a*b/(3*x**6) - B*a*c/(2*x**4) - B*b**2/(4*x**4) - B*b*c/x**2 + B*c**2*log(x) - C*
a**2/(7*x**7) - 2*C*a*b/(5*x**5) - 2*C*a*c/(3*x**3) - C*b**2/(3*x**3) - 2*C*b*c/x + C*c**2*x)/d**10, Eq(m, -10
)), ((-A*a**2/(8*x**8) - A*a*b/(3*x**6) - A*a*c/(2*x**4) - A*b**2/(4*x**4) - A*b*c/x**2 + A*c**2*log(x) - B*a*
*2/(7*x**7) - 2*B*a*b/(5*x**5) - 2*B*a*c/(3*x**3) - B*b**2/(3*x**3) - 2*B*b*c/x + B*c**2*x - C*a**2/(6*x**6) -
C*a*b/(2*x**4) - C*a*c/x**2 - C*b**2/(2*x**2) + 2*C*b*c*log(x) + C*c**2*x**2/2)/d**9, Eq(m, -9)), ((-A*a**2/(
7*x**7) - 2*A*a*b/(5*x**5) - 2*A*a*c/(3*x**3) - A*b**2/(3*x**3) - 2*A*b*c/x + A*c**2*x - B*a**2/(6*x**6) - B*a
*b/(2*x**4) - B*a*c/x**2 - B*b**2/(2*x**2) + 2*B*b*c*log(x) + B*c**2*x**2/2 - C*a**2/(5*x**5) - 2*C*a*b/(3*x**
3) - 2*C*a*c/x - C*b**2/x + 2*C*b*c*x + C*c**2*x**3/3)/d**8, Eq(m, -8)), ((-A*a**2/(6*x**6) - A*a*b/(2*x**4) -
A*a*c/x**2 - A*b**2/(2*x**2) + 2*A*b*c*log(x) + A*c**2*x**2/2 - B*a**2/(5*x**5) - 2*B*a*b/(3*x**3) - 2*B*a*c/
x - B*b**2/x + 2*B*b*c*x + B*c**2*x**3/3 - C*a**2/(4*x**4) - C*a*b/x**2 + 2*C*a*c*log(x) + C*b**2*log(x) + C*b
*c*x**2 + C*c**2*x**4/4)/d**7, Eq(m, -7)), ((-A*a**2/(5*x**5) - 2*A*a*b/(3*x**3) - 2*A*a*c/x - A*b**2/x + 2*A*
b*c*x + A*c**2*x**3/3 - B*a**2/(4*x**4) - B*a*b/x**2 + 2*B*a*c*log(x) + B*b**2*log(x) + B*b*c*x**2 + B*c**2*x*
*4/4 - C*a**2/(3*x**3) - 2*C*a*b/x + 2*C*a*c*x + C*b**2*x + 2*C*b*c*x**3/3 + C*c**2*x**5/5)/d**6, Eq(m, -6)),
((-A*a**2/(4*x**4) - A*a*b/x**2 + 2*A*a*c*log(x) + A*b**2*log(x) + A*b*c*x**2 + A*c**2*x**4/4 - B*a**2/(3*x**3
) - 2*B*a*b/x + 2*B*a*c*x + B*b**2*x + 2*B*b*c*x**3/3 + B*c**2*x**5/5 - C*a**2/(2*x**2) + 2*C*a*b*log(x) + C*a
*c*x**2 + C*b**2*x**2/2 + C*b*c*x**4/2 + C*c**2*x**6/6)/d**5, Eq(m, -5)), ((-A*a**2/(3*x**3) - 2*A*a*b/x + 2*A
*a*c*x + A*b**2*x + 2*A*b*c*x**3/3 + A*c**2*x**5/5 - B*a**2/(2*x**2) + 2*B*a*b*log(x) + B*a*c*x**2 + B*b**2*x*
*2/2 + B*b*c*x**4/2 + B*c**2*x**6/6 - C*a**2/x + 2*C*a*b*x + 2*C*a*c*x**3/3 + C*b**2*x**3/3 + 2*C*b*c*x**5/5 +
C*c**2*x**7/7)/d**4, Eq(m, -4)), ((-A*a**2/(2*x**2) + 2*A*a*b*log(x) + A*a*c*x**2 + A*b**2*x**2/2 + A*b*c*x**
4/2 + A*c**2*x**6/6 - B*a**2/x + 2*B*a*b*x + 2*B*a*c*x**3/3 + B*b**2*x**3/3 + 2*B*b*c*x**5/5 + B*c**2*x**7/7 +
C*a**2*log(x) + C*a*b*x**2 + C*a*c*x**4/2 + C*b**2*x**4/4 + C*b*c*x**6/3 + C*c**2*x**8/8)/d**3, Eq(m, -3)), (
(-A*a**2/x + 2*A*a*b*x + 2*A*a*c*x**3/3 + A*b**2*x**3/3 + 2*A*b*c*x**5/5 + A*c**2*x**7/7 + B*a**2*log(x) + B*a
*b*x**2 + B*a*c*x**4/2 + B*b**2*x**4/4 + B*b*c*x**6/3 + B*c**2*x**8/8 + C*a**2*x + 2*C*a*b*x**3/3 + 2*C*a*c*x*
*5/5 + C*b**2*x**5/5 + 2*C*b*c*x**7/7 + C*c**2*x**9/9)/d**2, Eq(m, -2)), ((A*a**2*log(x) + A*a*b*x**2 + A*a*c*
x**4/2 + A*b**2*x**4/4 + A*b*c*x**6/3 + A*c**2*x**8/8 + B*a**2*x + 2*B*a*b*x**3/3 + 2*B*a*c*x**5/5 + B*b**2*x*
*5/5 + 2*B*b*c*x**7/7 + B*c**2*x**9/9 + C*a**2*x**2/2 + C*a*b*x**4/2 + C*a*c*x**6/3 + C*b**2*x**6/6 + C*b*c*x*
*8/4 + C*c**2*x**10/10)/d, Eq(m, -1)), (A*a**2*m**10*x*(d*x)**m/(m**11 + 66*m**10 + 1925*m**9 + 32670*m**8 + 3
57423*m**7 + 2637558*m**6 + 13339535*m**5 + 45995730*m**4 + 105258076*m**3 + 150917976*m**2 + 120543840*m + 39
916800) + 65*A*a**2*m**9*x*(d*x)**m/(m**11 + 66*m**10 + 1925*m**9 + 32670*m**8 + 357423*m**7 + 2637558*m**6 +
13339535*m**5 + 45995730*m**4 + 105258076*m**3 + 150917976*m**2 + 120543840*m + 39916800) + 1860*A*a**2*m**8*x
*(d*x)**m/(m**11 + 66*m**10 + 1925*m**9 + 32670*m**8 + 357423*m**7 + 2637558*m**6 + 13339535*m**5 + 45995730*m
**4 + 105258076*m**3 + 150917976*m**2 + 120543840*m + 39916800) + 30810*A*a**2*m**7*x*(d*x)**m/(m**11 + 66*m**
10 + 1925*m**9 + 32670*m**8 + 357423*m**7 + 2637558*m**6 + 13339535*m**5 + 45995730*m**4 + 105258076*m**3 + 15
0917976*m**2 + 120543840*m + 39916800) + 326613*A*a**2*m**6*x*(d*x)**m/(m**11 + 66*m**10 + 1925*m**9 + 32670*m
**8 + 357423*m**7 + 2637558*m**6 + 13339535*m**5 + 45995730*m**4 + 105258076*m**3 + 150917976*m**2 + 120543840
*m + 39916800) + 2310945*A*a**2*m**5*x*(d*x)**m/(m**11 + 66*m**10 + 1925*m**9 + 32670*m**8 + 357423*m**7 + 263
7558*m**6 + 13339535*m**5 + 45995730*m**4 + 105258076*m**3 + 150917976*m**2 + 120543840*m + 39916800) + 110285
90*A*a**2*m**4*x*(d*x)**m/(m**11 + 66*m**10 + 1925*m**9 + 32670*m**8 + 357423*m**7 + 2637558*m**6 + 13339535*m
**5 + 45995730*m**4 + 105258076*m**3 + 150917976*m**2 + 120543840*m + 39916800) + 34967140*A*a**2*m**3*x*(d*x)
**m/(m**11 + 66*m**10 + 1925*m**9 + 32670*m**8 + 357423*m**7 + 2637558*m**6 + 13339535*m**5 + 45995730*m**4 +
105258076*m**3 + 150917976*m**2 + 120543840*m + 39916800) + 70290936*A*a**2*m**2*x*(d*x)**m/(m**11 + 66*m**10
+ 1925*m**9 + 32670*m**8 + 357423*m**7 + 263755...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3203 vs. \(2 (260) = 520\).
time = 5.22, size = 3203, normalized size = 12.32 \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)^2,x, algorithm="giac")

[Out]

((d*x)^m*C*c^2*m^10*x^11 + (d*x)^m*B*c^2*m^10*x^10 + 55*(d*x)^m*C*c^2*m^9*x^11 + 2*(d*x)^m*C*b*c*m^10*x^9 + (d
*x)^m*A*c^2*m^10*x^9 + 56*(d*x)^m*B*c^2*m^9*x^10 + 1320*(d*x)^m*C*c^2*m^8*x^11 + 2*(d*x)^m*B*b*c*m^10*x^8 + 11
4*(d*x)^m*C*b*c*m^9*x^9 + 57*(d*x)^m*A*c^2*m^9*x^9 + 1365*(d*x)^m*B*c^2*m^8*x^10 + 18150*(d*x)^m*C*c^2*m^7*x^1
1 + (d*x)^m*C*b^2*m^10*x^7 + 2*(d*x)^m*C*a*c*m^10*x^7 + 2*(d*x)^m*A*b*c*m^10*x^7 + 116*(d*x)^m*B*b*c*m^9*x^8 +
2824*(d*x)^m*C*b*c*m^8*x^9 + 1412*(d*x)^m*A*c^2*m^8*x^9 + 19020*(d*x)^m*B*c^2*m^7*x^10 + 157773*(d*x)^m*C*c^2
*m^6*x^11 + (d*x)^m*B*b^2*m^10*x^6 + 2*(d*x)^m*B*a*c*m^10*x^6 + 59*(d*x)^m*C*b^2*m^9*x^7 + 118*(d*x)^m*C*a*c*m
^9*x^7 + 118*(d*x)^m*A*b*c*m^9*x^7 + 2922*(d*x)^m*B*b*c*m^8*x^8 + 39924*(d*x)^m*C*b*c*m^7*x^9 + 19962*(d*x)^m*
A*c^2*m^7*x^9 + 167223*(d*x)^m*B*c^2*m^6*x^10 + 902055*(d*x)^m*C*c^2*m^5*x^11 + 2*(d*x)^m*C*a*b*m^10*x^5 + (d*
x)^m*A*b^2*m^10*x^5 + 2*(d*x)^m*A*a*c*m^10*x^5 + 60*(d*x)^m*B*b^2*m^9*x^6 + 120*(d*x)^m*B*a*c*m^9*x^6 + 1512*(
d*x)^m*C*b^2*m^8*x^7 + 3024*(d*x)^m*C*a*c*m^8*x^7 + 3024*(d*x)^m*A*b*c*m^8*x^7 + 41964*(d*x)^m*B*b*c*m^7*x^8 +
355530*(d*x)^m*C*b*c*m^6*x^9 + 177765*(d*x)^m*A*c^2*m^6*x^9 + 965328*(d*x)^m*B*c^2*m^5*x^10 + 3416930*(d*x)^m
*C*c^2*m^4*x^11 + 2*(d*x)^m*B*a*b*m^10*x^4 + 122*(d*x)^m*C*a*b*m^9*x^5 + 61*(d*x)^m*A*b^2*m^9*x^5 + 122*(d*x)^
m*A*a*c*m^9*x^5 + 1565*(d*x)^m*B*b^2*m^8*x^6 + 3130*(d*x)^m*B*a*c*m^8*x^6 + 22086*(d*x)^m*C*b^2*m^7*x^7 + 4417
2*(d*x)^m*C*a*c*m^7*x^7 + 44172*(d*x)^m*A*b*c*m^7*x^7 + 379134*(d*x)^m*B*b*c*m^6*x^8 + 2075346*(d*x)^m*C*b*c*m
^5*x^9 + 1037673*(d*x)^m*A*c^2*m^5*x^9 + 3686255*(d*x)^m*B*c^2*m^4*x^10 + 8409500*(d*x)^m*C*c^2*m^3*x^11 + (d*
x)^m*C*a^2*m^10*x^3 + 2*(d*x)^m*A*a*b*m^10*x^3 + 124*(d*x)^m*B*a*b*m^9*x^4 + 3240*(d*x)^m*C*a*b*m^8*x^5 + 1620
*(d*x)^m*A*b^2*m^8*x^5 + 3240*(d*x)^m*A*a*c*m^8*x^5 + 23280*(d*x)^m*B*b^2*m^7*x^6 + 46560*(d*x)^m*B*a*c*m^7*x^
6 + 202821*(d*x)^m*C*b^2*m^6*x^7 + 405642*(d*x)^m*C*a*c*m^6*x^7 + 405642*(d*x)^m*A*b*c*m^6*x^7 + 2242044*(d*x)
^m*B*b*c*m^5*x^8 + 8000956*(d*x)^m*C*b*c*m^4*x^9 + 4000478*(d*x)^m*A*c^2*m^4*x^9 + 9133180*(d*x)^m*B*c^2*m^3*x
^10 + 12753576*(d*x)^m*C*c^2*m^2*x^11 + (d*x)^m*B*a^2*m^10*x^2 + 63*(d*x)^m*C*a^2*m^9*x^3 + 126*(d*x)^m*A*a*b*
m^9*x^3 + 3354*(d*x)^m*B*a*b*m^8*x^4 + 49140*(d*x)^m*C*a*b*m^7*x^5 + 24570*(d*x)^m*A*b^2*m^7*x^5 + 49140*(d*x)
^m*A*a*c*m^7*x^5 + 217743*(d*x)^m*B*b^2*m^6*x^6 + 435486*(d*x)^m*B*a*c*m^6*x^6 + 1217811*(d*x)^m*C*b^2*m^5*x^7
+ 2435622*(d*x)^m*C*a*c*m^5*x^7 + 2435622*(d*x)^m*A*b*c*m^5*x^7 + 8742718*(d*x)^m*B*b*c*m^4*x^8 + 19982856*(d
*x)^m*C*b*c*m^3*x^9 + 9991428*(d*x)^m*A*c^2*m^3*x^9 + 13926276*(d*x)^m*B*c^2*m^2*x^10 + 10628640*(d*x)^m*C*c^2
*m*x^11 + (d*x)^m*A*a^2*m^10*x + 64*(d*x)^m*B*a^2*m^9*x^2 + 1736*(d*x)^m*C*a^2*m^8*x^3 + 3472*(d*x)^m*A*a*b*m^
8*x^3 + 51924*(d*x)^m*B*a*b*m^7*x^4 + 469146*(d*x)^m*C*a*b*m^6*x^5 + 234573*(d*x)^m*A*b^2*m^6*x^5 + 469146*(d*
x)^m*A*a*c*m^6*x^5 + 1331100*(d*x)^m*B*b^2*m^5*x^6 + 2662200*(d*x)^m*B*a*c*m^5*x^6 + 4814858*(d*x)^m*C*b^2*m^4
*x^7 + 9629716*(d*x)^m*C*a*c*m^4*x^7 + 9629716*(d*x)^m*A*b*c*m^4*x^7 + 22049716*(d*x)^m*B*b*c*m^3*x^8 + 306704
48*(d*x)^m*C*b*c*m^2*x^9 + 15335224*(d*x)^m*A*c^2*m^2*x^9 + 11655216*(d*x)^m*B*c^2*m*x^10 + 3628800*(d*x)^m*C*
c^2*x^11 + 65*(d*x)^m*A*a^2*m^9*x + 1797*(d*x)^m*B*a^2*m^8*x^2 + 27462*(d*x)^m*C*a^2*m^7*x^3 + 54924*(d*x)^m*A
*a*b*m^7*x^3 + 507150*(d*x)^m*B*a*b*m^6*x^4 + 2929386*(d*x)^m*C*a*b*m^5*x^5 + 1464693*(d*x)^m*A*b^2*m^5*x^5 +
2929386*(d*x)^m*A*a*c*m^5*x^5 + 5352935*(d*x)^m*B*b^2*m^4*x^6 + 10705870*(d*x)^m*B*a*c*m^4*x^6 + 12291724*(d*x
)^m*C*b^2*m^3*x^7 + 24583448*(d*x)^m*C*a*c*m^3*x^7 + 24583448*(d*x)^m*A*b*c*m^3*x^7 + 34118424*(d*x)^m*B*b*c*m
^2*x^8 + 25801920*(d*x)^m*C*b*c*m*x^9 + 12900960*(d*x)^m*A*c^2*m*x^9 + 3991680*(d*x)^m*B*c^2*x^10 + 1860*(d*x)
^m*A*a^2*m^8*x + 29076*(d*x)^m*B*a^2*m^7*x^2 + 275037*(d*x)^m*C*a^2*m^6*x^3 + 550074*(d*x)^m*A*a*b*m^6*x^3 + 3
246516*(d*x)^m*B*a*b*m^5*x^4 + 12032140*(d*x)^m*C*a*b*m^4*x^5 + 6016070*(d*x)^m*A*b^2*m^4*x^5 + 12032140*(d*x)
^m*A*a*c*m^4*x^5 + 13878120*(d*x)^m*B*b^2*m^3*x^6 + 27756240*(d*x)^m*B*a*c*m^3*x^6 + 19216008*(d*x)^m*C*b^2*m^
2*x^7 + 38432016*(d*x)^m*C*a*c*m^2*x^7 + 38432016*(d*x)^m*A*b*c*m^2*x^7 + 28888560*(d*x)^m*B*b*c*m*x^8 + 88704
00*(d*x)^m*C*b*c*x^9 + 4435200*(d*x)^m*A*c^2*x^9 + 30810*(d*x)^m*A*a^2*m^7*x + 299271*(d*x)^m*B*a^2*m^6*x^2 +
1812447*(d*x)^m*C*a^2*m^5*x^3 + 3624894*(d*x)^m*A*a*b*m^5*x^3 + 13693006*(d*x)^m*B*a*b*m^4*x^4 + 31830760*(d*x
)^m*C*a*b*m^3*x^5 + 15915380*(d*x)^m*A*b^2*m^3*x^5 + 31830760*(d*x)^m*A*a*c*m^3*x^5 + 21989356*(d*x)^m*B*b^2*m
^2*x^6 + 43978712*(d*x)^m*B*a*c*m^2*x^6 + 16405920*(d*x)^m*C*b^2*m*x^7 + 32811840*(d*x)^m*C*a*c*m*x^7 + 328118
40*(d*x)^m*A*b*c*m*x^7 + 9979200*(d*x)^m*B*b*c*x^8 + 326613*(d*x)^m*A*a^2*m^6*x + 2039016*(d*x)^m*B*a^2*m^5*x^
2 + 7902194*(d*x)^m*C*a^2*m^4*x^3 + 15804388*(d*x)^m*A*a*b*m^4*x^3 + 37219436*(d*x)^m*B*a*b*m^3*x^4 + 51362352
*(d*x)^m*C*a*b*m^2*x^5 + 25681176*(d*x)^m*A*b^2*m^2*x^5 + 51362352*(d*x)^m*A*a*c*m^2*x^5 + 18981840*(d*x)^m*B*
b^2*m*x^6 + 37963680*(d*x)^m*B*a*c*m*x^6 + 5702...

________________________________________________________________________________________

Mupad [B]
time = 1.81, size = 1314, normalized size = 5.05 \begin {gather*} \frac {x^5\,{\left (d\,x\right )}^m\,\left (A\,b^2+2\,C\,a\,b+2\,A\,a\,c\right )\,\left (m^{10}+61\,m^9+1620\,m^8+24570\,m^7+234573\,m^6+1464693\,m^5+6016070\,m^4+15915380\,m^3+25681176\,m^2+22512096\,m+7983360\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {x^7\,{\left (d\,x\right )}^m\,\left (C\,b^2+2\,A\,c\,b+2\,C\,a\,c\right )\,\left (m^{10}+59\,m^9+1512\,m^8+22086\,m^7+202821\,m^6+1217811\,m^5+4814858\,m^4+12291724\,m^3+19216008\,m^2+16405920\,m+5702400\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {B\,x^6\,{\left (d\,x\right )}^m\,\left (b^2+2\,a\,c\right )\,\left (m^{10}+60\,m^9+1565\,m^8+23280\,m^7+217743\,m^6+1331100\,m^5+5352935\,m^4+13878120\,m^3+21989356\,m^2+18981840\,m+6652800\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {A\,a^2\,x\,{\left (d\,x\right )}^m\,\left (m^{10}+65\,m^9+1860\,m^8+30810\,m^7+326613\,m^6+2310945\,m^5+11028590\,m^4+34967140\,m^3+70290936\,m^2+80627040\,m+39916800\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {c\,x^9\,{\left (d\,x\right )}^m\,\left (A\,c+2\,C\,b\right )\,\left (m^{10}+57\,m^9+1412\,m^8+19962\,m^7+177765\,m^6+1037673\,m^5+4000478\,m^4+9991428\,m^3+15335224\,m^2+12900960\,m+4435200\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {a\,x^3\,{\left (d\,x\right )}^m\,\left (2\,A\,b+C\,a\right )\,\left (m^{10}+63\,m^9+1736\,m^8+27462\,m^7+275037\,m^6+1812447\,m^5+7902194\,m^4+22289148\,m^3+38390632\,m^2+35746080\,m+13305600\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {B\,c^2\,x^{10}\,{\left (d\,x\right )}^m\,\left (m^{10}+56\,m^9+1365\,m^8+19020\,m^7+167223\,m^6+965328\,m^5+3686255\,m^4+9133180\,m^3+13926276\,m^2+11655216\,m+3991680\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {C\,c^2\,x^{11}\,{\left (d\,x\right )}^m\,\left (m^{10}+55\,m^9+1320\,m^8+18150\,m^7+157773\,m^6+902055\,m^5+3416930\,m^4+8409500\,m^3+12753576\,m^2+10628640\,m+3628800\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {B\,a^2\,x^2\,{\left (d\,x\right )}^m\,\left (m^{10}+64\,m^9+1797\,m^8+29076\,m^7+299271\,m^6+2039016\,m^5+9261503\,m^4+27472724\,m^3+50312628\,m^2+50292720\,m+19958400\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {2\,B\,b\,c\,x^8\,{\left (d\,x\right )}^m\,\left (m^{10}+58\,m^9+1461\,m^8+20982\,m^7+189567\,m^6+1121022\,m^5+4371359\,m^4+11024858\,m^3+17059212\,m^2+14444280\,m+4989600\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800}+\frac {2\,B\,a\,b\,x^4\,{\left (d\,x\right )}^m\,\left (m^{10}+62\,m^9+1677\,m^8+25962\,m^7+253575\,m^6+1623258\,m^5+6846503\,m^4+18609718\,m^3+30819204\,m^2+27641160\,m+9979200\right )}{m^{11}+66\,m^{10}+1925\,m^9+32670\,m^8+357423\,m^7+2637558\,m^6+13339535\,m^5+45995730\,m^4+105258076\,m^3+150917976\,m^2+120543840\,m+39916800} \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)^2,x)

[Out]

(x^5*(d*x)^m*(A*b^2 + 2*A*a*c + 2*C*a*b)*(22512096*m + 25681176*m^2 + 15915380*m^3 + 6016070*m^4 + 1464693*m^5
+ 234573*m^6 + 24570*m^7 + 1620*m^8 + 61*m^9 + m^10 + 7983360))/(120543840*m + 150917976*m^2 + 105258076*m^3
+ 45995730*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) +
(x^7*(d*x)^m*(C*b^2 + 2*A*b*c + 2*C*a*c)*(16405920*m + 19216008*m^2 + 12291724*m^3 + 4814858*m^4 + 1217811*m^
5 + 202821*m^6 + 22086*m^7 + 1512*m^8 + 59*m^9 + m^10 + 5702400))/(120543840*m + 150917976*m^2 + 105258076*m^3
+ 45995730*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800)
+ (B*x^6*(d*x)^m*(2*a*c + b^2)*(18981840*m + 21989356*m^2 + 13878120*m^3 + 5352935*m^4 + 1331100*m^5 + 217743*
m^6 + 23280*m^7 + 1565*m^8 + 60*m^9 + m^10 + 6652800))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730
*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (A*a^2*x*
(d*x)^m*(80627040*m + 70290936*m^2 + 34967140*m^3 + 11028590*m^4 + 2310945*m^5 + 326613*m^6 + 30810*m^7 + 1860
*m^8 + 65*m^9 + m^10 + 39916800))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 13339535*m^5 +
2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (c*x^9*(d*x)^m*(A*c + 2*C*b)*(
12900960*m + 15335224*m^2 + 9991428*m^3 + 4000478*m^4 + 1037673*m^5 + 177765*m^6 + 19962*m^7 + 1412*m^8 + 57*m
^9 + m^10 + 4435200))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 13339535*m^5 + 2637558*m^6
+ 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (a*x^3*(d*x)^m*(2*A*b + C*a)*(35746080*m +
38390632*m^2 + 22289148*m^3 + 7902194*m^4 + 1812447*m^5 + 275037*m^6 + 27462*m^7 + 1736*m^8 + 63*m^9 + m^10 +
13305600))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*
m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (B*c^2*x^10*(d*x)^m*(11655216*m + 13926276*m^2 + 913
3180*m^3 + 3686255*m^4 + 965328*m^5 + 167223*m^6 + 19020*m^7 + 1365*m^8 + 56*m^9 + m^10 + 3991680))/(120543840
*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925
*m^9 + 66*m^10 + m^11 + 39916800) + (C*c^2*x^11*(d*x)^m*(10628640*m + 12753576*m^2 + 8409500*m^3 + 3416930*m^4
+ 902055*m^5 + 157773*m^6 + 18150*m^7 + 1320*m^8 + 55*m^9 + m^10 + 3628800))/(120543840*m + 150917976*m^2 + 1
05258076*m^3 + 45995730*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11
+ 39916800) + (B*a^2*x^2*(d*x)^m*(50292720*m + 50312628*m^2 + 27472724*m^3 + 9261503*m^4 + 2039016*m^5 + 29927
1*m^6 + 29076*m^7 + 1797*m^8 + 64*m^9 + m^10 + 19958400))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995
730*m^4 + 13339535*m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (2*B*b
*c*x^8*(d*x)^m*(14444280*m + 17059212*m^2 + 11024858*m^3 + 4371359*m^4 + 1121022*m^5 + 189567*m^6 + 20982*m^7
+ 1461*m^8 + 58*m^9 + m^10 + 4989600))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 13339535*
m^5 + 2637558*m^6 + 357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800) + (2*B*a*b*x^4*(d*x)^m*(276
41160*m + 30819204*m^2 + 18609718*m^3 + 6846503*m^4 + 1623258*m^5 + 253575*m^6 + 25962*m^7 + 1677*m^8 + 62*m^9
+ m^10 + 9979200))/(120543840*m + 150917976*m^2 + 105258076*m^3 + 45995730*m^4 + 13339535*m^5 + 2637558*m^6 +
357423*m^7 + 32670*m^8 + 1925*m^9 + 66*m^10 + m^11 + 39916800)

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