### 3.38 $$\int (d x)^m (A+B x+C x^2) (a+b x^2+c x^4)^2 \, dx$$

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

[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))

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Rubi [A]  time = 0.222501, antiderivative size = 260, normalized size of antiderivative = 1., 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 = {1628} $\frac{a^2 A (d x)^{m+1}}{d (m+1)}+\frac{a^2 B (d x)^{m+2}}{d^2 (m+2)}+\frac{(d x)^{m+5} \left (A \left (2 a c+b^2\right )+2 a b C\right )}{d^5 (m+5)}+\frac{(d x)^{m+7} \left (C \left (2 a c+b^2\right )+2 A b c\right )}{d^7 (m+7)}+\frac{a (d x)^{m+3} (a C+2 A b)}{d^3 (m+3)}+\frac{B \left (2 a c+b^2\right ) (d x)^{m+6}}{d^6 (m+6)}+\frac{2 a b B (d x)^{m+4}}{d^4 (m+4)}+\frac{c (d x)^{m+9} (A c+2 b C)}{d^9 (m+9)}+\frac{2 b B c (d x)^{m+8}}{d^8 (m+8)}+\frac{B c^2 (d x)^{m+10}}{d^{10} (m+10)}+\frac{c^2 C (d x)^{m+11}}{d^{11} (m+11)}$

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 1628

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*}

Mathematica [A]  time = 0.38257, size = 185, normalized size = 0.71 $x (d x)^m \left (\frac{a^2 A}{m+1}+\frac{a^2 B x}{m+2}+\frac{x^6 \left (C \left (2 a c+b^2\right )+2 A b c\right )}{m+7}+\frac{x^4 \left (A \left (2 a c+b^2\right )+2 a b C\right )}{m+5}+\frac{a x^2 (a C+2 A b)}{m+3}+\frac{B x^5 \left (2 a c+b^2\right )}{m+6}+\frac{2 a b B x^3}{m+4}+\frac{c x^8 (A c+2 b C)}{m+9}+\frac{2 b B c x^7}{m+8}+\frac{B c^2 x^9}{m+10}+\frac{c^2 C x^{10}}{m+11}\right )$

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]

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

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Maple [B]  time = 0.01, size = 2187, normalized size = 8.4 \begin{align*} \text{result too large to display} \end{align*}

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)

[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
+34118424*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)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.82465, size = 4469, normalized size = 17.19 \begin{align*} \text{result too large to display} \end{align*}

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.19524, size = 4324, normalized size = 16.63 \begin{align*} \text{result too large to display} \end{align*}

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 + 5702400*(d*x)^m*C*b^2*x^7 + 11404800*(d*x)^m*C*a*c*x^7 + 11404800*(
d*x)^m*A*b*c*x^7 + 2310945*(d*x)^m*A*a^2*m^5*x + 9261503*(d*x)^m*B*a^2*m^4*x^2 + 22289148*(d*x)^m*C*a^2*m^3*x^
3 + 44578296*(d*x)^m*A*a*b*m^3*x^3 + 61638408*(d*x)^m*B*a*b*m^2*x^4 + 45024192*(d*x)^m*C*a*b*m*x^5 + 22512096*
(d*x)^m*A*b^2*m*x^5 + 45024192*(d*x)^m*A*a*c*m*x^5 + 6652800*(d*x)^m*B*b^2*x^6 + 13305600*(d*x)^m*B*a*c*x^6 +
11028590*(d*x)^m*A*a^2*m^4*x + 27472724*(d*x)^m*B*a^2*m^3*x^2 + 38390632*(d*x)^m*C*a^2*m^2*x^3 + 76781264*(d*x
)^m*A*a*b*m^2*x^3 + 55282320*(d*x)^m*B*a*b*m*x^4 + 15966720*(d*x)^m*C*a*b*x^5 + 7983360*(d*x)^m*A*b^2*x^5 + 15
966720*(d*x)^m*A*a*c*x^5 + 34967140*(d*x)^m*A*a^2*m^3*x + 50312628*(d*x)^m*B*a^2*m^2*x^2 + 35746080*(d*x)^m*C*
a^2*m*x^3 + 71492160*(d*x)^m*A*a*b*m*x^3 + 19958400*(d*x)^m*B*a*b*x^4 + 70290936*(d*x)^m*A*a^2*m^2*x + 5029272
0*(d*x)^m*B*a^2*m*x^2 + 13305600*(d*x)^m*C*a^2*x^3 + 26611200*(d*x)^m*A*a*b*x^3 + 80627040*(d*x)^m*A*a^2*m*x +
19958400*(d*x)^m*B*a^2*x^2 + 39916800*(d*x)^m*A*a^2*x)/(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)