3.6.85 \(\int (d+e x)^m (a+c x^2)^3 \, dx\)

Optimal. Leaf size=223 \[ -\frac {4 c^2 d \left (3 a e^2+5 c d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {3 c^2 \left (a e^2+5 c d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}+\frac {\left (a e^2+c d^2\right )^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {6 c d \left (a e^2+c d^2\right )^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac {c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

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Rubi [A]  time = 0.13, antiderivative size = 223, 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 {4 c^2 d \left (3 a e^2+5 c d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {3 c^2 \left (a e^2+5 c d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}+\frac {\left (a e^2+c d^2\right )^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {6 c d \left (a e^2+c d^2\right )^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac {c^3 (d+e x)^{m+7}}{e^7 (m+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

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

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Mathematica [A]  time = 0.66, size = 379, normalized size = 1.70 \begin {gather*} \frac {(d+e x)^{m+1} \left (\frac {6 \left ((m+6) \left (a e^2+c d^2\right ) \left (4 (m+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 )-4 c d (m+1) (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+1) (m+2) (m+3) (m+4) \left (a+c x^2\right )^2\right )-c d (m+1) (d+e x) \left (4 (m+5) \left (a e^2+c d^2\right ) \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 )-4 c d (m+2) (d+e x) \left (a e^2 \left (m^2+9 m+20\right )+c \left (2 d^2-2 d e (m+3) x+e^2 \left (m^2+7 m+12\right ) x^2\right )\right )+e^4 (m+2) (m+3) (m+4) (m+5) \left (a+c x^2\right )^2\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)}+\left (a+c x^2\right )^3\right )}{e (m+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*((a + c*x^2)^3 + (6*((c*d^2 + a*e^2)*(6 + m)*(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(a + c*x^
2)^2 + 4*(c*d^2 + a*e^2)*(4 + m)*(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
)) - 4*c*d*(1 + m)*(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)))
- c*d*(1 + m)*(d + e*x)*(e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + c*x^2)^2 + 4*(c*d^2 + a*e^2)*(5 + m)*(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)) - 4*c*d*(2 + m)*(d + e*x)*(a*e^2*(2
0 + 9*m + m^2) + c*(2*d^2 - 2*d*e*(3 + m)*x + e^2*(12 + 7*m + m^2)*x^2)))))/(e^6*(1 + m)*(2 + m)*(3 + m)*(4 +
m)*(5 + m)*(6 + m))))/(e*(7 + m))

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IntegrateAlgebraic [F]  time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a+c x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

Defer[IntegrateAlgebraic][(d + e*x)^m*(a + c*x^2)^3, x]

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fricas [B]  time = 0.44, size = 1250, normalized size = 5.61

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)^3,x, algorithm="fricas")

[Out]

(a^3*d*e^6*m^6 + 27*a^3*d*e^6*m^5 + 720*c^3*d^7 + 3024*a*c^2*d^5*e^2 + 5040*a^2*c*d^3*e^4 + 5040*a^3*d*e^6 + (
c^3*e^7*m^6 + 21*c^3*e^7*m^5 + 175*c^3*e^7*m^4 + 735*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3
*e^7)*x^7 + (c^3*d*e^6*m^6 + 15*c^3*d*e^6*m^5 + 85*c^3*d*e^6*m^4 + 225*c^3*d*e^6*m^3 + 274*c^3*d*e^6*m^2 + 120
*c^3*d*e^6*m)*x^6 + 3*(a*c^2*e^7*m^6 + 1008*a*c^2*e^7 - (2*c^3*d^2*e^5 - 23*a*c^2*e^7)*m^5 - (20*c^3*d^2*e^5 -
207*a*c^2*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 185*a*c^2*e^7)*m^3 - 4*(25*c^3*d^2*e^5 - 536*a*c^2*e^7)*m^2 - 12*(4*
c^3*d^2*e^5 - 201*a*c^2*e^7)*m)*x^5 + (6*a^2*c*d^3*e^4 + 295*a^3*d*e^6)*m^4 + 3*(a*c^2*d*e^6*m^6 + 19*a*c^2*d*
e^6*m^5 + (10*c^3*d^3*e^4 + 131*a*c^2*d*e^6)*m^4 + (60*c^3*d^3*e^4 + 401*a*c^2*d*e^6)*m^3 + 10*(11*c^3*d^3*e^4
+ 54*a*c^2*d*e^6)*m^2 + 12*(5*c^3*d^3*e^4 + 21*a*c^2*d*e^6)*m)*x^4 + 3*(44*a^2*c*d^3*e^4 + 555*a^3*d*e^6)*m^3
+ 3*(a^2*c*e^7*m^6 + 1680*a^2*c*e^7 - (4*a*c^2*d^2*e^5 - 25*a^2*c*e^7)*m^5 - (64*a*c^2*d^2*e^5 - 247*a^2*c*e^
7)*m^4 - (40*c^3*d^4*e^3 + 332*a*c^2*d^2*e^5 - 1219*a^2*c*e^7)*m^3 - 8*(15*c^3*d^4*e^3 + 76*a*c^2*d^2*e^5 - 38
9*a^2*c*e^7)*m^2 - 4*(20*c^3*d^4*e^3 + 84*a*c^2*d^2*e^5 - 949*a^2*c*e^7)*m)*x^3 + 2*(36*a*c^2*d^5*e^2 + 537*a^
2*c*d^3*e^4 + 2552*a^3*d*e^6)*m^2 + 3*(a^2*c*d*e^6*m^6 + 23*a^2*c*d*e^6*m^5 + 3*(4*a*c^2*d^3*e^4 + 67*a^2*c*d*
e^6)*m^4 + (168*a*c^2*d^3*e^4 + 817*a^2*c*d*e^6)*m^3 + 2*(60*c^3*d^5*e^2 + 330*a*c^2*d^3*e^4 + 739*a^2*c*d*e^6
)*m^2 + 24*(5*c^3*d^5*e^2 + 21*a*c^2*d^3*e^4 + 35*a^2*c*d*e^6)*m)*x^2 + 12*(78*a*c^2*d^5*e^2 + 319*a^2*c*d^3*e
^4 + 669*a^3*d*e^6)*m + (a^3*e^7*m^6 + 5040*a^3*e^7 - 3*(2*a^2*c*d^2*e^5 - 9*a^3*e^7)*m^5 - (132*a^2*c*d^2*e^5
- 295*a^3*e^7)*m^4 - 3*(24*a*c^2*d^4*e^3 + 358*a^2*c*d^2*e^5 - 555*a^3*e^7)*m^3 - 4*(234*a*c^2*d^4*e^3 + 957*
a^2*c*d^2*e^5 - 1276*a^3*e^7)*m^2 - 36*(20*c^3*d^6*e + 84*a*c^2*d^4*e^3 + 140*a^2*c*d^2*e^5 - 223*a^3*e^7)*m)*
x)*(e*x + d)^m/(e^7*m^7 + 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m
+ 5040*e^7)

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giac [B]  time = 0.25, size = 2080, normalized size = 9.33

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)^3,x, algorithm="giac")

[Out]

((x*e + d)^m*c^3*m^6*x^7*e^7 + (x*e + d)^m*c^3*d*m^6*x^6*e^6 + 21*(x*e + d)^m*c^3*m^5*x^7*e^7 + 15*(x*e + d)^m
*c^3*d*m^5*x^6*e^6 - 6*(x*e + d)^m*c^3*d^2*m^5*x^5*e^5 + 3*(x*e + d)^m*a*c^2*m^6*x^5*e^7 + 175*(x*e + d)^m*c^3
*m^4*x^7*e^7 + 3*(x*e + d)^m*a*c^2*d*m^6*x^4*e^6 + 85*(x*e + d)^m*c^3*d*m^4*x^6*e^6 - 60*(x*e + d)^m*c^3*d^2*m
^4*x^5*e^5 + 30*(x*e + d)^m*c^3*d^3*m^4*x^4*e^4 + 69*(x*e + d)^m*a*c^2*m^5*x^5*e^7 + 735*(x*e + d)^m*c^3*m^3*x
^7*e^7 + 57*(x*e + d)^m*a*c^2*d*m^5*x^4*e^6 + 225*(x*e + d)^m*c^3*d*m^3*x^6*e^6 - 12*(x*e + d)^m*a*c^2*d^2*m^5
*x^3*e^5 - 210*(x*e + d)^m*c^3*d^2*m^3*x^5*e^5 + 180*(x*e + d)^m*c^3*d^3*m^3*x^4*e^4 - 120*(x*e + d)^m*c^3*d^4
*m^3*x^3*e^3 + 3*(x*e + d)^m*a^2*c*m^6*x^3*e^7 + 621*(x*e + d)^m*a*c^2*m^4*x^5*e^7 + 1624*(x*e + d)^m*c^3*m^2*
x^7*e^7 + 3*(x*e + d)^m*a^2*c*d*m^6*x^2*e^6 + 393*(x*e + d)^m*a*c^2*d*m^4*x^4*e^6 + 274*(x*e + d)^m*c^3*d*m^2*
x^6*e^6 - 192*(x*e + d)^m*a*c^2*d^2*m^4*x^3*e^5 - 300*(x*e + d)^m*c^3*d^2*m^2*x^5*e^5 + 36*(x*e + d)^m*a*c^2*d
^3*m^4*x^2*e^4 + 330*(x*e + d)^m*c^3*d^3*m^2*x^4*e^4 - 360*(x*e + d)^m*c^3*d^4*m^2*x^3*e^3 + 360*(x*e + d)^m*c
^3*d^5*m^2*x^2*e^2 + 75*(x*e + d)^m*a^2*c*m^5*x^3*e^7 + 2775*(x*e + d)^m*a*c^2*m^3*x^5*e^7 + 1764*(x*e + d)^m*
c^3*m*x^7*e^7 + 69*(x*e + d)^m*a^2*c*d*m^5*x^2*e^6 + 1203*(x*e + d)^m*a*c^2*d*m^3*x^4*e^6 + 120*(x*e + d)^m*c^
3*d*m*x^6*e^6 - 6*(x*e + d)^m*a^2*c*d^2*m^5*x*e^5 - 996*(x*e + d)^m*a*c^2*d^2*m^3*x^3*e^5 - 144*(x*e + d)^m*c^
3*d^2*m*x^5*e^5 + 504*(x*e + d)^m*a*c^2*d^3*m^3*x^2*e^4 + 180*(x*e + d)^m*c^3*d^3*m*x^4*e^4 - 72*(x*e + d)^m*a
*c^2*d^4*m^3*x*e^3 - 240*(x*e + d)^m*c^3*d^4*m*x^3*e^3 + 360*(x*e + d)^m*c^3*d^5*m*x^2*e^2 - 720*(x*e + d)^m*c
^3*d^6*m*x*e + (x*e + d)^m*a^3*m^6*x*e^7 + 741*(x*e + d)^m*a^2*c*m^4*x^3*e^7 + 6432*(x*e + d)^m*a*c^2*m^2*x^5*
e^7 + 720*(x*e + d)^m*c^3*x^7*e^7 + (x*e + d)^m*a^3*d*m^6*e^6 + 603*(x*e + d)^m*a^2*c*d*m^4*x^2*e^6 + 1620*(x*
e + d)^m*a*c^2*d*m^2*x^4*e^6 - 132*(x*e + d)^m*a^2*c*d^2*m^4*x*e^5 - 1824*(x*e + d)^m*a*c^2*d^2*m^2*x^3*e^5 +
6*(x*e + d)^m*a^2*c*d^3*m^4*e^4 + 1980*(x*e + d)^m*a*c^2*d^3*m^2*x^2*e^4 - 936*(x*e + d)^m*a*c^2*d^4*m^2*x*e^3
+ 72*(x*e + d)^m*a*c^2*d^5*m^2*e^2 + 720*(x*e + d)^m*c^3*d^7 + 27*(x*e + d)^m*a^3*m^5*x*e^7 + 3657*(x*e + d)^
m*a^2*c*m^3*x^3*e^7 + 7236*(x*e + d)^m*a*c^2*m*x^5*e^7 + 27*(x*e + d)^m*a^3*d*m^5*e^6 + 2451*(x*e + d)^m*a^2*c
*d*m^3*x^2*e^6 + 756*(x*e + d)^m*a*c^2*d*m*x^4*e^6 - 1074*(x*e + d)^m*a^2*c*d^2*m^3*x*e^5 - 1008*(x*e + d)^m*a
*c^2*d^2*m*x^3*e^5 + 132*(x*e + d)^m*a^2*c*d^3*m^3*e^4 + 1512*(x*e + d)^m*a*c^2*d^3*m*x^2*e^4 - 3024*(x*e + d)
^m*a*c^2*d^4*m*x*e^3 + 936*(x*e + d)^m*a*c^2*d^5*m*e^2 + 295*(x*e + d)^m*a^3*m^4*x*e^7 + 9336*(x*e + d)^m*a^2*
c*m^2*x^3*e^7 + 3024*(x*e + d)^m*a*c^2*x^5*e^7 + 295*(x*e + d)^m*a^3*d*m^4*e^6 + 4434*(x*e + d)^m*a^2*c*d*m^2*
x^2*e^6 - 3828*(x*e + d)^m*a^2*c*d^2*m^2*x*e^5 + 1074*(x*e + d)^m*a^2*c*d^3*m^2*e^4 + 3024*(x*e + d)^m*a*c^2*d
^5*e^2 + 1665*(x*e + d)^m*a^3*m^3*x*e^7 + 11388*(x*e + d)^m*a^2*c*m*x^3*e^7 + 1665*(x*e + d)^m*a^3*d*m^3*e^6 +
2520*(x*e + d)^m*a^2*c*d*m*x^2*e^6 - 5040*(x*e + d)^m*a^2*c*d^2*m*x*e^5 + 3828*(x*e + d)^m*a^2*c*d^3*m*e^4 +
5104*(x*e + d)^m*a^3*m^2*x*e^7 + 5040*(x*e + d)^m*a^2*c*x^3*e^7 + 5104*(x*e + d)^m*a^3*d*m^2*e^6 + 5040*(x*e +
d)^m*a^2*c*d^3*e^4 + 8028*(x*e + d)^m*a^3*m*x*e^7 + 8028*(x*e + d)^m*a^3*d*m*e^6 + 5040*(x*e + d)^m*a^3*x*e^7
+ 5040*(x*e + d)^m*a^3*d*e^6)/(m^7*e^7 + 28*m^6*e^7 + 322*m^5*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^2*e
^7 + 13068*m*e^7 + 5040*e^7)

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maple [B]  time = 0.05, size = 1140, normalized size = 5.11 \begin {gather*} \frac {\left (c^{3} e^{6} m^{6} x^{6}+21 c^{3} e^{6} m^{5} x^{6}+3 a \,c^{2} e^{6} m^{6} x^{4}-6 c^{3} d \,e^{5} m^{5} x^{5}+175 c^{3} e^{6} m^{4} x^{6}+69 a \,c^{2} e^{6} m^{5} x^{4}-90 c^{3} d \,e^{5} m^{4} x^{5}+735 c^{3} e^{6} m^{3} x^{6}+3 a^{2} c \,e^{6} m^{6} x^{2}-12 a \,c^{2} d \,e^{5} m^{5} x^{3}+621 a \,c^{2} e^{6} m^{4} x^{4}+30 c^{3} d^{2} e^{4} m^{4} x^{4}-510 c^{3} d \,e^{5} m^{3} x^{5}+1624 c^{3} e^{6} m^{2} x^{6}+75 a^{2} c \,e^{6} m^{5} x^{2}-228 a \,c^{2} d \,e^{5} m^{4} x^{3}+2775 a \,c^{2} e^{6} m^{3} x^{4}+300 c^{3} d^{2} e^{4} m^{3} x^{4}-1350 c^{3} d \,e^{5} m^{2} x^{5}+1764 c^{3} e^{6} m \,x^{6}+a^{3} e^{6} m^{6}-6 a^{2} c d \,e^{5} m^{5} x +741 a^{2} c \,e^{6} m^{4} x^{2}+36 a \,c^{2} d^{2} e^{4} m^{4} x^{2}-1572 a \,c^{2} d \,e^{5} m^{3} x^{3}+6432 a \,c^{2} e^{6} m^{2} x^{4}-120 c^{3} d^{3} e^{3} m^{3} x^{3}+1050 c^{3} d^{2} e^{4} m^{2} x^{4}-1644 c^{3} d \,e^{5} m \,x^{5}+720 c^{3} e^{6} x^{6}+27 a^{3} e^{6} m^{5}-138 a^{2} c d \,e^{5} m^{4} x +3657 a^{2} c \,e^{6} m^{3} x^{2}+576 a \,c^{2} d^{2} e^{4} m^{3} x^{2}-4812 a \,c^{2} d \,e^{5} m^{2} x^{3}+7236 a \,c^{2} e^{6} m \,x^{4}-720 c^{3} d^{3} e^{3} m^{2} x^{3}+1500 c^{3} d^{2} e^{4} m \,x^{4}-720 c^{3} d \,e^{5} x^{5}+295 a^{3} e^{6} m^{4}+6 a^{2} c \,d^{2} e^{4} m^{4}-1206 a^{2} c d \,e^{5} m^{3} x +9336 a^{2} c \,e^{6} m^{2} x^{2}-72 a \,c^{2} d^{3} e^{3} m^{3} x +2988 a \,c^{2} d^{2} e^{4} m^{2} x^{2}-6480 a \,c^{2} d \,e^{5} m \,x^{3}+3024 a \,c^{2} e^{6} x^{4}+360 c^{3} d^{4} e^{2} m^{2} x^{2}-1320 c^{3} d^{3} e^{3} m \,x^{3}+720 c^{3} d^{2} e^{4} x^{4}+1665 a^{3} e^{6} m^{3}+132 a^{2} c \,d^{2} e^{4} m^{3}-4902 a^{2} c d \,e^{5} m^{2} x +11388 a^{2} c \,e^{6} m \,x^{2}-1008 a \,c^{2} d^{3} e^{3} m^{2} x +5472 a \,c^{2} d^{2} e^{4} m \,x^{2}-3024 a \,c^{2} d \,e^{5} x^{3}+1080 c^{3} d^{4} e^{2} m \,x^{2}-720 c^{3} d^{3} e^{3} x^{3}+5104 a^{3} e^{6} m^{2}+1074 a^{2} c \,d^{2} e^{4} m^{2}-8868 a^{2} c d \,e^{5} m x +5040 a^{2} c \,e^{6} x^{2}+72 a \,c^{2} d^{4} e^{2} m^{2}-3960 a \,c^{2} d^{3} e^{3} m x +3024 a \,c^{2} d^{2} e^{4} x^{2}-720 c^{3} d^{5} e m x +720 c^{3} d^{4} e^{2} x^{2}+8028 a^{3} e^{6} m +3828 a^{2} c \,d^{2} e^{4} m -5040 a^{2} c d \,e^{5} x +936 a \,c^{2} d^{4} e^{2} m -3024 a \,c^{2} d^{3} e^{3} x -720 c^{3} d^{5} e x +5040 a^{3} e^{6}+5040 a^{2} c \,d^{2} e^{4}+3024 a \,c^{2} d^{4} e^{2}+720 c^{3} d^{6}\right ) \left (e x +d \right )^{m +1}}{\left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right ) e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*(c*x^2+a)^3,x)

[Out]

(e*x+d)^(m+1)*(c^3*e^6*m^6*x^6+21*c^3*e^6*m^5*x^6+3*a*c^2*e^6*m^6*x^4-6*c^3*d*e^5*m^5*x^5+175*c^3*e^6*m^4*x^6+
69*a*c^2*e^6*m^5*x^4-90*c^3*d*e^5*m^4*x^5+735*c^3*e^6*m^3*x^6+3*a^2*c*e^6*m^6*x^2-12*a*c^2*d*e^5*m^5*x^3+621*a
*c^2*e^6*m^4*x^4+30*c^3*d^2*e^4*m^4*x^4-510*c^3*d*e^5*m^3*x^5+1624*c^3*e^6*m^2*x^6+75*a^2*c*e^6*m^5*x^2-228*a*
c^2*d*e^5*m^4*x^3+2775*a*c^2*e^6*m^3*x^4+300*c^3*d^2*e^4*m^3*x^4-1350*c^3*d*e^5*m^2*x^5+1764*c^3*e^6*m*x^6+a^3
*e^6*m^6-6*a^2*c*d*e^5*m^5*x+741*a^2*c*e^6*m^4*x^2+36*a*c^2*d^2*e^4*m^4*x^2-1572*a*c^2*d*e^5*m^3*x^3+6432*a*c^
2*e^6*m^2*x^4-120*c^3*d^3*e^3*m^3*x^3+1050*c^3*d^2*e^4*m^2*x^4-1644*c^3*d*e^5*m*x^5+720*c^3*e^6*x^6+27*a^3*e^6
*m^5-138*a^2*c*d*e^5*m^4*x+3657*a^2*c*e^6*m^3*x^2+576*a*c^2*d^2*e^4*m^3*x^2-4812*a*c^2*d*e^5*m^2*x^3+7236*a*c^
2*e^6*m*x^4-720*c^3*d^3*e^3*m^2*x^3+1500*c^3*d^2*e^4*m*x^4-720*c^3*d*e^5*x^5+295*a^3*e^6*m^4+6*a^2*c*d^2*e^4*m
^4-1206*a^2*c*d*e^5*m^3*x+9336*a^2*c*e^6*m^2*x^2-72*a*c^2*d^3*e^3*m^3*x+2988*a*c^2*d^2*e^4*m^2*x^2-6480*a*c^2*
d*e^5*m*x^3+3024*a*c^2*e^6*x^4+360*c^3*d^4*e^2*m^2*x^2-1320*c^3*d^3*e^3*m*x^3+720*c^3*d^2*e^4*x^4+1665*a^3*e^6
*m^3+132*a^2*c*d^2*e^4*m^3-4902*a^2*c*d*e^5*m^2*x+11388*a^2*c*e^6*m*x^2-1008*a*c^2*d^3*e^3*m^2*x+5472*a*c^2*d^
2*e^4*m*x^2-3024*a*c^2*d*e^5*x^3+1080*c^3*d^4*e^2*m*x^2-720*c^3*d^3*e^3*x^3+5104*a^3*e^6*m^2+1074*a^2*c*d^2*e^
4*m^2-8868*a^2*c*d*e^5*m*x+5040*a^2*c*e^6*x^2+72*a*c^2*d^4*e^2*m^2-3960*a*c^2*d^3*e^3*m*x+3024*a*c^2*d^2*e^4*x
^2-720*c^3*d^5*e*m*x+720*c^3*d^4*e^2*x^2+8028*a^3*e^6*m+3828*a^2*c*d^2*e^4*m-5040*a^2*c*d*e^5*x+936*a*c^2*d^4*
e^2*m-3024*a*c^2*d^3*e^3*x-720*c^3*d^5*e*x+5040*a^3*e^6+5040*a^2*c*d^2*e^4+3024*a*c^2*d^4*e^2+720*c^3*d^6)/e^7
/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)

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maxima [B]  time = 1.56, size = 472, normalized size = 2.12 \begin {gather*} \frac {{\left (e x + d\right )}^{m + 1} a^{3}}{e {\left (m + 1\right )}} + \frac {3 \, {\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^{2} c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {3 \, {\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} a c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d e^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} e^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} e^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} e^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} e^{2} x^{2} - 720 \, d^{6} e m x + 720 \, d^{7}\right )} {\left (e x + d\right )}^{m} c^{3}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

(e*x + d)^(m + 1)*a^3/(e*(m + 1)) + 3*((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^2*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3
+ 11*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*a*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + ((m^6 + 21*m^5 + 175*m^4 + 735*
m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*e^6*x^6 - 6*(m^
5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4*x^4 - 120*(m^3 + 3*
m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m*c^3/((m^7 + 28*m^6 +
322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^7)

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mupad [B]  time = 1.05, size = 1144, normalized size = 5.13 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (a^3\,d\,e^6\,m^6+27\,a^3\,d\,e^6\,m^5+295\,a^3\,d\,e^6\,m^4+1665\,a^3\,d\,e^6\,m^3+5104\,a^3\,d\,e^6\,m^2+8028\,a^3\,d\,e^6\,m+5040\,a^3\,d\,e^6+6\,a^2\,c\,d^3\,e^4\,m^4+132\,a^2\,c\,d^3\,e^4\,m^3+1074\,a^2\,c\,d^3\,e^4\,m^2+3828\,a^2\,c\,d^3\,e^4\,m+5040\,a^2\,c\,d^3\,e^4+72\,a\,c^2\,d^5\,e^2\,m^2+936\,a\,c^2\,d^5\,e^2\,m+3024\,a\,c^2\,d^5\,e^2+720\,c^3\,d^7\right )}{e^7\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}-\frac {x\,{\left (d+e\,x\right )}^m\,\left (-a^3\,e^7\,m^6-27\,a^3\,e^7\,m^5-295\,a^3\,e^7\,m^4-1665\,a^3\,e^7\,m^3-5104\,a^3\,e^7\,m^2-8028\,a^3\,e^7\,m-5040\,a^3\,e^7+6\,a^2\,c\,d^2\,e^5\,m^5+132\,a^2\,c\,d^2\,e^5\,m^4+1074\,a^2\,c\,d^2\,e^5\,m^3+3828\,a^2\,c\,d^2\,e^5\,m^2+5040\,a^2\,c\,d^2\,e^5\,m+72\,a\,c^2\,d^4\,e^3\,m^3+936\,a\,c^2\,d^4\,e^3\,m^2+3024\,a\,c^2\,d^4\,e^3\,m+720\,c^3\,d^6\,e\,m\right )}{e^7\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {c^3\,x^7\,{\left (d+e\,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}+\frac {3\,c^2\,x^5\,{\left (d+e\,x\right )}^m\,\left (-2\,c\,d^2\,m+a\,e^2\,m^2+13\,a\,e^2\,m+42\,a\,e^2\right )\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{e^2\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {3\,c\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (a^2\,e^4\,m^4+22\,a^2\,e^4\,m^3+179\,a^2\,e^4\,m^2+638\,a^2\,e^4\,m+840\,a^2\,e^4-4\,a\,c\,d^2\,e^2\,m^3-52\,a\,c\,d^2\,e^2\,m^2-168\,a\,c\,d^2\,e^2\,m-40\,c^2\,d^4\,m\right )}{e^4\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {c^3\,d\,m\,x^6\,{\left (d+e\,x\right )}^m\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{e\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {3\,c^2\,d\,m\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (10\,c\,d^2+a\,e^2\,m^2+13\,a\,e^2\,m+42\,a\,e^2\right )}{e^3\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {3\,c\,d\,m\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^4\,m^4+22\,a^2\,e^4\,m^3+179\,a^2\,e^4\,m^2+638\,a^2\,e^4\,m+840\,a^2\,e^4+12\,a\,c\,d^2\,e^2\,m^2+156\,a\,c\,d^2\,e^2\,m+504\,a\,c\,d^2\,e^2+120\,c^2\,d^4\right )}{e^5\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(d + e*x)^m,x)

[Out]

((d + e*x)^m*(720*c^3*d^7 + 5040*a^3*d*e^6 + 3024*a*c^2*d^5*e^2 + 5040*a^2*c*d^3*e^4 + 5104*a^3*d*e^6*m^2 + 16
65*a^3*d*e^6*m^3 + 295*a^3*d*e^6*m^4 + 27*a^3*d*e^6*m^5 + a^3*d*e^6*m^6 + 8028*a^3*d*e^6*m + 936*a*c^2*d^5*e^2
*m + 3828*a^2*c*d^3*e^4*m + 72*a*c^2*d^5*e^2*m^2 + 1074*a^2*c*d^3*e^4*m^2 + 132*a^2*c*d^3*e^4*m^3 + 6*a^2*c*d^
3*e^4*m^4))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) - (x*(d + e*x)^m
*(720*c^3*d^6*e*m - 8028*a^3*e^7*m - 5104*a^3*e^7*m^2 - 1665*a^3*e^7*m^3 - 295*a^3*e^7*m^4 - 27*a^3*e^7*m^5 -
a^3*e^7*m^6 - 5040*a^3*e^7 + 3024*a*c^2*d^4*e^3*m + 5040*a^2*c*d^2*e^5*m + 936*a*c^2*d^4*e^3*m^2 + 3828*a^2*c*
d^2*e^5*m^2 + 72*a*c^2*d^4*e^3*m^3 + 1074*a^2*c*d^2*e^5*m^3 + 132*a^2*c*d^2*e^5*m^4 + 6*a^2*c*d^2*e^5*m^5))/(e
^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (c^3*x^7*(d + e*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 +
28*m^6 + m^7 + 5040) + (3*c^2*x^5*(d + e*x)^m*(42*a*e^2 + a*e^2*m^2 + 13*a*e^2*m - 2*c*d^2*m)*(50*m + 35*m^2 +
10*m^3 + m^4 + 24))/(e^2*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c*
x^3*(d + e*x)^m*(3*m + m^2 + 2)*(840*a^2*e^4 + 638*a^2*e^4*m - 40*c^2*d^4*m + 179*a^2*e^4*m^2 + 22*a^2*e^4*m^3
+ a^2*e^4*m^4 - 168*a*c*d^2*e^2*m - 52*a*c*d^2*e^2*m^2 - 4*a*c*d^2*e^2*m^3))/(e^4*(13068*m + 13132*m^2 + 6769
*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (c^3*d*m*x^6*(d + e*x)^m*(274*m + 225*m^2 + 85*m^3 + 15*m^
4 + m^5 + 120))/(e*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c^2*d*m*x
^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(42*a*e^2 + 10*c*d^2 + a*e^2*m^2 + 13*a*e^2*m))/(e^3*(13068*m + 13132*
m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c*d*m*x^2*(m + 1)*(d + e*x)^m*(840*a^2*e^4 +
120*c^2*d^4 + 638*a^2*e^4*m + 179*a^2*e^4*m^2 + 22*a^2*e^4*m^3 + a^2*e^4*m^4 + 504*a*c*d^2*e^2 + 156*a*c*d^2*e
^2*m + 12*a*c*d^2*e^2*m^2))/(e^5*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*(c*x**2+a)**3,x)

[Out]

Timed out

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