3.6.86 \(\int (d+e x)^m (a+c x^2)^2 \, dx\)

Optimal. Leaf size=140 \[ \frac {\left (a e^2+c d^2\right )^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+2}}{e^5 (m+2)}+\frac {2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}-\frac {4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \begin {gather*} \frac {\left (a e^2+c d^2\right )^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+2}}{e^5 (m+2)}+\frac {2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}-\frac {4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a + c*x^2)^2,x]

[Out]

((c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*c*d*(c*d^2 + a*e^2)*(d + e*x)^(2 + m))/(e^5*(2 + m))
+ (2*c*(3*c*d^2 + a*e^2)*(d + e*x)^(3 + m))/(e^5*(3 + m)) - (4*c^2*d*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (c^2*(
d + e*x)^(5 + m))/(e^5*(5 + m))

Rule 697

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

Rubi steps

\begin {align*} \int (d+e x)^m \left (a+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2+a e^2\right )^2 (d+e x)^m}{e^4}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{1+m}}{e^4}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{2+m}}{e^4}-\frac {4 c^2 d (d+e x)^{3+m}}{e^4}+\frac {c^2 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {\left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 c^2 d (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 176, normalized size = 1.26 \begin {gather*} \frac {(d+e x)^{m+1} \left (\frac {4 \left (a e^2+c d^2\right ) \left (a e^2 \left (m^2+5 m+6\right )+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )}{e^4 (m+1) (m+2) (m+3)}-\frac {4 c d (d+e x) \left (a e^2 \left (m^2+7 m+12\right )+c \left (2 d^2-2 d e (m+2) x+e^2 \left (m^2+5 m+6\right ) x^2\right )\right )}{e^4 (m+2) (m+3) (m+4)}+\left (a+c x^2\right )^2\right )}{e (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*((a + c*x^2)^2 + (4*(c*d^2 + a*e^2)*(a*e^2*(6 + 5*m + m^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e
^2*(2 + 3*m + m^2)*x^2)))/(e^4*(1 + m)*(2 + m)*(3 + m)) - (4*c*d*(d + e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2
- 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)))/(e^4*(2 + m)*(3 + m)*(4 + m))))/(e*(5 + m))

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a+c x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.43, size = 520, normalized size = 3.71 \begin {gather*} \frac {{\left (a^{2} d e^{4} m^{4} + 14 \, a^{2} d e^{4} m^{3} + 24 \, c^{2} d^{5} + 80 \, a c d^{3} e^{2} + 120 \, a^{2} d e^{4} + {\left (c^{2} e^{5} m^{4} + 10 \, c^{2} e^{5} m^{3} + 35 \, c^{2} e^{5} m^{2} + 50 \, c^{2} e^{5} m + 24 \, c^{2} e^{5}\right )} x^{5} + {\left (c^{2} d e^{4} m^{4} + 6 \, c^{2} d e^{4} m^{3} + 11 \, c^{2} d e^{4} m^{2} + 6 \, c^{2} d e^{4} m\right )} x^{4} + 2 \, {\left (a c e^{5} m^{4} + 40 \, a c e^{5} - 2 \, {\left (c^{2} d^{2} e^{3} - 6 \, a c e^{5}\right )} m^{3} - {\left (6 \, c^{2} d^{2} e^{3} - 49 \, a c e^{5}\right )} m^{2} - 2 \, {\left (2 \, c^{2} d^{2} e^{3} - 39 \, a c e^{5}\right )} m\right )} x^{3} + {\left (4 \, a c d^{3} e^{2} + 71 \, a^{2} d e^{4}\right )} m^{2} + 2 \, {\left (a c d e^{4} m^{4} + 10 \, a c d e^{4} m^{3} + {\left (6 \, c^{2} d^{3} e^{2} + 29 \, a c d e^{4}\right )} m^{2} + 2 \, {\left (3 \, c^{2} d^{3} e^{2} + 10 \, a c d e^{4}\right )} m\right )} x^{2} + 2 \, {\left (18 \, a c d^{3} e^{2} + 77 \, a^{2} d e^{4}\right )} m + {\left (a^{2} e^{5} m^{4} + 120 \, a^{2} e^{5} - 2 \, {\left (2 \, a c d^{2} e^{3} - 7 \, a^{2} e^{5}\right )} m^{3} - {\left (36 \, a c d^{2} e^{3} - 71 \, a^{2} e^{5}\right )} m^{2} - 2 \, {\left (12 \, c^{2} d^{4} e + 40 \, a c d^{2} e^{3} - 77 \, a^{2} e^{5}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*d*e^4*m^4 + 14*a^2*d*e^4*m^3 + 24*c^2*d^5 + 80*a*c*d^3*e^2 + 120*a^2*d*e^4 + (c^2*e^5*m^4 + 10*c^2*e^5*m^
3 + 35*c^2*e^5*m^2 + 50*c^2*e^5*m + 24*c^2*e^5)*x^5 + (c^2*d*e^4*m^4 + 6*c^2*d*e^4*m^3 + 11*c^2*d*e^4*m^2 + 6*
c^2*d*e^4*m)*x^4 + 2*(a*c*e^5*m^4 + 40*a*c*e^5 - 2*(c^2*d^2*e^3 - 6*a*c*e^5)*m^3 - (6*c^2*d^2*e^3 - 49*a*c*e^5
)*m^2 - 2*(2*c^2*d^2*e^3 - 39*a*c*e^5)*m)*x^3 + (4*a*c*d^3*e^2 + 71*a^2*d*e^4)*m^2 + 2*(a*c*d*e^4*m^4 + 10*a*c
*d*e^4*m^3 + (6*c^2*d^3*e^2 + 29*a*c*d*e^4)*m^2 + 2*(3*c^2*d^3*e^2 + 10*a*c*d*e^4)*m)*x^2 + 2*(18*a*c*d^3*e^2
+ 77*a^2*d*e^4)*m + (a^2*e^5*m^4 + 120*a^2*e^5 - 2*(2*a*c*d^2*e^3 - 7*a^2*e^5)*m^3 - (36*a*c*d^2*e^3 - 71*a^2*
e^5)*m^2 - 2*(12*c^2*d^4*e + 40*a*c*d^2*e^3 - 77*a^2*e^5)*m)*x)*(e*x + d)^m/(e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^3
+ 225*e^5*m^2 + 274*e^5*m + 120*e^5)

________________________________________________________________________________________

giac [B]  time = 0.19, size = 848, normalized size = 6.06 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c^{2} m^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} c^{2} d m^{4} x^{4} e^{4} + 10 \, {\left (x e + d\right )}^{m} c^{2} m^{3} x^{5} e^{5} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m^{3} x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{3} x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} a c m^{4} x^{3} e^{5} + 35 \, {\left (x e + d\right )}^{m} c^{2} m^{2} x^{5} e^{5} + 2 \, {\left (x e + d\right )}^{m} a c d m^{4} x^{2} e^{4} + 11 \, {\left (x e + d\right )}^{m} c^{2} d m^{2} x^{4} e^{4} - 12 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{2} e^{2} + 24 \, {\left (x e + d\right )}^{m} a c m^{3} x^{3} e^{5} + 50 \, {\left (x e + d\right )}^{m} c^{2} m x^{5} e^{5} + 20 \, {\left (x e + d\right )}^{m} a c d m^{3} x^{2} e^{4} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} a c d^{2} m^{3} x e^{3} - 8 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m x^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m x e + {\left (x e + d\right )}^{m} a^{2} m^{4} x e^{5} + 98 \, {\left (x e + d\right )}^{m} a c m^{2} x^{3} e^{5} + 24 \, {\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + {\left (x e + d\right )}^{m} a^{2} d m^{4} e^{4} + 58 \, {\left (x e + d\right )}^{m} a c d m^{2} x^{2} e^{4} - 36 \, {\left (x e + d\right )}^{m} a c d^{2} m^{2} x e^{3} + 4 \, {\left (x e + d\right )}^{m} a c d^{3} m^{2} e^{2} + 24 \, {\left (x e + d\right )}^{m} c^{2} d^{5} + 14 \, {\left (x e + d\right )}^{m} a^{2} m^{3} x e^{5} + 156 \, {\left (x e + d\right )}^{m} a c m x^{3} e^{5} + 14 \, {\left (x e + d\right )}^{m} a^{2} d m^{3} e^{4} + 40 \, {\left (x e + d\right )}^{m} a c d m x^{2} e^{4} - 80 \, {\left (x e + d\right )}^{m} a c d^{2} m x e^{3} + 36 \, {\left (x e + d\right )}^{m} a c d^{3} m e^{2} + 71 \, {\left (x e + d\right )}^{m} a^{2} m^{2} x e^{5} + 80 \, {\left (x e + d\right )}^{m} a c x^{3} e^{5} + 71 \, {\left (x e + d\right )}^{m} a^{2} d m^{2} e^{4} + 80 \, {\left (x e + d\right )}^{m} a c d^{3} e^{2} + 154 \, {\left (x e + d\right )}^{m} a^{2} m x e^{5} + 154 \, {\left (x e + d\right )}^{m} a^{2} d m e^{4} + 120 \, {\left (x e + d\right )}^{m} a^{2} x e^{5} + 120 \, {\left (x e + d\right )}^{m} a^{2} d e^{4}}{m^{5} e^{5} + 15 \, m^{4} e^{5} + 85 \, m^{3} e^{5} + 225 \, m^{2} e^{5} + 274 \, m e^{5} + 120 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c^2*m^4*x^5*e^5 + (x*e + d)^m*c^2*d*m^4*x^4*e^4 + 10*(x*e + d)^m*c^2*m^3*x^5*e^5 + 6*(x*e + d)^m*
c^2*d*m^3*x^4*e^4 - 4*(x*e + d)^m*c^2*d^2*m^3*x^3*e^3 + 2*(x*e + d)^m*a*c*m^4*x^3*e^5 + 35*(x*e + d)^m*c^2*m^2
*x^5*e^5 + 2*(x*e + d)^m*a*c*d*m^4*x^2*e^4 + 11*(x*e + d)^m*c^2*d*m^2*x^4*e^4 - 12*(x*e + d)^m*c^2*d^2*m^2*x^3
*e^3 + 12*(x*e + d)^m*c^2*d^3*m^2*x^2*e^2 + 24*(x*e + d)^m*a*c*m^3*x^3*e^5 + 50*(x*e + d)^m*c^2*m*x^5*e^5 + 20
*(x*e + d)^m*a*c*d*m^3*x^2*e^4 + 6*(x*e + d)^m*c^2*d*m*x^4*e^4 - 4*(x*e + d)^m*a*c*d^2*m^3*x*e^3 - 8*(x*e + d)
^m*c^2*d^2*m*x^3*e^3 + 12*(x*e + d)^m*c^2*d^3*m*x^2*e^2 - 24*(x*e + d)^m*c^2*d^4*m*x*e + (x*e + d)^m*a^2*m^4*x
*e^5 + 98*(x*e + d)^m*a*c*m^2*x^3*e^5 + 24*(x*e + d)^m*c^2*x^5*e^5 + (x*e + d)^m*a^2*d*m^4*e^4 + 58*(x*e + d)^
m*a*c*d*m^2*x^2*e^4 - 36*(x*e + d)^m*a*c*d^2*m^2*x*e^3 + 4*(x*e + d)^m*a*c*d^3*m^2*e^2 + 24*(x*e + d)^m*c^2*d^
5 + 14*(x*e + d)^m*a^2*m^3*x*e^5 + 156*(x*e + d)^m*a*c*m*x^3*e^5 + 14*(x*e + d)^m*a^2*d*m^3*e^4 + 40*(x*e + d)
^m*a*c*d*m*x^2*e^4 - 80*(x*e + d)^m*a*c*d^2*m*x*e^3 + 36*(x*e + d)^m*a*c*d^3*m*e^2 + 71*(x*e + d)^m*a^2*m^2*x*
e^5 + 80*(x*e + d)^m*a*c*x^3*e^5 + 71*(x*e + d)^m*a^2*d*m^2*e^4 + 80*(x*e + d)^m*a*c*d^3*e^2 + 154*(x*e + d)^m
*a^2*m*x*e^5 + 154*(x*e + d)^m*a^2*d*m*e^4 + 120*(x*e + d)^m*a^2*x*e^5 + 120*(x*e + d)^m*a^2*d*e^4)/(m^5*e^5 +
15*m^4*e^5 + 85*m^3*e^5 + 225*m^2*e^5 + 274*m*e^5 + 120*e^5)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 420, normalized size = 3.00 \begin {gather*} \frac {\left (c^{2} e^{4} m^{4} x^{4}+10 c^{2} e^{4} m^{3} x^{4}+2 a c \,e^{4} m^{4} x^{2}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+24 a c \,e^{4} m^{3} x^{2}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}+a^{2} e^{4} m^{4}-4 a c d \,e^{3} m^{3} x +98 a c \,e^{4} m^{2} x^{2}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} e^{4} x^{4}+14 a^{2} e^{4} m^{3}-40 a c d \,e^{3} m^{2} x +156 a c \,e^{4} m \,x^{2}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 c^{2} d \,e^{3} x^{3}+71 a^{2} e^{4} m^{2}+4 a c \,d^{2} e^{2} m^{2}-116 a c d \,e^{3} m x +80 a c \,e^{4} x^{2}-24 c^{2} d^{3} e m x +24 c^{2} d^{2} e^{2} x^{2}+154 a^{2} e^{4} m +36 a c \,d^{2} e^{2} m -80 a c d \,e^{3} x -24 c^{2} d^{3} e x +120 a^{2} e^{4}+80 a c \,d^{2} e^{2}+24 c^{2} d^{4}\right ) \left (e x +d \right )^{m +1}}{\left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x+d)^(m+1)*(c^2*e^4*m^4*x^4+10*c^2*e^4*m^3*x^4+2*a*c*e^4*m^4*x^2-4*c^2*d*e^3*m^3*x^3+35*c^2*e^4*m^2*x^4+24*
a*c*e^4*m^3*x^2-24*c^2*d*e^3*m^2*x^3+50*c^2*e^4*m*x^4+a^2*e^4*m^4-4*a*c*d*e^3*m^3*x+98*a*c*e^4*m^2*x^2+12*c^2*
d^2*e^2*m^2*x^2-44*c^2*d*e^3*m*x^3+24*c^2*e^4*x^4+14*a^2*e^4*m^3-40*a*c*d*e^3*m^2*x+156*a*c*e^4*m*x^2+36*c^2*d
^2*e^2*m*x^2-24*c^2*d*e^3*x^3+71*a^2*e^4*m^2+4*a*c*d^2*e^2*m^2-116*a*c*d*e^3*m*x+80*a*c*e^4*x^2-24*c^2*d^3*e*m
*x+24*c^2*d^2*e^2*x^2+154*a^2*e^4*m+36*a*c*d^2*e^2*m-80*a*c*d*e^3*x-24*c^2*d^3*e*x+120*a^2*e^4+80*a*c*d^2*e^2+
24*c^2*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)

________________________________________________________________________________________

maxima [A]  time = 1.53, size = 235, normalized size = 1.68 \begin {gather*} \frac {{\left (e x + d\right )}^{m + 1} a^{2}}{e {\left (m + 1\right )}} + \frac {2 \, {\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^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x + d)^(m + 1)*a^2/(e*(m + 1)) + 2*((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^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 1
1*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)
*(e*x + d)^m*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)

________________________________________________________________________________________

mupad [B]  time = 0.70, size = 496, normalized size = 3.54 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c^2\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {d\,\left (a^2\,e^4\,m^4+14\,a^2\,e^4\,m^3+71\,a^2\,e^4\,m^2+154\,a^2\,e^4\,m+120\,a^2\,e^4+4\,a\,c\,d^2\,e^2\,m^2+36\,a\,c\,d^2\,e^2\,m+80\,a\,c\,d^2\,e^2+24\,c^2\,d^4\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,\left (a^2\,e^5\,m^4+14\,a^2\,e^5\,m^3+71\,a^2\,e^5\,m^2+154\,a^2\,e^5\,m+120\,a^2\,e^5-4\,a\,c\,d^2\,e^3\,m^3-36\,a\,c\,d^2\,e^3\,m^2-80\,a\,c\,d^2\,e^3\,m-24\,c^2\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,c\,x^3\,\left (m^2+3\,m+2\right )\,\left (-2\,c\,d^2\,m+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c^2\,d\,m\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,c\,d\,m\,x^2\,\left (m+1\right )\,\left (6\,c\,d^2+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^2*(d + e*x)^m,x)

[Out]

(d + e*x)^m*((c^2*x^5*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (
d*(120*a^2*e^4 + 24*c^2*d^4 + 154*a^2*e^4*m + 71*a^2*e^4*m^2 + 14*a^2*e^4*m^3 + a^2*e^4*m^4 + 80*a*c*d^2*e^2 +
36*a*c*d^2*e^2*m + 4*a*c*d^2*e^2*m^2))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*(120*a^2*e^
5 + 154*a^2*e^5*m + 71*a^2*e^5*m^2 + 14*a^2*e^5*m^3 + a^2*e^5*m^4 - 24*c^2*d^4*e*m - 80*a*c*d^2*e^3*m - 36*a*c
*d^2*e^3*m^2 - 4*a*c*d^2*e^3*m^3))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (2*c*x^3*(3*m + m^2
+ 2)*(20*a*e^2 + a*e^2*m^2 + 9*a*e^2*m - 2*c*d^2*m))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) +
(c^2*d*m*x^4*(11*m + 6*m^2 + m^3 + 6))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (2*c*d*m*x^2*(m +
1)*(20*a*e^2 + 6*c*d^2 + a*e^2*m^2 + 9*a*e^2*m))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)))

________________________________________________________________________________________

sympy [A]  time = 6.58, size = 5097, normalized size = 36.41

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(a**2*x + 2*a*c*x**3/3 + c**2*x**5/5), Eq(e, 0)), (-3*a**2*e**4/(12*d**4*e**5 + 48*d**3*e**6*x
+ 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 2*a*c*d**2*e**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d*
*2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 8*a*c*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x
**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12*a*c*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 4
8*d*e**8*x**3 + 12*e**9*x**4) + 12*c**2*d**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 +
48*d*e**8*x**3 + 12*e**9*x**4) + 25*c**2*d**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*
x**3 + 12*e**9*x**4) + 48*c**2*d**3*e*x*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d
*e**8*x**3 + 12*e**9*x**4) + 88*c**2*d**3*e*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x
**3 + 12*e**9*x**4) + 72*c**2*d**2*e**2*x**2*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 +
48*d*e**8*x**3 + 12*e**9*x**4) + 108*c**2*d**2*e**2*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 +
48*d*e**8*x**3 + 12*e**9*x**4) + 48*c**2*d*e**3*x**3*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e*
*7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*c**2*d*e**3*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x
**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*c**2*e**4*x**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2
*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4), Eq(m, -5)), (-a**2*e**4/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x
**2 + 3*e**8*x**3) - 2*a*c*d**2*e**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 6*a*c*d*e**
3*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 6*a*c*e**4*x**2/(3*d**3*e**5 + 9*d**2*e**6*x
+ 9*d*e**7*x**2 + 3*e**8*x**3) - 12*c**2*d**4*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e
**8*x**3) - 22*c**2*d**4/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*c**2*d**3*e*x*log(d/
e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 54*c**2*d**3*e*x/(3*d**3*e**5 + 9*d**2*e*
*6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*c**2*d**2*e**2*x**2*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e
**7*x**2 + 3*e**8*x**3) - 36*c**2*d**2*e**2*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) -
12*c**2*d*e**3*x**3*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 3*c**2*e**4*x*
*4/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3), Eq(m, -4)), (-a**2*e**4/(2*d**2*e**5 + 4*d*e**
6*x + 2*e**7*x**2) + 4*a*c*d**2*e**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 6*a*c*d**2*e**2/(
2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 8*a*c*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2)
+ 8*a*c*d*e**3*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*a*c*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e
**6*x + 2*e**7*x**2) + 12*c**2*d**4*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 18*c**2*d**4/(2*d*
*2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) +
24*c**2*d**3*e*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d**2*e**2*x**2*log(d/e + x)/(2*d**2*e**5
+ 4*d*e**6*x + 2*e**7*x**2) - 4*c**2*d*e**3*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + c**2*e**4*x**4/(2*
d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2), Eq(m, -3)), (-3*a**2*e**4/(3*d*e**5 + 3*e**6*x) - 12*a*c*d**2*e**2*log(
d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*a*c*d**2*e**2/(3*d*e**5 + 3*e**6*x) - 12*a*c*d*e**3*x*log(d/e + x)/(3*d*e*
*5 + 3*e**6*x) + 6*a*c*e**4*x**2/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**4*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*
c**2*d**4/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**3*e*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 6*c**2*d**2*e**2*x**2/
(3*d*e**5 + 3*e**6*x) - 2*c**2*d*e**3*x**3/(3*d*e**5 + 3*e**6*x) + c**2*e**4*x**4/(3*d*e**5 + 3*e**6*x), Eq(m,
-2)), (a**2*log(d/e + x)/e + 2*a*c*d**2*log(d/e + x)/e**3 - 2*a*c*d*x/e**2 + a*c*x**2/e + c**2*d**4*log(d/e +
x)/e**5 - c**2*d**3*x/e**4 + c**2*d**2*x**2/(2*e**3) - c**2*d*x**3/(3*e**2) + c**2*x**4/(4*e), Eq(m, -1)), (a
**2*d*e**4*m**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
+ 14*a**2*d*e**4*m**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 12
0*e**5) + 71*a**2*d*e**4*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5) + 154*a**2*d*e**4*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274
*e**5*m + 120*e**5) + 120*a**2*d*e**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) + a**2*e**5*m**4*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**
2 + 274*e**5*m + 120*e**5) + 14*a**2*e**5*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e
**5*m**2 + 274*e**5*m + 120*e**5) + 71*a**2*e**5*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 154*a**2*e**5*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*
m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*a**2*e**5*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e*
*5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 4*a*c*d**3*e**2*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 36*a*c*d**3*e**2*m*(d + e*x)**m/(e**5*m**5 + 15*e**
5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 80*a*c*d**3*e**2*(d + e*x)**m/(e**5*m**5 + 15
*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*a*c*d**2*e**3*m**3*x*(d + e*x)**m/(e**5
*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 36*a*c*d**2*e**3*m**2*x*(d + e*
x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 80*a*c*d**2*e**3*m*x
*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*a*c*d*e**4
*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 20
*a*c*d*e**4*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120
*e**5) + 58*a*c*d*e**4*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e
**5*m + 120*e**5) + 40*a*c*d*e**4*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + 2*a*c*e**5*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e
**5*m**2 + 274*e**5*m + 120*e**5) + 24*a*c*e**5*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**
3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 98*a*c*e**5*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85
*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 156*a*c*e**5*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m*
*4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 80*a*c*e**5*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**
5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*c**2*d**5*(d + e*x)**m/(e**5*m**5 + 15*e**
5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 24*c**2*d**4*e*m*x*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*c**2*d**3*e**2*m**2*x**2*(d + e*x)**
m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*c**2*d**3*e**2*m*x**2
*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*c**2*d**2*
e**3*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
- 12*c**2*d**2*e**3*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5
*m + 120*e**5) - 8*c**2*d**2*e**3*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + c**2*d*e**4*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*
e**5*m**2 + 274*e**5*m + 120*e**5) + 6*c**2*d*e**4*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*
m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 11*c**2*d*e**4*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**
4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*c**2*d*e**4*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + c**2*e**5*m**4*x**5*(d + e*x)**m/(e**5*m**
5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*c**2*e**5*m**3*x**5*(d + e*x)**m
/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 35*c**2*e**5*m**2*x**5*(d
+ e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 50*c**2*e**5*m*
x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*c**2*
e**5*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5), True
))

________________________________________________________________________________________