3.13.100 \(\int (A+B x) (d+e x)^m (a+c x^2) \, dx\)

Optimal. Leaf size=126 \[ -\frac {\left (a e^2+c d^2\right ) (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac {(d+e x)^{m+2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 (m+2)}-\frac {c (3 B d-A e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac {B c (d+e x)^{m+4}}{e^4 (m+4)} \]

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Rubi [A]  time = 0.07, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {772} \begin {gather*} -\frac {\left (a e^2+c d^2\right ) (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac {(d+e x)^{m+2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 (m+2)}-\frac {c (3 B d-A e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac {B c (d+e x)^{m+4}}{e^4 (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x
)^(2 + m))/(e^4*(2 + m)) - (c*(3*B*d - A*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (B*c*(d + e*x)^(4 + m))/(e^4*(4
+ m))

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2+a e^2\right ) (d+e x)^m}{e^3}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{1+m}}{e^3}+\frac {c (-3 B d+A e) (d+e x)^{2+m}}{e^3}+\frac {B c (d+e x)^{3+m}}{e^3}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {c (3 B d-A e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {B c (d+e x)^{4+m}}{e^4 (4+m)}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 122, normalized size = 0.97 \begin {gather*} \frac {(d+e x)^{m+1} \left ((A e-B d) \left (\frac {a e^2+c d^2}{m+1}+\frac {c (d+e x)^2}{m+3}-\frac {2 c d (d+e x)}{m+2}\right )+B (d+e x) \left (\frac {a e^2+c d^2}{m+2}+\frac {c (d+e x)^2}{m+4}-\frac {2 c d (d+e x)}{m+3}\right )\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*((-(B*d) + A*e)*((c*d^2 + a*e^2)/(1 + m) - (2*c*d*(d + e*x))/(2 + m) + (c*(d + e*x)^2)/(3 +
m)) + B*(d + e*x)*((c*d^2 + a*e^2)/(2 + m) - (2*c*d*(d + e*x))/(3 + m) + (c*(d + e*x)^2)/(4 + m))))/e^4

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IntegrateAlgebraic [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.45, size = 434, normalized size = 3.44 \begin {gather*} \frac {{\left (A a d e^{3} m^{3} - 6 \, B c d^{4} + 8 \, A c d^{3} e - 12 \, B a d^{2} e^{2} + 24 \, A a d e^{3} + {\left (B c e^{4} m^{3} + 6 \, B c e^{4} m^{2} + 11 \, B c e^{4} m + 6 \, B c e^{4}\right )} x^{4} + {\left (8 \, A c e^{4} + {\left (B c d e^{3} + A c e^{4}\right )} m^{3} + {\left (3 \, B c d e^{3} + 7 \, A c e^{4}\right )} m^{2} + 2 \, {\left (B c d e^{3} + 7 \, A c e^{4}\right )} m\right )} x^{3} - {\left (B a d^{2} e^{2} - 9 \, A a d e^{3}\right )} m^{2} + {\left (12 \, B a e^{4} + {\left (A c d e^{3} + B a e^{4}\right )} m^{3} - {\left (3 \, B c d^{2} e^{2} - 5 \, A c d e^{3} - 8 \, B a e^{4}\right )} m^{2} - {\left (3 \, B c d^{2} e^{2} - 4 \, A c d e^{3} - 19 \, B a e^{4}\right )} m\right )} x^{2} + {\left (2 \, A c d^{3} e - 7 \, B a d^{2} e^{2} + 26 \, A a d e^{3}\right )} m + {\left (24 \, A a e^{4} + {\left (B a d e^{3} + A a e^{4}\right )} m^{3} - {\left (2 \, A c d^{2} e^{2} - 7 \, B a d e^{3} - 9 \, A a e^{4}\right )} m^{2} + 2 \, {\left (3 \, B c d^{3} e - 4 \, A c d^{2} e^{2} + 6 \, B a d e^{3} + 13 \, A a e^{4}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m*(c*x^2+a),x, algorithm="fricas")

[Out]

(A*a*d*e^3*m^3 - 6*B*c*d^4 + 8*A*c*d^3*e - 12*B*a*d^2*e^2 + 24*A*a*d*e^3 + (B*c*e^4*m^3 + 6*B*c*e^4*m^2 + 11*B
*c*e^4*m + 6*B*c*e^4)*x^4 + (8*A*c*e^4 + (B*c*d*e^3 + A*c*e^4)*m^3 + (3*B*c*d*e^3 + 7*A*c*e^4)*m^2 + 2*(B*c*d*
e^3 + 7*A*c*e^4)*m)*x^3 - (B*a*d^2*e^2 - 9*A*a*d*e^3)*m^2 + (12*B*a*e^4 + (A*c*d*e^3 + B*a*e^4)*m^3 - (3*B*c*d
^2*e^2 - 5*A*c*d*e^3 - 8*B*a*e^4)*m^2 - (3*B*c*d^2*e^2 - 4*A*c*d*e^3 - 19*B*a*e^4)*m)*x^2 + (2*A*c*d^3*e - 7*B
*a*d^2*e^2 + 26*A*a*d*e^3)*m + (24*A*a*e^4 + (B*a*d*e^3 + A*a*e^4)*m^3 - (2*A*c*d^2*e^2 - 7*B*a*d*e^3 - 9*A*a*
e^4)*m^2 + 2*(3*B*c*d^3*e - 4*A*c*d^2*e^2 + 6*B*a*d*e^3 + 13*A*a*e^4)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^3
+ 35*e^4*m^2 + 50*e^4*m + 24*e^4)

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giac [B]  time = 0.18, size = 770, normalized size = 6.11 \begin {gather*} \frac {{\left (x e + d\right )}^{m} B c m^{3} x^{4} e^{4} + {\left (x e + d\right )}^{m} B c d m^{3} x^{3} e^{3} + {\left (x e + d\right )}^{m} A c m^{3} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} B c m^{2} x^{4} e^{4} + {\left (x e + d\right )}^{m} A c d m^{3} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} B c d m^{2} x^{3} e^{3} - 3 \, {\left (x e + d\right )}^{m} B c d^{2} m^{2} x^{2} e^{2} + {\left (x e + d\right )}^{m} B a m^{3} x^{2} e^{4} + 7 \, {\left (x e + d\right )}^{m} A c m^{2} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} B c m x^{4} e^{4} + {\left (x e + d\right )}^{m} B a d m^{3} x e^{3} + 5 \, {\left (x e + d\right )}^{m} A c d m^{2} x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} B c d m x^{3} e^{3} - 2 \, {\left (x e + d\right )}^{m} A c d^{2} m^{2} x e^{2} - 3 \, {\left (x e + d\right )}^{m} B c d^{2} m x^{2} e^{2} + 6 \, {\left (x e + d\right )}^{m} B c d^{3} m x e + {\left (x e + d\right )}^{m} A a m^{3} x e^{4} + 8 \, {\left (x e + d\right )}^{m} B a m^{2} x^{2} e^{4} + 14 \, {\left (x e + d\right )}^{m} A c m x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} B c x^{4} e^{4} + {\left (x e + d\right )}^{m} A a d m^{3} e^{3} + 7 \, {\left (x e + d\right )}^{m} B a d m^{2} x e^{3} + 4 \, {\left (x e + d\right )}^{m} A c d m x^{2} e^{3} - {\left (x e + d\right )}^{m} B a d^{2} m^{2} e^{2} - 8 \, {\left (x e + d\right )}^{m} A c d^{2} m x e^{2} + 2 \, {\left (x e + d\right )}^{m} A c d^{3} m e - 6 \, {\left (x e + d\right )}^{m} B c d^{4} + 9 \, {\left (x e + d\right )}^{m} A a m^{2} x e^{4} + 19 \, {\left (x e + d\right )}^{m} B a m x^{2} e^{4} + 8 \, {\left (x e + d\right )}^{m} A c x^{3} e^{4} + 9 \, {\left (x e + d\right )}^{m} A a d m^{2} e^{3} + 12 \, {\left (x e + d\right )}^{m} B a d m x e^{3} - 7 \, {\left (x e + d\right )}^{m} B a d^{2} m e^{2} + 8 \, {\left (x e + d\right )}^{m} A c d^{3} e + 26 \, {\left (x e + d\right )}^{m} A a m x e^{4} + 12 \, {\left (x e + d\right )}^{m} B a x^{2} e^{4} + 26 \, {\left (x e + d\right )}^{m} A a d m e^{3} - 12 \, {\left (x e + d\right )}^{m} B a d^{2} e^{2} + 24 \, {\left (x e + d\right )}^{m} A a x e^{4} + 24 \, {\left (x e + d\right )}^{m} A a d e^{3}}{m^{4} e^{4} + 10 \, m^{3} e^{4} + 35 \, m^{2} e^{4} + 50 \, m e^{4} + 24 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m*(c*x^2+a),x, algorithm="giac")

[Out]

((x*e + d)^m*B*c*m^3*x^4*e^4 + (x*e + d)^m*B*c*d*m^3*x^3*e^3 + (x*e + d)^m*A*c*m^3*x^3*e^4 + 6*(x*e + d)^m*B*c
*m^2*x^4*e^4 + (x*e + d)^m*A*c*d*m^3*x^2*e^3 + 3*(x*e + d)^m*B*c*d*m^2*x^3*e^3 - 3*(x*e + d)^m*B*c*d^2*m^2*x^2
*e^2 + (x*e + d)^m*B*a*m^3*x^2*e^4 + 7*(x*e + d)^m*A*c*m^2*x^3*e^4 + 11*(x*e + d)^m*B*c*m*x^4*e^4 + (x*e + d)^
m*B*a*d*m^3*x*e^3 + 5*(x*e + d)^m*A*c*d*m^2*x^2*e^3 + 2*(x*e + d)^m*B*c*d*m*x^3*e^3 - 2*(x*e + d)^m*A*c*d^2*m^
2*x*e^2 - 3*(x*e + d)^m*B*c*d^2*m*x^2*e^2 + 6*(x*e + d)^m*B*c*d^3*m*x*e + (x*e + d)^m*A*a*m^3*x*e^4 + 8*(x*e +
d)^m*B*a*m^2*x^2*e^4 + 14*(x*e + d)^m*A*c*m*x^3*e^4 + 6*(x*e + d)^m*B*c*x^4*e^4 + (x*e + d)^m*A*a*d*m^3*e^3 +
7*(x*e + d)^m*B*a*d*m^2*x*e^3 + 4*(x*e + d)^m*A*c*d*m*x^2*e^3 - (x*e + d)^m*B*a*d^2*m^2*e^2 - 8*(x*e + d)^m*A
*c*d^2*m*x*e^2 + 2*(x*e + d)^m*A*c*d^3*m*e - 6*(x*e + d)^m*B*c*d^4 + 9*(x*e + d)^m*A*a*m^2*x*e^4 + 19*(x*e + d
)^m*B*a*m*x^2*e^4 + 8*(x*e + d)^m*A*c*x^3*e^4 + 9*(x*e + d)^m*A*a*d*m^2*e^3 + 12*(x*e + d)^m*B*a*d*m*x*e^3 - 7
*(x*e + d)^m*B*a*d^2*m*e^2 + 8*(x*e + d)^m*A*c*d^3*e + 26*(x*e + d)^m*A*a*m*x*e^4 + 12*(x*e + d)^m*B*a*x^2*e^4
+ 26*(x*e + d)^m*A*a*d*m*e^3 - 12*(x*e + d)^m*B*a*d^2*e^2 + 24*(x*e + d)^m*A*a*x*e^4 + 24*(x*e + d)^m*A*a*d*e
^3)/(m^4*e^4 + 10*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4 + 24*e^4)

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maple [B]  time = 0.05, size = 338, normalized size = 2.68 \begin {gather*} \frac {\left (B c \,e^{3} m^{3} x^{3}+A c \,e^{3} m^{3} x^{2}+6 B c \,e^{3} m^{2} x^{3}+7 A c \,e^{3} m^{2} x^{2}+B a \,e^{3} m^{3} x -3 B c d \,e^{2} m^{2} x^{2}+11 B c \,e^{3} m \,x^{3}+A a \,e^{3} m^{3}-2 A c d \,e^{2} m^{2} x +14 A c \,e^{3} m \,x^{2}+8 B a \,e^{3} m^{2} x -9 B c d \,e^{2} m \,x^{2}+6 B c \,x^{3} e^{3}+9 A a \,e^{3} m^{2}-10 A c d \,e^{2} m x +8 A c \,e^{3} x^{2}-B a d \,e^{2} m^{2}+19 B a \,e^{3} m x +6 B c \,d^{2} e m x -6 B c d \,e^{2} x^{2}+26 A a \,e^{3} m +2 A c \,d^{2} e m -8 A c d \,e^{2} x -7 B a d \,e^{2} m +12 B a \,e^{3} x +6 B c \,d^{2} e x +24 a A \,e^{3}+8 A c \,d^{2} e -12 a B d \,e^{2}-6 B c \,d^{3}\right ) \left (e x +d \right )^{m +1}}{\left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right ) e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^m*(c*x^2+a),x)

[Out]

(e*x+d)^(m+1)*(B*c*e^3*m^3*x^3+A*c*e^3*m^3*x^2+6*B*c*e^3*m^2*x^3+7*A*c*e^3*m^2*x^2+B*a*e^3*m^3*x-3*B*c*d*e^2*m
^2*x^2+11*B*c*e^3*m*x^3+A*a*e^3*m^3-2*A*c*d*e^2*m^2*x+14*A*c*e^3*m*x^2+8*B*a*e^3*m^2*x-9*B*c*d*e^2*m*x^2+6*B*c
*e^3*x^3+9*A*a*e^3*m^2-10*A*c*d*e^2*m*x+8*A*c*e^3*x^2-B*a*d*e^2*m^2+19*B*a*e^3*m*x+6*B*c*d^2*e*m*x-6*B*c*d*e^2
*x^2+26*A*a*e^3*m+2*A*c*d^2*e*m-8*A*c*d*e^2*x-7*B*a*d*e^2*m+12*B*a*e^3*x+6*B*c*d^2*e*x+24*A*a*e^3+8*A*c*d^2*e-
12*B*a*d*e^2-6*B*c*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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maxima [A]  time = 0.67, size = 238, normalized size = 1.89 \begin {gather*} \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^m*(c*x^2+a),x, algorithm="maxima")

[Out]

(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*A*a/(e*(m + 1)) +
((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*c/((m^3 + 6*m^2 + 11*m + 6
)*e^3) + ((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e
*m*x - 6*d^4)*(e*x + d)^m*B*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4)

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mupad [B]  time = 2.04, size = 446, normalized size = 3.54 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (-6\,B\,c\,d^4+2\,A\,c\,d^3\,e\,m+8\,A\,c\,d^3\,e-B\,a\,d^2\,e^2\,m^2-7\,B\,a\,d^2\,e^2\,m-12\,B\,a\,d^2\,e^2+A\,a\,d\,e^3\,m^3+9\,A\,a\,d\,e^3\,m^2+26\,A\,a\,d\,e^3\,m+24\,A\,a\,d\,e^3\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (6\,B\,c\,d^3\,e\,m-2\,A\,c\,d^2\,e^2\,m^2-8\,A\,c\,d^2\,e^2\,m+B\,a\,d\,e^3\,m^3+7\,B\,a\,d\,e^3\,m^2+12\,B\,a\,d\,e^3\,m+A\,a\,e^4\,m^3+9\,A\,a\,e^4\,m^2+26\,A\,a\,e^4\,m+24\,A\,a\,e^4\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (-3\,B\,c\,d^2\,m+A\,c\,d\,e\,m^2+4\,A\,c\,d\,e\,m+B\,a\,e^2\,m^2+7\,B\,a\,e^2\,m+12\,B\,a\,e^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {B\,c\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {c\,x^3\,{\left (d+e\,x\right )}^m\,\left (4\,A\,e+A\,e\,m+B\,d\,m\right )\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)*(A + B*x)*(d + e*x)^m,x)

[Out]

((d + e*x)^m*(24*A*a*d*e^3 - 6*B*c*d^4 + 8*A*c*d^3*e - 12*B*a*d^2*e^2 - B*a*d^2*e^2*m^2 + 26*A*a*d*e^3*m + 2*A
*c*d^3*e*m + 9*A*a*d*e^3*m^2 + A*a*d*e^3*m^3 - 7*B*a*d^2*e^2*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (
x*(d + e*x)^m*(24*A*a*e^4 + 26*A*a*e^4*m + 9*A*a*e^4*m^2 + A*a*e^4*m^3 - 2*A*c*d^2*e^2*m^2 + 12*B*a*d*e^3*m +
6*B*c*d^3*e*m + 7*B*a*d*e^3*m^2 + B*a*d*e^3*m^3 - 8*A*c*d^2*e^2*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))
+ (x^2*(m + 1)*(d + e*x)^m*(12*B*a*e^2 + 7*B*a*e^2*m - 3*B*c*d^2*m + B*a*e^2*m^2 + 4*A*c*d*e*m + A*c*d*e*m^2))
/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (B*c*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 1
0*m^3 + m^4 + 24) + (c*x^3*(d + e*x)^m*(4*A*e + A*e*m + B*d*m)*(3*m + m^2 + 2))/(e*(50*m + 35*m^2 + 10*m^3 + m
^4 + 24))

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sympy [A]  time = 4.75, size = 3958, normalized size = 31.41

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**m*(c*x**2+a),x)

[Out]

Piecewise((d**m*(A*a*x + A*c*x**3/3 + B*a*x**2/2 + B*c*x**4/4), Eq(e, 0)), (-2*A*a*e**3/(6*d**3*e**4 + 18*d**2
*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*A*c*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7
*x**3) - 6*A*c*d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*c*e**3*x**2/(6*d**
3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - B*a*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6
*x**2 + 6*e**7*x**3) - 3*B*a*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*B*c*d**3
*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*B*c*d**3/(6*d**3*e**4 + 18*d*
*2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*c*d**2*e*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*
e**6*x**2 + 6*e**7*x**3) + 27*B*c*d**2*e*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*
B*c*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*c*d*e**2*x**
2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*B*c*e**3*x**3*log(d/e + x)/(6*d**3*e**4 +
18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3), Eq(m, -4)), (-A*a*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2
) + 2*A*c*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*A*c*d**2*e/(2*d**2*e**4 + 4*d*e**5*
x + 2*e**6*x**2) + 4*A*c*d*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*A*c*d*e**2*x/(2*d*
*2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*A*c*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) -
B*a*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*B*a*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) -
6*B*c*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 9*B*c*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e*
*6*x**2) - 12*B*c*d**2*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*B*c*d**2*e*x/(2*d**2*e**
4 + 4*d*e**5*x + 2*e**6*x**2) - 6*B*c*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*B*
c*e**3*x**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2), Eq(m, -3)), (-2*A*a*e**3/(2*d*e**4 + 2*e**5*x) - 4*A*c*d
**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*c*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*A*c*d*e**2*x*log(d/e + x)/(2
*d*e**4 + 2*e**5*x) + 2*A*c*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 2*B*a*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x)
+ 2*B*a*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*B*a*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3*log(d/e +
x)/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x
) - 3*B*c*d*e**2*x**2/(2*d*e**4 + 2*e**5*x) + B*c*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (A*a*log(d/e +
x)/e + A*c*d**2*log(d/e + x)/e**3 - A*c*d*x/e**2 + A*c*x**2/(2*e) - B*a*d*log(d/e + x)/e**2 + B*a*x/e - B*c*d*
*3*log(d/e + x)/e**4 + B*c*d**2*x/e**3 - B*c*d*x**2/(2*e**2) + B*c*x**3/(3*e), Eq(m, -1)), (A*a*d*e**3*m**3*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 9*A*a*d*e**3*m**2*(d + e*x)**m/(e
**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*A*a*d*e**3*m*(d + e*x)**m/(e**4*m**4 + 10*e
**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*A*a*d*e**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**
4*m**2 + 50*e**4*m + 24*e**4) + A*a*e**4*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**
4*m + 24*e**4) + 9*A*a*e**4*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + 26*A*a*e**4*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*A*a*e**4
*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*A*c*d**3*e*m*(d + e*x)**m/
(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*A*c*d**3*e*(d + e*x)**m/(e**4*m**4 + 10*e*
*4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 2*A*c*d**2*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3
+ 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 8*A*c*d**2*e**2*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m
**2 + 50*e**4*m + 24*e**4) + A*c*d*e**3*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) + 5*A*c*d*e**3*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m +
24*e**4) + 4*A*c*d*e**3*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) +
A*c*e**4*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*A*c*e**4*m
**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 14*A*c*e**4*m*x**3*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*A*c*e**4*x**3*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - B*a*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e
**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 7*B*a*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35
*e**4*m**2 + 50*e**4*m + 24*e**4) - 12*B*a*d**2*e**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) + B*a*d*e**3*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24
*e**4) + 7*B*a*d*e**3*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12
*B*a*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*a*e**4*m**3*x
**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*B*a*e**4*m**2*x**2*(d + e
*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 19*B*a*e**4*m*x**2*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*B*a*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**
4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 6*B*c*d**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**
2 + 50*e**4*m + 24*e**4) + 6*B*c*d**3*e*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m
+ 24*e**4) - 3*B*c*d**2*e**2*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*
e**4) - 3*B*c*d**2*e**2*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) +
B*c*d*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 3*B*c*d*e*
*3*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*B*c*d*e**3*m*x**
3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*c*e**4*m**3*x**4*(d + e*x)*
*m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*c*e**4*m**2*x**4*(d + e*x)**m/(e**4*m
**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 11*B*c*e**4*m*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**
4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*c*e**4*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**
4*m**2 + 50*e**4*m + 24*e**4), True))

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