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

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

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Rubi [A]  time = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.10, size = 90, normalized size = 0.66 \begin {gather*} 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 ) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.29, size = 444, normalized size = 3.24 \begin {gather*} \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 {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),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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giac [B]  time = 0.53, size = 914, normalized size = 6.67 \begin {gather*} \frac {\left (d x\right )^{m} C c m^{6} x^{7} + \left (d x\right )^{m} B c m^{6} x^{6} + 21 \, \left (d x\right )^{m} C c m^{5} x^{7} + \left (d x\right )^{m} C b m^{6} x^{5} + \left (d x\right )^{m} A c m^{6} x^{5} + 22 \, \left (d x\right )^{m} B c m^{5} x^{6} + 175 \, \left (d x\right )^{m} C c m^{4} x^{7} + \left (d x\right )^{m} B b m^{6} x^{4} + 23 \, \left (d x\right )^{m} C b m^{5} x^{5} + 23 \, \left (d x\right )^{m} A c m^{5} x^{5} + 190 \, \left (d x\right )^{m} B c m^{4} x^{6} + 735 \, \left (d x\right )^{m} C c m^{3} x^{7} + \left (d x\right )^{m} C a m^{6} x^{3} + \left (d x\right )^{m} A b m^{6} x^{3} + 24 \, \left (d x\right )^{m} B b m^{5} x^{4} + 207 \, \left (d x\right )^{m} C b m^{4} x^{5} + 207 \, \left (d x\right )^{m} A c m^{4} x^{5} + 820 \, \left (d x\right )^{m} B c m^{3} x^{6} + 1624 \, \left (d x\right )^{m} C c m^{2} x^{7} + \left (d x\right )^{m} B a m^{6} x^{2} + 25 \, \left (d x\right )^{m} C a m^{5} x^{3} + 25 \, \left (d x\right )^{m} A b m^{5} x^{3} + 226 \, \left (d x\right )^{m} B b m^{4} x^{4} + 925 \, \left (d x\right )^{m} C b m^{3} x^{5} + 925 \, \left (d x\right )^{m} A c m^{3} x^{5} + 1849 \, \left (d x\right )^{m} B c m^{2} x^{6} + 1764 \, \left (d x\right )^{m} C c m x^{7} + \left (d x\right )^{m} A a m^{6} x + 26 \, \left (d x\right )^{m} B a m^{5} x^{2} + 247 \, \left (d x\right )^{m} C a m^{4} x^{3} + 247 \, \left (d x\right )^{m} A b m^{4} x^{3} + 1056 \, \left (d x\right )^{m} B b m^{3} x^{4} + 2144 \, \left (d x\right )^{m} C b m^{2} x^{5} + 2144 \, \left (d x\right )^{m} A c m^{2} x^{5} + 2038 \, \left (d x\right )^{m} B c m x^{6} + 720 \, \left (d x\right )^{m} C c x^{7} + 27 \, \left (d x\right )^{m} A a m^{5} x + 270 \, \left (d x\right )^{m} B a m^{4} x^{2} + 1219 \, \left (d x\right )^{m} C a m^{3} x^{3} + 1219 \, \left (d x\right )^{m} A b m^{3} x^{3} + 2545 \, \left (d x\right )^{m} B b m^{2} x^{4} + 2412 \, \left (d x\right )^{m} C b m x^{5} + 2412 \, \left (d x\right )^{m} A c m x^{5} + 840 \, \left (d x\right )^{m} B c x^{6} + 295 \, \left (d x\right )^{m} A a m^{4} x + 1420 \, \left (d x\right )^{m} B a m^{3} x^{2} + 3112 \, \left (d x\right )^{m} C a m^{2} x^{3} + 3112 \, \left (d x\right )^{m} A b m^{2} x^{3} + 2952 \, \left (d x\right )^{m} B b m x^{4} + 1008 \, \left (d x\right )^{m} C b x^{5} + 1008 \, \left (d x\right )^{m} A c x^{5} + 1665 \, \left (d x\right )^{m} A a m^{3} x + 3929 \, \left (d x\right )^{m} B a m^{2} x^{2} + 3796 \, \left (d x\right )^{m} C a m x^{3} + 3796 \, \left (d x\right )^{m} A b m x^{3} + 1260 \, \left (d x\right )^{m} B b x^{4} + 5104 \, \left (d x\right )^{m} A a m^{2} x + 5274 \, \left (d x\right )^{m} B a m x^{2} + 1680 \, \left (d x\right )^{m} C a x^{3} + 1680 \, \left (d x\right )^{m} A b x^{3} + 8028 \, \left (d x\right )^{m} A a m x + 2520 \, \left (d x\right )^{m} B a x^{2} + 5040 \, \left (d x\right )^{m} A a x}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \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),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)

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maple [B]  time = 0.00, size = 585, normalized size = 4.27 \begin {gather*} \frac {\left (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 \right ) x \left (d x \right )^{m}}{\left (m +7\right ) \left (m +6\right ) \left (m +5\right ) \left (m +4\right ) \left (m +3\right ) \left (m +2\right ) \left (m +1\right )} \end {gather*}

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/(m+7)/(m+6)/(m+5)/(m+4)/(m+3)/(m+2)/(m+1)

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maxima [A]  time = 0.82, size = 155, normalized size = 1.13 \begin {gather*} \frac {C c d^{m} x^{7} x^{m}}{m + 7} + \frac {B c d^{m} x^{6} x^{m}}{m + 6} + \frac {C b d^{m} x^{5} x^{m}}{m + 5} + \frac {A c d^{m} x^{5} x^{m}}{m + 5} + \frac {B b d^{m} x^{4} x^{m}}{m + 4} + \frac {C a d^{m} x^{3} x^{m}}{m + 3} + \frac {A b d^{m} x^{3} x^{m}}{m + 3} + \frac {B a d^{m} x^{2} x^{m}}{m + 2} + \frac {\left (d x\right )^{m + 1} A a}{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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.07, size = 527, normalized size = 3.85 \begin {gather*} \frac {x^3\,{\left (d\,x\right )}^m\,\left (A\,b+C\,a\right )\,\left (m^6+25\,m^5+247\,m^4+1219\,m^3+3112\,m^2+3796\,m+1680\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {x^5\,{\left (d\,x\right )}^m\,\left (A\,c+C\,b\right )\,\left (m^6+23\,m^5+207\,m^4+925\,m^3+2144\,m^2+2412\,m+1008\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {A\,a\,x\,{\left (d\,x\right )}^m\,\left (m^6+27\,m^5+295\,m^4+1665\,m^3+5104\,m^2+8028\,m+5040\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {B\,a\,x^2\,{\left (d\,x\right )}^m\,\left (m^6+26\,m^5+270\,m^4+1420\,m^3+3929\,m^2+5274\,m+2520\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {B\,b\,x^4\,{\left (d\,x\right )}^m\,\left (m^6+24\,m^5+226\,m^4+1056\,m^3+2545\,m^2+2952\,m+1260\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {B\,c\,x^6\,{\left (d\,x\right )}^m\,\left (m^6+22\,m^5+190\,m^4+820\,m^3+1849\,m^2+2038\,m+840\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {C\,c\,x^7\,{\left (d\,x\right )}^m\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040} \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),x)

[Out]

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

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sympy [A]  time = 2.58, size = 3735, normalized size = 27.26

result too large to display

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))

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