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

Optimal. Leaf size=137 \[ \frac{(d x)^{m+3} (a C+A b)}{d^3 (m+3)}+\frac{a A (d x)^{m+1}}{d (m+1)}+\frac{a B (d x)^{m+2}}{d^2 (m+2)}+\frac{(d x)^{m+5} (A c+b C)}{d^5 (m+5)}+\frac{b B (d x)^{m+4}}{d^4 (m+4)}+\frac{B c (d x)^{m+6}}{d^6 (m+6)}+\frac{c C (d x)^{m+7}}{d^7 (m+7)} \]

[Out]

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

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Rubi [A]  time = 0.0880793, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1628} \[ \frac{(d x)^{m+3} (a C+A b)}{d^3 (m+3)}+\frac{a A (d x)^{m+1}}{d (m+1)}+\frac{a B (d x)^{m+2}}{d^2 (m+2)}+\frac{(d x)^{m+5} (A c+b C)}{d^5 (m+5)}+\frac{b B (d x)^{m+4}}{d^4 (m+4)}+\frac{B c (d x)^{m+6}}{d^6 (m+6)}+\frac{c C (d x)^{m+7}}{d^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.121717, size = 90, normalized size = 0.66 \[ x (d x)^m \left (\frac{x^2 (a C+A b)}{m+3}+\frac{a A}{m+1}+\frac{a B x}{m+2}+\frac{x^4 (A c+b C)}{m+5}+\frac{b B x^3}{m+4}+\frac{B c x^5}{m+6}+\frac{c C x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.004, size = 585, normalized size = 4.3 \begin{align*}{\frac{ \left ( Cc{m}^{6}{x}^{6}+Bc{m}^{6}{x}^{5}+21\,Cc{m}^{5}{x}^{6}+Ac{m}^{6}{x}^{4}+22\,Bc{m}^{5}{x}^{5}+Cb{m}^{6}{x}^{4}+175\,Cc{m}^{4}{x}^{6}+23\,Ac{m}^{5}{x}^{4}+Bb{m}^{6}{x}^{3}+190\,Bc{m}^{4}{x}^{5}+23\,Cb{m}^{5}{x}^{4}+735\,Cc{m}^{3}{x}^{6}+Ab{m}^{6}{x}^{2}+207\,Ac{m}^{4}{x}^{4}+24\,Bb{m}^{5}{x}^{3}+820\,Bc{m}^{3}{x}^{5}+Ca{m}^{6}{x}^{2}+207\,Cb{m}^{4}{x}^{4}+1624\,Cc{m}^{2}{x}^{6}+25\,Ab{m}^{5}{x}^{2}+925\,Ac{m}^{3}{x}^{4}+Ba{m}^{6}x+226\,Bb{m}^{4}{x}^{3}+1849\,Bc{m}^{2}{x}^{5}+25\,Ca{m}^{5}{x}^{2}+925\,Cb{m}^{3}{x}^{4}+1764\,Ccm{x}^{6}+Aa{m}^{6}+247\,Ab{m}^{4}{x}^{2}+2144\,Ac{m}^{2}{x}^{4}+26\,Ba{m}^{5}x+1056\,Bb{m}^{3}{x}^{3}+2038\,Bcm{x}^{5}+247\,Ca{m}^{4}{x}^{2}+2144\,Cb{m}^{2}{x}^{4}+720\,cC{x}^{6}+27\,Aa{m}^{5}+1219\,Ab{m}^{3}{x}^{2}+2412\,Acm{x}^{4}+270\,Ba{m}^{4}x+2545\,Bb{m}^{2}{x}^{3}+840\,Bc{x}^{5}+1219\,Ca{m}^{3}{x}^{2}+2412\,Cbm{x}^{4}+295\,Aa{m}^{4}+3112\,Ab{m}^{2}{x}^{2}+1008\,A{x}^{4}c+1420\,Ba{m}^{3}x+2952\,Bbm{x}^{3}+3112\,Ca{m}^{2}{x}^{2}+1008\,C{x}^{4}b+1665\,Aa{m}^{3}+3796\,Abm{x}^{2}+3929\,Ba{m}^{2}x+1260\,bB{x}^{3}+3796\,Cam{x}^{2}+5104\,Aa{m}^{2}+1680\,A{x}^{2}b+5274\,Bamx+1680\,C{x}^{2}a+8028\,Aam+2520\,aBx+5040\,Aa \right ) x \left ( dx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \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),x)

[Out]

x*(C*c*m^6*x^6+B*c*m^6*x^5+21*C*c*m^5*x^6+A*c*m^6*x^4+22*B*c*m^5*x^5+C*b*m^6*x^4+175*C*c*m^4*x^6+23*A*c*m^5*x^
4+B*b*m^6*x^3+190*B*c*m^4*x^5+23*C*b*m^5*x^4+735*C*c*m^3*x^6+A*b*m^6*x^2+207*A*c*m^4*x^4+24*B*b*m^5*x^3+820*B*
c*m^3*x^5+C*a*m^6*x^2+207*C*b*m^4*x^4+1624*C*c*m^2*x^6+25*A*b*m^5*x^2+925*A*c*m^3*x^4+B*a*m^6*x+226*B*b*m^4*x^
3+1849*B*c*m^2*x^5+25*C*a*m^5*x^2+925*C*b*m^3*x^4+1764*C*c*m*x^6+A*a*m^6+247*A*b*m^4*x^2+2144*A*c*m^2*x^4+26*B
*a*m^5*x+1056*B*b*m^3*x^3+2038*B*c*m*x^5+247*C*a*m^4*x^2+2144*C*b*m^2*x^4+720*C*c*x^6+27*A*a*m^5+1219*A*b*m^3*
x^2+2412*A*c*m*x^4+270*B*a*m^4*x+2545*B*b*m^2*x^3+840*B*c*x^5+1219*C*a*m^3*x^2+2412*C*b*m*x^4+295*A*a*m^4+3112
*A*b*m^2*x^2+1008*A*c*x^4+1420*B*a*m^3*x+2952*B*b*m*x^3+3112*C*a*m^2*x^2+1008*C*b*x^4+1665*A*a*m^3+3796*A*b*m*
x^2+3929*B*a*m^2*x+1260*B*b*x^3+3796*C*a*m*x^2+5104*A*a*m^2+1680*A*b*x^2+5274*B*a*m*x+1680*C*a*x^2+8028*A*a*m+
2520*B*a*x+5040*A*a)*(d*x)^m/(7+m)/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.5775, size = 1181, normalized size = 8.62 \begin{align*} \frac{{\left ({\left (C c m^{6} + 21 \, C c m^{5} + 175 \, C c m^{4} + 735 \, C c m^{3} + 1624 \, C c m^{2} + 1764 \, C c m + 720 \, C c\right )} x^{7} +{\left (B c m^{6} + 22 \, B c m^{5} + 190 \, B c m^{4} + 820 \, B c m^{3} + 1849 \, B c m^{2} + 2038 \, B c m + 840 \, B c\right )} x^{6} +{\left ({\left (C b + A c\right )} m^{6} + 23 \,{\left (C b + A c\right )} m^{5} + 207 \,{\left (C b + A c\right )} m^{4} + 925 \,{\left (C b + A c\right )} m^{3} + 2144 \,{\left (C b + A c\right )} m^{2} + 1008 \, C b + 1008 \, A c + 2412 \,{\left (C b + A c\right )} m\right )} x^{5} +{\left (B b m^{6} + 24 \, B b m^{5} + 226 \, B b m^{4} + 1056 \, B b m^{3} + 2545 \, B b m^{2} + 2952 \, B b m + 1260 \, B b\right )} x^{4} +{\left ({\left (C a + A b\right )} m^{6} + 25 \,{\left (C a + A b\right )} m^{5} + 247 \,{\left (C a + A b\right )} m^{4} + 1219 \,{\left (C a + A b\right )} m^{3} + 3112 \,{\left (C a + A b\right )} m^{2} + 1680 \, C a + 1680 \, A b + 3796 \,{\left (C a + A b\right )} m\right )} x^{3} +{\left (B a m^{6} + 26 \, B a m^{5} + 270 \, B a m^{4} + 1420 \, B a m^{3} + 3929 \, B a m^{2} + 5274 \, B a m + 2520 \, B a\right )} x^{2} +{\left (A a m^{6} + 27 \, A a m^{5} + 295 \, A a m^{4} + 1665 \, A a m^{3} + 5104 \, A a m^{2} + 8028 \, A a m + 5040 \, A a\right )} x\right )} \left (d x\right )^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \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),x, algorithm="fricas")

[Out]

((C*c*m^6 + 21*C*c*m^5 + 175*C*c*m^4 + 735*C*c*m^3 + 1624*C*c*m^2 + 1764*C*c*m + 720*C*c)*x^7 + (B*c*m^6 + 22*
B*c*m^5 + 190*B*c*m^4 + 820*B*c*m^3 + 1849*B*c*m^2 + 2038*B*c*m + 840*B*c)*x^6 + ((C*b + A*c)*m^6 + 23*(C*b +
A*c)*m^5 + 207*(C*b + A*c)*m^4 + 925*(C*b + A*c)*m^3 + 2144*(C*b + A*c)*m^2 + 1008*C*b + 1008*A*c + 2412*(C*b
+ A*c)*m)*x^5 + (B*b*m^6 + 24*B*b*m^5 + 226*B*b*m^4 + 1056*B*b*m^3 + 2545*B*b*m^2 + 2952*B*b*m + 1260*B*b)*x^4
+ ((C*a + A*b)*m^6 + 25*(C*a + A*b)*m^5 + 247*(C*a + A*b)*m^4 + 1219*(C*a + A*b)*m^3 + 3112*(C*a + A*b)*m^2 +
1680*C*a + 1680*A*b + 3796*(C*a + A*b)*m)*x^3 + (B*a*m^6 + 26*B*a*m^5 + 270*B*a*m^4 + 1420*B*a*m^3 + 3929*B*a
*m^2 + 5274*B*a*m + 2520*B*a)*x^2 + (A*a*m^6 + 27*A*a*m^5 + 295*A*a*m^4 + 1665*A*a*m^3 + 5104*A*a*m^2 + 8028*A
*a*m + 5040*A*a)*x)*(d*x)^m/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)

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Sympy [A]  time = 2.79044, size = 3735, normalized size = 27.26 \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),x)

[Out]

Piecewise(((-A*a/(6*x**6) - A*b/(4*x**4) - A*c/(2*x**2) - B*a/(5*x**5) - B*b/(3*x**3) - B*c/x - C*a/(4*x**4) -
C*b/(2*x**2) + C*c*log(x))/d**7, Eq(m, -7)), ((-A*a/(5*x**5) - A*b/(3*x**3) - A*c/x - B*a/(4*x**4) - B*b/(2*x
**2) + B*c*log(x) - C*a/(3*x**3) - C*b/x + C*c*x)/d**6, Eq(m, -6)), ((-A*a/(4*x**4) - A*b/(2*x**2) + A*c*log(x
) - B*a/(3*x**3) - B*b/x + B*c*x - C*a/(2*x**2) + C*b*log(x) + C*c*x**2/2)/d**5, Eq(m, -5)), ((-A*a/(3*x**3) -
A*b/x + A*c*x - B*a/(2*x**2) + B*b*log(x) + B*c*x**2/2 - C*a/x + C*b*x + C*c*x**3/3)/d**4, Eq(m, -4)), ((-A*a
/(2*x**2) + A*b*log(x) + A*c*x**2/2 - B*a/x + B*b*x + B*c*x**3/3 + C*a*log(x) + C*b*x**2/2 + C*c*x**4/4)/d**3,
Eq(m, -3)), ((-A*a/x + A*b*x + A*c*x**3/3 + B*a*log(x) + B*b*x**2/2 + B*c*x**4/4 + C*a*x + C*b*x**3/3 + C*c*x
**5/5)/d**2, Eq(m, -2)), ((A*a*log(x) + A*b*x**2/2 + A*c*x**4/4 + B*a*x + B*b*x**3/3 + B*c*x**5/5 + C*a*x**2/2
+ C*b*x**4/4 + C*c*x**6/6)/d, Eq(m, -1)), (A*a*d**m*m**6*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769
*m**3 + 13132*m**2 + 13068*m + 5040) + 27*A*a*d**m*m**5*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m
**3 + 13132*m**2 + 13068*m + 5040) + 295*A*a*d**m*m**4*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m*
*3 + 13132*m**2 + 13068*m + 5040) + 1665*A*a*d**m*m**3*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m*
*3 + 13132*m**2 + 13068*m + 5040) + 5104*A*a*d**m*m**2*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m*
*3 + 13132*m**2 + 13068*m + 5040) + 8028*A*a*d**m*m*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3
+ 13132*m**2 + 13068*m + 5040) + 5040*A*a*d**m*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 131
32*m**2 + 13068*m + 5040) + A*b*d**m*m**6*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132
*m**2 + 13068*m + 5040) + 25*A*b*d**m*m**5*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 1313
2*m**2 + 13068*m + 5040) + 247*A*b*d**m*m**4*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13
132*m**2 + 13068*m + 5040) + 1219*A*b*d**m*m**3*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 +
13132*m**2 + 13068*m + 5040) + 3112*A*b*d**m*m**2*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**
3 + 13132*m**2 + 13068*m + 5040) + 3796*A*b*d**m*m*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**
3 + 13132*m**2 + 13068*m + 5040) + 1680*A*b*d**m*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3
+ 13132*m**2 + 13068*m + 5040) + A*c*d**m*m**6*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 +
13132*m**2 + 13068*m + 5040) + 23*A*c*d**m*m**5*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 +
13132*m**2 + 13068*m + 5040) + 207*A*c*d**m*m**4*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3
+ 13132*m**2 + 13068*m + 5040) + 925*A*c*d**m*m**3*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m*
*3 + 13132*m**2 + 13068*m + 5040) + 2144*A*c*d**m*m**2*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769
*m**3 + 13132*m**2 + 13068*m + 5040) + 2412*A*c*d**m*m*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769
*m**3 + 13132*m**2 + 13068*m + 5040) + 1008*A*c*d**m*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m
**3 + 13132*m**2 + 13068*m + 5040) + B*a*d**m*m**6*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**
3 + 13132*m**2 + 13068*m + 5040) + 26*B*a*d**m*m**5*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m*
*3 + 13132*m**2 + 13068*m + 5040) + 270*B*a*d**m*m**4*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*
m**3 + 13132*m**2 + 13068*m + 5040) + 1420*B*a*d**m*m**3*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 67
69*m**3 + 13132*m**2 + 13068*m + 5040) + 3929*B*a*d**m*m**2*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 +
6769*m**3 + 13132*m**2 + 13068*m + 5040) + 5274*B*a*d**m*m*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 +
6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2520*B*a*d**m*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6
769*m**3 + 13132*m**2 + 13068*m + 5040) + B*b*d**m*m**6*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 676
9*m**3 + 13132*m**2 + 13068*m + 5040) + 24*B*b*d**m*m**5*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 67
69*m**3 + 13132*m**2 + 13068*m + 5040) + 226*B*b*d**m*m**4*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 +
6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1056*B*b*d**m*m**3*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4
+ 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2545*B*b*d**m*m**2*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m
**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2952*B*b*d**m*m*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m
**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1260*B*b*d**m*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**
4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + B*c*d**m*m**6*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4
+ 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 22*B*c*d**m*m**5*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4
+ 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 190*B*c*d**m*m**4*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m*
*4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 820*B*c*d**m*m**3*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*
m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1849*B*c*d**m*m**2*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 19
60*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2038*B*c*d**m*m*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 19
60*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 840*B*c*d**m*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*
m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + C*a*d**m*m**6*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m*
*4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 25*C*a*d**m*m**5*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m
**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 247*C*a*d**m*m**4*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960
*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1219*C*a*d**m*m**3*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1
960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 3112*C*a*d**m*m**2*x**3*x**m/(m**7 + 28*m**6 + 322*m**5
+ 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 3796*C*a*d**m*m*x**3*x**m/(m**7 + 28*m**6 + 322*m**5
+ 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1680*C*a*d**m*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 +
1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + C*b*d**m*m**6*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 19
60*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 23*C*b*d**m*m**5*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1
960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 207*C*b*d**m*m**4*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 +
1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 925*C*b*d**m*m**3*x**5*x**m/(m**7 + 28*m**6 + 322*m**5
+ 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2144*C*b*d**m*m**2*x**5*x**m/(m**7 + 28*m**6 + 322*m
**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2412*C*b*d**m*m*x**5*x**m/(m**7 + 28*m**6 + 322*m
**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1008*C*b*d**m*x**5*x**m/(m**7 + 28*m**6 + 322*m**
5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + C*c*d**m*m**6*x**7*x**m/(m**7 + 28*m**6 + 322*m**5
+ 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 21*C*c*d**m*m**5*x**7*x**m/(m**7 + 28*m**6 + 322*m**5
+ 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 175*C*c*d**m*m**4*x**7*x**m/(m**7 + 28*m**6 + 322*m*
*5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 735*C*c*d**m*m**3*x**7*x**m/(m**7 + 28*m**6 + 322*
m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1624*C*c*d**m*m**2*x**7*x**m/(m**7 + 28*m**6 + 3
22*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1764*C*c*d**m*m*x**7*x**m/(m**7 + 28*m**6 + 3
22*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 720*C*c*d**m*x**7*x**m/(m**7 + 28*m**6 + 322*
m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040), True))

________________________________________________________________________________________

Giac [B]  time = 1.12333, size = 1234, normalized size = 9.01 \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),x, algorithm="giac")

[Out]

((d*x)^m*C*c*m^6*x^7 + (d*x)^m*B*c*m^6*x^6 + 21*(d*x)^m*C*c*m^5*x^7 + (d*x)^m*C*b*m^6*x^5 + (d*x)^m*A*c*m^6*x^
5 + 22*(d*x)^m*B*c*m^5*x^6 + 175*(d*x)^m*C*c*m^4*x^7 + (d*x)^m*B*b*m^6*x^4 + 23*(d*x)^m*C*b*m^5*x^5 + 23*(d*x)
^m*A*c*m^5*x^5 + 190*(d*x)^m*B*c*m^4*x^6 + 735*(d*x)^m*C*c*m^3*x^7 + (d*x)^m*C*a*m^6*x^3 + (d*x)^m*A*b*m^6*x^3
+ 24*(d*x)^m*B*b*m^5*x^4 + 207*(d*x)^m*C*b*m^4*x^5 + 207*(d*x)^m*A*c*m^4*x^5 + 820*(d*x)^m*B*c*m^3*x^6 + 1624
*(d*x)^m*C*c*m^2*x^7 + (d*x)^m*B*a*m^6*x^2 + 25*(d*x)^m*C*a*m^5*x^3 + 25*(d*x)^m*A*b*m^5*x^3 + 226*(d*x)^m*B*b
*m^4*x^4 + 925*(d*x)^m*C*b*m^3*x^5 + 925*(d*x)^m*A*c*m^3*x^5 + 1849*(d*x)^m*B*c*m^2*x^6 + 1764*(d*x)^m*C*c*m*x
^7 + (d*x)^m*A*a*m^6*x + 26*(d*x)^m*B*a*m^5*x^2 + 247*(d*x)^m*C*a*m^4*x^3 + 247*(d*x)^m*A*b*m^4*x^3 + 1056*(d*
x)^m*B*b*m^3*x^4 + 2144*(d*x)^m*C*b*m^2*x^5 + 2144*(d*x)^m*A*c*m^2*x^5 + 2038*(d*x)^m*B*c*m*x^6 + 720*(d*x)^m*
C*c*x^7 + 27*(d*x)^m*A*a*m^5*x + 270*(d*x)^m*B*a*m^4*x^2 + 1219*(d*x)^m*C*a*m^3*x^3 + 1219*(d*x)^m*A*b*m^3*x^3
+ 2545*(d*x)^m*B*b*m^2*x^4 + 2412*(d*x)^m*C*b*m*x^5 + 2412*(d*x)^m*A*c*m*x^5 + 840*(d*x)^m*B*c*x^6 + 295*(d*x
)^m*A*a*m^4*x + 1420*(d*x)^m*B*a*m^3*x^2 + 3112*(d*x)^m*C*a*m^2*x^3 + 3112*(d*x)^m*A*b*m^2*x^3 + 2952*(d*x)^m*
B*b*m*x^4 + 1008*(d*x)^m*C*b*x^5 + 1008*(d*x)^m*A*c*x^5 + 1665*(d*x)^m*A*a*m^3*x + 3929*(d*x)^m*B*a*m^2*x^2 +
3796*(d*x)^m*C*a*m*x^3 + 3796*(d*x)^m*A*b*m*x^3 + 1260*(d*x)^m*B*b*x^4 + 5104*(d*x)^m*A*a*m^2*x + 5274*(d*x)^m
*B*a*m*x^2 + 1680*(d*x)^m*C*a*x^3 + 1680*(d*x)^m*A*b*x^3 + 8028*(d*x)^m*A*a*m*x + 2520*(d*x)^m*B*a*x^2 + 5040*
(d*x)^m*A*a*x)/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)