3.18.100 \(\int (d+e x)^4 (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=272 \[ \frac {c (d+e x)^9 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {(d+e x)^8 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{8 e^7}+\frac {3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac {3 c^2 (d+e x)^{10} (2 c d-b e)}{10 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7} \]

________________________________________________________________________________________

Rubi [A]  time = 0.51, antiderivative size = 272, 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 = {698} \begin {gather*} \frac {c (d+e x)^9 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^7}-\frac {(d+e x)^8 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{8 e^7}+\frac {3 (d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac {(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^3}{5 e^7}-\frac {3 c^2 (d+e x)^{10} (2 c d-b e)}{10 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a + b*x + c*x^2)^3,x]

[Out]

((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^5)/(5*e^7) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^6)/(2*e^7
) + (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^7)/(7*e^7) - ((2*c*d - b*e)
*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^8)/(8*e^7) + (c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d -
a*e))*(d + e*x)^9)/(3*e^7) - (3*c^2*(2*c*d - b*e)*(d + e*x)^10)/(10*e^7) + (c^3*(d + e*x)^11)/(11*e^7)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^4}{e^6}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{e^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^6}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^7}{e^6}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^8}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^9}{e^6}+\frac {c^3 (d+e x)^{10}}{e^6}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^5}{5 e^7}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}{2 e^7}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^7}{7 e^7}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^8}{8 e^7}+\frac {c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^9}{3 e^7}-\frac {3 c^2 (2 c d-b e) (d+e x)^{10}}{10 e^7}+\frac {c^3 (d+e x)^{11}}{11 e^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 497, normalized size = 1.83 \begin {gather*} a^3 d^4 x+\frac {1}{7} x^7 \left (3 c e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+b^2 e^3 (3 a e+4 b d)+6 c^2 d^2 e (3 a e+2 b d)+c^3 d^4\right )+\frac {1}{5} x^5 \left (a \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+3 b^2 \left (6 a d^2 e^2+c d^4\right )+12 a b d e \left (a e^2+2 c d^2\right )+4 b^3 d^3 e\right )+\frac {1}{2} x^6 \left (b \left (a^2 e^4+12 a c d^2 e^2+c^2 d^4\right )+4 b^2 \left (a d e^3+c d^3 e\right )+4 a c d e \left (a e^2+c d^2\right )+2 b^3 d^2 e^2\right )+\frac {1}{4} d x^4 \left (4 a^2 e \left (a e^2+3 c d^2\right )+12 a b^2 d^2 e+6 a b d \left (3 a e^2+c d^2\right )+b^3 d^3\right )+\frac {1}{2} a^2 d^3 x^2 (4 a e+3 b d)+\frac {1}{8} e x^8 \left (6 c^2 d e (2 a e+3 b d)+6 b c e^2 (a e+2 b d)+b^3 e^3+4 c^3 d^3\right )+\frac {1}{3} c e^2 x^9 \left (c e (a e+4 b d)+b^2 e^2+2 c^2 d^2\right )+a d^2 x^3 \left (4 a b d e+a \left (2 a e^2+c d^2\right )+b^2 d^2\right )+\frac {1}{10} c^2 e^3 x^{10} (3 b e+4 c d)+\frac {1}{11} c^3 e^4 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(a + b*x + c*x^2)^3,x]

[Out]

a^3*d^4*x + (a^2*d^3*(3*b*d + 4*a*e)*x^2)/2 + a*d^2*(b^2*d^2 + 4*a*b*d*e + a*(c*d^2 + 2*a*e^2))*x^3 + (d*(b^3*
d^3 + 12*a*b^2*d^2*e + 4*a^2*e*(3*c*d^2 + a*e^2) + 6*a*b*d*(c*d^2 + 3*a*e^2))*x^4)/4 + ((4*b^3*d^3*e + 12*a*b*
d*e*(2*c*d^2 + a*e^2) + 3*b^2*(c*d^4 + 6*a*d^2*e^2) + a*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4))*x^5)/5 + ((2*b
^3*d^2*e^2 + 4*a*c*d*e*(c*d^2 + a*e^2) + 4*b^2*(c*d^3*e + a*d*e^3) + b*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4))*x
^6)/2 + ((c^3*d^4 + 6*c^2*d^2*e*(2*b*d + 3*a*e) + b^2*e^3*(4*b*d + 3*a*e) + 3*c*e^2*(6*b^2*d^2 + 8*a*b*d*e + a
^2*e^2))*x^7)/7 + (e*(4*c^3*d^3 + b^3*e^3 + 6*b*c*e^2*(2*b*d + a*e) + 6*c^2*d*e*(3*b*d + 2*a*e))*x^8)/8 + (c*e
^2*(2*c^2*d^2 + b^2*e^2 + c*e*(4*b*d + a*e))*x^9)/3 + (c^2*e^3*(4*c*d + 3*b*e)*x^10)/10 + (c^3*e^4*x^11)/11

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^4 \left (a+b x+c x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^4*(a + b*x + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^4*(a + b*x + c*x^2)^3, x]

________________________________________________________________________________________

fricas [B]  time = 0.35, size = 624, normalized size = 2.29 \begin {gather*} \frac {1}{11} x^{11} e^{4} c^{3} + \frac {2}{5} x^{10} e^{3} d c^{3} + \frac {3}{10} x^{10} e^{4} c^{2} b + \frac {2}{3} x^{9} e^{2} d^{2} c^{3} + \frac {4}{3} x^{9} e^{3} d c^{2} b + \frac {1}{3} x^{9} e^{4} c b^{2} + \frac {1}{3} x^{9} e^{4} c^{2} a + \frac {1}{2} x^{8} e d^{3} c^{3} + \frac {9}{4} x^{8} e^{2} d^{2} c^{2} b + \frac {3}{2} x^{8} e^{3} d c b^{2} + \frac {1}{8} x^{8} e^{4} b^{3} + \frac {3}{2} x^{8} e^{3} d c^{2} a + \frac {3}{4} x^{8} e^{4} c b a + \frac {1}{7} x^{7} d^{4} c^{3} + \frac {12}{7} x^{7} e d^{3} c^{2} b + \frac {18}{7} x^{7} e^{2} d^{2} c b^{2} + \frac {4}{7} x^{7} e^{3} d b^{3} + \frac {18}{7} x^{7} e^{2} d^{2} c^{2} a + \frac {24}{7} x^{7} e^{3} d c b a + \frac {3}{7} x^{7} e^{4} b^{2} a + \frac {3}{7} x^{7} e^{4} c a^{2} + \frac {1}{2} x^{6} d^{4} c^{2} b + 2 x^{6} e d^{3} c b^{2} + x^{6} e^{2} d^{2} b^{3} + 2 x^{6} e d^{3} c^{2} a + 6 x^{6} e^{2} d^{2} c b a + 2 x^{6} e^{3} d b^{2} a + 2 x^{6} e^{3} d c a^{2} + \frac {1}{2} x^{6} e^{4} b a^{2} + \frac {3}{5} x^{5} d^{4} c b^{2} + \frac {4}{5} x^{5} e d^{3} b^{3} + \frac {3}{5} x^{5} d^{4} c^{2} a + \frac {24}{5} x^{5} e d^{3} c b a + \frac {18}{5} x^{5} e^{2} d^{2} b^{2} a + \frac {18}{5} x^{5} e^{2} d^{2} c a^{2} + \frac {12}{5} x^{5} e^{3} d b a^{2} + \frac {1}{5} x^{5} e^{4} a^{3} + \frac {1}{4} x^{4} d^{4} b^{3} + \frac {3}{2} x^{4} d^{4} c b a + 3 x^{4} e d^{3} b^{2} a + 3 x^{4} e d^{3} c a^{2} + \frac {9}{2} x^{4} e^{2} d^{2} b a^{2} + x^{4} e^{3} d a^{3} + x^{3} d^{4} b^{2} a + x^{3} d^{4} c a^{2} + 4 x^{3} e d^{3} b a^{2} + 2 x^{3} e^{2} d^{2} a^{3} + \frac {3}{2} x^{2} d^{4} b a^{2} + 2 x^{2} e d^{3} a^{3} + x d^{4} a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/11*x^11*e^4*c^3 + 2/5*x^10*e^3*d*c^3 + 3/10*x^10*e^4*c^2*b + 2/3*x^9*e^2*d^2*c^3 + 4/3*x^9*e^3*d*c^2*b + 1/3
*x^9*e^4*c*b^2 + 1/3*x^9*e^4*c^2*a + 1/2*x^8*e*d^3*c^3 + 9/4*x^8*e^2*d^2*c^2*b + 3/2*x^8*e^3*d*c*b^2 + 1/8*x^8
*e^4*b^3 + 3/2*x^8*e^3*d*c^2*a + 3/4*x^8*e^4*c*b*a + 1/7*x^7*d^4*c^3 + 12/7*x^7*e*d^3*c^2*b + 18/7*x^7*e^2*d^2
*c*b^2 + 4/7*x^7*e^3*d*b^3 + 18/7*x^7*e^2*d^2*c^2*a + 24/7*x^7*e^3*d*c*b*a + 3/7*x^7*e^4*b^2*a + 3/7*x^7*e^4*c
*a^2 + 1/2*x^6*d^4*c^2*b + 2*x^6*e*d^3*c*b^2 + x^6*e^2*d^2*b^3 + 2*x^6*e*d^3*c^2*a + 6*x^6*e^2*d^2*c*b*a + 2*x
^6*e^3*d*b^2*a + 2*x^6*e^3*d*c*a^2 + 1/2*x^6*e^4*b*a^2 + 3/5*x^5*d^4*c*b^2 + 4/5*x^5*e*d^3*b^3 + 3/5*x^5*d^4*c
^2*a + 24/5*x^5*e*d^3*c*b*a + 18/5*x^5*e^2*d^2*b^2*a + 18/5*x^5*e^2*d^2*c*a^2 + 12/5*x^5*e^3*d*b*a^2 + 1/5*x^5
*e^4*a^3 + 1/4*x^4*d^4*b^3 + 3/2*x^4*d^4*c*b*a + 3*x^4*e*d^3*b^2*a + 3*x^4*e*d^3*c*a^2 + 9/2*x^4*e^2*d^2*b*a^2
+ x^4*e^3*d*a^3 + x^3*d^4*b^2*a + x^3*d^4*c*a^2 + 4*x^3*e*d^3*b*a^2 + 2*x^3*e^2*d^2*a^3 + 3/2*x^2*d^4*b*a^2 +
2*x^2*e*d^3*a^3 + x*d^4*a^3

________________________________________________________________________________________

giac [B]  time = 0.20, size = 604, normalized size = 2.22 \begin {gather*} \frac {1}{11} \, c^{3} x^{11} e^{4} + \frac {2}{5} \, c^{3} d x^{10} e^{3} + \frac {2}{3} \, c^{3} d^{2} x^{9} e^{2} + \frac {1}{2} \, c^{3} d^{3} x^{8} e + \frac {1}{7} \, c^{3} d^{4} x^{7} + \frac {3}{10} \, b c^{2} x^{10} e^{4} + \frac {4}{3} \, b c^{2} d x^{9} e^{3} + \frac {9}{4} \, b c^{2} d^{2} x^{8} e^{2} + \frac {12}{7} \, b c^{2} d^{3} x^{7} e + \frac {1}{2} \, b c^{2} d^{4} x^{6} + \frac {1}{3} \, b^{2} c x^{9} e^{4} + \frac {1}{3} \, a c^{2} x^{9} e^{4} + \frac {3}{2} \, b^{2} c d x^{8} e^{3} + \frac {3}{2} \, a c^{2} d x^{8} e^{3} + \frac {18}{7} \, b^{2} c d^{2} x^{7} e^{2} + \frac {18}{7} \, a c^{2} d^{2} x^{7} e^{2} + 2 \, b^{2} c d^{3} x^{6} e + 2 \, a c^{2} d^{3} x^{6} e + \frac {3}{5} \, b^{2} c d^{4} x^{5} + \frac {3}{5} \, a c^{2} d^{4} x^{5} + \frac {1}{8} \, b^{3} x^{8} e^{4} + \frac {3}{4} \, a b c x^{8} e^{4} + \frac {4}{7} \, b^{3} d x^{7} e^{3} + \frac {24}{7} \, a b c d x^{7} e^{3} + b^{3} d^{2} x^{6} e^{2} + 6 \, a b c d^{2} x^{6} e^{2} + \frac {4}{5} \, b^{3} d^{3} x^{5} e + \frac {24}{5} \, a b c d^{3} x^{5} e + \frac {1}{4} \, b^{3} d^{4} x^{4} + \frac {3}{2} \, a b c d^{4} x^{4} + \frac {3}{7} \, a b^{2} x^{7} e^{4} + \frac {3}{7} \, a^{2} c x^{7} e^{4} + 2 \, a b^{2} d x^{6} e^{3} + 2 \, a^{2} c d x^{6} e^{3} + \frac {18}{5} \, a b^{2} d^{2} x^{5} e^{2} + \frac {18}{5} \, a^{2} c d^{2} x^{5} e^{2} + 3 \, a b^{2} d^{3} x^{4} e + 3 \, a^{2} c d^{3} x^{4} e + a b^{2} d^{4} x^{3} + a^{2} c d^{4} x^{3} + \frac {1}{2} \, a^{2} b x^{6} e^{4} + \frac {12}{5} \, a^{2} b d x^{5} e^{3} + \frac {9}{2} \, a^{2} b d^{2} x^{4} e^{2} + 4 \, a^{2} b d^{3} x^{3} e + \frac {3}{2} \, a^{2} b d^{4} x^{2} + \frac {1}{5} \, a^{3} x^{5} e^{4} + a^{3} d x^{4} e^{3} + 2 \, a^{3} d^{2} x^{3} e^{2} + 2 \, a^{3} d^{3} x^{2} e + a^{3} d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

1/11*c^3*x^11*e^4 + 2/5*c^3*d*x^10*e^3 + 2/3*c^3*d^2*x^9*e^2 + 1/2*c^3*d^3*x^8*e + 1/7*c^3*d^4*x^7 + 3/10*b*c^
2*x^10*e^4 + 4/3*b*c^2*d*x^9*e^3 + 9/4*b*c^2*d^2*x^8*e^2 + 12/7*b*c^2*d^3*x^7*e + 1/2*b*c^2*d^4*x^6 + 1/3*b^2*
c*x^9*e^4 + 1/3*a*c^2*x^9*e^4 + 3/2*b^2*c*d*x^8*e^3 + 3/2*a*c^2*d*x^8*e^3 + 18/7*b^2*c*d^2*x^7*e^2 + 18/7*a*c^
2*d^2*x^7*e^2 + 2*b^2*c*d^3*x^6*e + 2*a*c^2*d^3*x^6*e + 3/5*b^2*c*d^4*x^5 + 3/5*a*c^2*d^4*x^5 + 1/8*b^3*x^8*e^
4 + 3/4*a*b*c*x^8*e^4 + 4/7*b^3*d*x^7*e^3 + 24/7*a*b*c*d*x^7*e^3 + b^3*d^2*x^6*e^2 + 6*a*b*c*d^2*x^6*e^2 + 4/5
*b^3*d^3*x^5*e + 24/5*a*b*c*d^3*x^5*e + 1/4*b^3*d^4*x^4 + 3/2*a*b*c*d^4*x^4 + 3/7*a*b^2*x^7*e^4 + 3/7*a^2*c*x^
7*e^4 + 2*a*b^2*d*x^6*e^3 + 2*a^2*c*d*x^6*e^3 + 18/5*a*b^2*d^2*x^5*e^2 + 18/5*a^2*c*d^2*x^5*e^2 + 3*a*b^2*d^3*
x^4*e + 3*a^2*c*d^3*x^4*e + a*b^2*d^4*x^3 + a^2*c*d^4*x^3 + 1/2*a^2*b*x^6*e^4 + 12/5*a^2*b*d*x^5*e^3 + 9/2*a^2
*b*d^2*x^4*e^2 + 4*a^2*b*d^3*x^3*e + 3/2*a^2*b*d^4*x^2 + 1/5*a^3*x^5*e^4 + a^3*d*x^4*e^3 + 2*a^3*d^2*x^3*e^2 +
2*a^3*d^3*x^2*e + a^3*d^4*x

________________________________________________________________________________________

maple [B]  time = 0.04, size = 631, normalized size = 2.32 \begin {gather*} \frac {c^{3} e^{4} x^{11}}{11}+\frac {\left (3 e^{4} b \,c^{2}+4 d \,e^{3} c^{3}\right ) x^{10}}{10}+\frac {\left (12 b \,c^{2} d \,e^{3}+6 c^{3} d^{2} e^{2}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) e^{4}\right ) x^{9}}{9}+a^{3} d^{4} x +\frac {\left (18 b \,c^{2} d^{2} e^{2}+4 c^{3} d^{3} e +4 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d \,e^{3}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) e^{4}\right ) x^{8}}{8}+\frac {\left (12 b \,c^{2} d^{3} e +c^{3} d^{4}+6 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d^{2} e^{2}+4 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d \,e^{3}+\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) e^{4}\right ) x^{7}}{7}+\frac {\left (3 a^{2} b \,e^{4}+3 b \,c^{2} d^{4}+4 \left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d^{3} e +6 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d^{2} e^{2}+4 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d \,e^{3}\right ) x^{6}}{6}+\frac {\left (a^{3} e^{4}+12 a^{2} b d \,e^{3}+\left (a \,c^{2}+2 b^{2} c +\left (2 a c +b^{2}\right ) c \right ) d^{4}+4 \left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d^{3} e +6 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d^{2} e^{2}\right ) x^{5}}{5}+\frac {\left (4 a^{3} d \,e^{3}+18 a^{2} b \,d^{2} e^{2}+\left (4 a b c +\left (2 a c +b^{2}\right ) b \right ) d^{4}+4 \left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d^{3} e \right ) x^{4}}{4}+\frac {\left (6 a^{3} d^{2} e^{2}+12 a^{2} b \,d^{3} e +\left (a^{2} c +2 a \,b^{2}+\left (2 a c +b^{2}\right ) a \right ) d^{4}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{3}+3 d^{4} a^{2} b \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(c*x^2+b*x+a)^3,x)

[Out]

1/11*c^3*e^4*x^11+1/10*(3*b*c^2*e^4+4*c^3*d*e^3)*x^10+1/9*(6*d^2*e^2*c^3+12*d*e^3*b*c^2+e^4*(a*c^2+2*b^2*c+c*(
2*a*c+b^2)))*x^9+1/8*(4*d^3*e*c^3+18*d^2*e^2*b*c^2+4*d*e^3*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+e^4*(4*a*b*c+b*(2*a*c
+b^2)))*x^8+1/7*(d^4*c^3+12*d^3*e*b*c^2+6*d^2*e^2*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+4*d*e^3*(4*a*b*c+b*(2*a*c+b^2)
)+e^4*(a*(2*a*c+b^2)+2*a*b^2+c*a^2))*x^7+1/6*(3*d^4*b*c^2+4*d^3*e*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+6*d^2*e^2*(4*a
*b*c+b*(2*a*c+b^2))+4*d*e^3*(a*(2*a*c+b^2)+2*a*b^2+c*a^2)+3*e^4*a^2*b)*x^6+1/5*(d^4*(a*c^2+2*b^2*c+c*(2*a*c+b^
2))+4*d^3*e*(4*a*b*c+b*(2*a*c+b^2))+6*d^2*e^2*(a*(2*a*c+b^2)+2*a*b^2+c*a^2)+12*d*e^3*a^2*b+e^4*a^3)*x^5+1/4*(d
^4*(4*a*b*c+b*(2*a*c+b^2))+4*d^3*e*(a*(2*a*c+b^2)+2*a*b^2+c*a^2)+18*d^2*e^2*a^2*b+4*d*e^3*a^3)*x^4+1/3*(d^4*(a
*(2*a*c+b^2)+2*a*b^2+c*a^2)+12*d^3*e*a^2*b+6*d^2*e^2*a^3)*x^3+1/2*(4*a^3*d^3*e+3*a^2*b*d^4)*x^2+d^4*a^3*x

________________________________________________________________________________________

maxima [A]  time = 1.05, size = 484, normalized size = 1.78 \begin {gather*} \frac {1}{11} \, c^{3} e^{4} x^{11} + \frac {1}{10} \, {\left (4 \, c^{3} d e^{3} + 3 \, b c^{2} e^{4}\right )} x^{10} + \frac {1}{3} \, {\left (2 \, c^{3} d^{2} e^{2} + 4 \, b c^{2} d e^{3} + {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (4 \, c^{3} d^{3} e + 18 \, b c^{2} d^{2} e^{2} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{8} + a^{3} d^{4} x + \frac {1}{7} \, {\left (c^{3} d^{4} + 12 \, b c^{2} d^{3} e + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} + 4 \, {\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x^{7} + \frac {1}{2} \, {\left (b c^{2} d^{4} + a^{2} b e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e + 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{2} + 4 \, {\left (a b^{2} + a^{2} c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (12 \, a^{2} b d e^{3} + a^{3} e^{4} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} + 4 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e + 18 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (18 \, a^{2} b d^{2} e^{2} + 4 \, a^{3} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{4} + 12 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e\right )} x^{4} + {\left (4 \, a^{2} b d^{3} e + 2 \, a^{3} d^{2} e^{2} + {\left (a b^{2} + a^{2} c\right )} d^{4}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{4} + 4 \, a^{3} d^{3} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

1/11*c^3*e^4*x^11 + 1/10*(4*c^3*d*e^3 + 3*b*c^2*e^4)*x^10 + 1/3*(2*c^3*d^2*e^2 + 4*b*c^2*d*e^3 + (b^2*c + a*c^
2)*e^4)*x^9 + 1/8*(4*c^3*d^3*e + 18*b*c^2*d^2*e^2 + 12*(b^2*c + a*c^2)*d*e^3 + (b^3 + 6*a*b*c)*e^4)*x^8 + a^3*
d^4*x + 1/7*(c^3*d^4 + 12*b*c^2*d^3*e + 18*(b^2*c + a*c^2)*d^2*e^2 + 4*(b^3 + 6*a*b*c)*d*e^3 + 3*(a*b^2 + a^2*
c)*e^4)*x^7 + 1/2*(b*c^2*d^4 + a^2*b*e^4 + 4*(b^2*c + a*c^2)*d^3*e + 2*(b^3 + 6*a*b*c)*d^2*e^2 + 4*(a*b^2 + a^
2*c)*d*e^3)*x^6 + 1/5*(12*a^2*b*d*e^3 + a^3*e^4 + 3*(b^2*c + a*c^2)*d^4 + 4*(b^3 + 6*a*b*c)*d^3*e + 18*(a*b^2
+ a^2*c)*d^2*e^2)*x^5 + 1/4*(18*a^2*b*d^2*e^2 + 4*a^3*d*e^3 + (b^3 + 6*a*b*c)*d^4 + 12*(a*b^2 + a^2*c)*d^3*e)*
x^4 + (4*a^2*b*d^3*e + 2*a^3*d^2*e^2 + (a*b^2 + a^2*c)*d^4)*x^3 + 1/2*(3*a^2*b*d^4 + 4*a^3*d^3*e)*x^2

________________________________________________________________________________________

mupad [B]  time = 0.78, size = 506, normalized size = 1.86 \begin {gather*} x^4\,\left (a^3\,d\,e^3+\frac {9\,a^2\,b\,d^2\,e^2}{2}+3\,c\,a^2\,d^3\,e+3\,a\,b^2\,d^3\,e+\frac {3\,c\,a\,b\,d^4}{2}+\frac {b^3\,d^4}{4}\right )+x^8\,\left (\frac {b^3\,e^4}{8}+\frac {3\,b^2\,c\,d\,e^3}{2}+\frac {9\,b\,c^2\,d^2\,e^2}{4}+\frac {3\,a\,b\,c\,e^4}{4}+\frac {c^3\,d^3\,e}{2}+\frac {3\,a\,c^2\,d\,e^3}{2}\right )+x^6\,\left (\frac {a^2\,b\,e^4}{2}+2\,a^2\,c\,d\,e^3+2\,a\,b^2\,d\,e^3+6\,a\,b\,c\,d^2\,e^2+2\,a\,c^2\,d^3\,e+b^3\,d^2\,e^2+2\,b^2\,c\,d^3\,e+\frac {b\,c^2\,d^4}{2}\right )+x^5\,\left (\frac {a^3\,e^4}{5}+\frac {12\,a^2\,b\,d\,e^3}{5}+\frac {18\,a^2\,c\,d^2\,e^2}{5}+\frac {18\,a\,b^2\,d^2\,e^2}{5}+\frac {24\,a\,b\,c\,d^3\,e}{5}+\frac {3\,a\,c^2\,d^4}{5}+\frac {4\,b^3\,d^3\,e}{5}+\frac {3\,b^2\,c\,d^4}{5}\right )+x^7\,\left (\frac {3\,a^2\,c\,e^4}{7}+\frac {3\,a\,b^2\,e^4}{7}+\frac {24\,a\,b\,c\,d\,e^3}{7}+\frac {18\,a\,c^2\,d^2\,e^2}{7}+\frac {4\,b^3\,d\,e^3}{7}+\frac {18\,b^2\,c\,d^2\,e^2}{7}+\frac {12\,b\,c^2\,d^3\,e}{7}+\frac {c^3\,d^4}{7}\right )+a^3\,d^4\,x+\frac {c^3\,e^4\,x^{11}}{11}+a\,d^2\,x^3\,\left (2\,a^2\,e^2+4\,a\,b\,d\,e+c\,a\,d^2+b^2\,d^2\right )+\frac {c\,e^2\,x^9\,\left (b^2\,e^2+4\,b\,c\,d\,e+2\,c^2\,d^2+a\,c\,e^2\right )}{3}+\frac {a^2\,d^3\,x^2\,\left (4\,a\,e+3\,b\,d\right )}{2}+\frac {c^2\,e^3\,x^{10}\,\left (3\,b\,e+4\,c\,d\right )}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4*(a + b*x + c*x^2)^3,x)

[Out]

x^4*((b^3*d^4)/4 + a^3*d*e^3 + (9*a^2*b*d^2*e^2)/2 + (3*a*b*c*d^4)/2 + 3*a*b^2*d^3*e + 3*a^2*c*d^3*e) + x^8*((
b^3*e^4)/8 + (c^3*d^3*e)/2 + (9*b*c^2*d^2*e^2)/4 + (3*a*b*c*e^4)/4 + (3*a*c^2*d*e^3)/2 + (3*b^2*c*d*e^3)/2) +
x^6*((a^2*b*e^4)/2 + (b*c^2*d^4)/2 + b^3*d^2*e^2 + 2*a*b^2*d*e^3 + 2*a*c^2*d^3*e + 2*a^2*c*d*e^3 + 2*b^2*c*d^3
*e + 6*a*b*c*d^2*e^2) + x^5*((a^3*e^4)/5 + (3*a*c^2*d^4)/5 + (3*b^2*c*d^4)/5 + (4*b^3*d^3*e)/5 + (18*a*b^2*d^2
*e^2)/5 + (18*a^2*c*d^2*e^2)/5 + (12*a^2*b*d*e^3)/5 + (24*a*b*c*d^3*e)/5) + x^7*((c^3*d^4)/7 + (3*a*b^2*e^4)/7
+ (3*a^2*c*e^4)/7 + (4*b^3*d*e^3)/7 + (18*a*c^2*d^2*e^2)/7 + (18*b^2*c*d^2*e^2)/7 + (12*b*c^2*d^3*e)/7 + (24*
a*b*c*d*e^3)/7) + a^3*d^4*x + (c^3*e^4*x^11)/11 + a*d^2*x^3*(2*a^2*e^2 + b^2*d^2 + a*c*d^2 + 4*a*b*d*e) + (c*e
^2*x^9*(b^2*e^2 + 2*c^2*d^2 + a*c*e^2 + 4*b*c*d*e))/3 + (a^2*d^3*x^2*(4*a*e + 3*b*d))/2 + (c^2*e^3*x^10*(3*b*e
+ 4*c*d))/10

________________________________________________________________________________________

sympy [B]  time = 0.16, size = 620, normalized size = 2.28 \begin {gather*} a^{3} d^{4} x + \frac {c^{3} e^{4} x^{11}}{11} + x^{10} \left (\frac {3 b c^{2} e^{4}}{10} + \frac {2 c^{3} d e^{3}}{5}\right ) + x^{9} \left (\frac {a c^{2} e^{4}}{3} + \frac {b^{2} c e^{4}}{3} + \frac {4 b c^{2} d e^{3}}{3} + \frac {2 c^{3} d^{2} e^{2}}{3}\right ) + x^{8} \left (\frac {3 a b c e^{4}}{4} + \frac {3 a c^{2} d e^{3}}{2} + \frac {b^{3} e^{4}}{8} + \frac {3 b^{2} c d e^{3}}{2} + \frac {9 b c^{2} d^{2} e^{2}}{4} + \frac {c^{3} d^{3} e}{2}\right ) + x^{7} \left (\frac {3 a^{2} c e^{4}}{7} + \frac {3 a b^{2} e^{4}}{7} + \frac {24 a b c d e^{3}}{7} + \frac {18 a c^{2} d^{2} e^{2}}{7} + \frac {4 b^{3} d e^{3}}{7} + \frac {18 b^{2} c d^{2} e^{2}}{7} + \frac {12 b c^{2} d^{3} e}{7} + \frac {c^{3} d^{4}}{7}\right ) + x^{6} \left (\frac {a^{2} b e^{4}}{2} + 2 a^{2} c d e^{3} + 2 a b^{2} d e^{3} + 6 a b c d^{2} e^{2} + 2 a c^{2} d^{3} e + b^{3} d^{2} e^{2} + 2 b^{2} c d^{3} e + \frac {b c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac {a^{3} e^{4}}{5} + \frac {12 a^{2} b d e^{3}}{5} + \frac {18 a^{2} c d^{2} e^{2}}{5} + \frac {18 a b^{2} d^{2} e^{2}}{5} + \frac {24 a b c d^{3} e}{5} + \frac {3 a c^{2} d^{4}}{5} + \frac {4 b^{3} d^{3} e}{5} + \frac {3 b^{2} c d^{4}}{5}\right ) + x^{4} \left (a^{3} d e^{3} + \frac {9 a^{2} b d^{2} e^{2}}{2} + 3 a^{2} c d^{3} e + 3 a b^{2} d^{3} e + \frac {3 a b c d^{4}}{2} + \frac {b^{3} d^{4}}{4}\right ) + x^{3} \left (2 a^{3} d^{2} e^{2} + 4 a^{2} b d^{3} e + a^{2} c d^{4} + a b^{2} d^{4}\right ) + x^{2} \left (2 a^{3} d^{3} e + \frac {3 a^{2} b d^{4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(c*x**2+b*x+a)**3,x)

[Out]

a**3*d**4*x + c**3*e**4*x**11/11 + x**10*(3*b*c**2*e**4/10 + 2*c**3*d*e**3/5) + x**9*(a*c**2*e**4/3 + b**2*c*e
**4/3 + 4*b*c**2*d*e**3/3 + 2*c**3*d**2*e**2/3) + x**8*(3*a*b*c*e**4/4 + 3*a*c**2*d*e**3/2 + b**3*e**4/8 + 3*b
**2*c*d*e**3/2 + 9*b*c**2*d**2*e**2/4 + c**3*d**3*e/2) + x**7*(3*a**2*c*e**4/7 + 3*a*b**2*e**4/7 + 24*a*b*c*d*
e**3/7 + 18*a*c**2*d**2*e**2/7 + 4*b**3*d*e**3/7 + 18*b**2*c*d**2*e**2/7 + 12*b*c**2*d**3*e/7 + c**3*d**4/7) +
x**6*(a**2*b*e**4/2 + 2*a**2*c*d*e**3 + 2*a*b**2*d*e**3 + 6*a*b*c*d**2*e**2 + 2*a*c**2*d**3*e + b**3*d**2*e**
2 + 2*b**2*c*d**3*e + b*c**2*d**4/2) + x**5*(a**3*e**4/5 + 12*a**2*b*d*e**3/5 + 18*a**2*c*d**2*e**2/5 + 18*a*b
**2*d**2*e**2/5 + 24*a*b*c*d**3*e/5 + 3*a*c**2*d**4/5 + 4*b**3*d**3*e/5 + 3*b**2*c*d**4/5) + x**4*(a**3*d*e**3
+ 9*a**2*b*d**2*e**2/2 + 3*a**2*c*d**3*e + 3*a*b**2*d**3*e + 3*a*b*c*d**4/2 + b**3*d**4/4) + x**3*(2*a**3*d**
2*e**2 + 4*a**2*b*d**3*e + a**2*c*d**4 + a*b**2*d**4) + x**2*(2*a**3*d**3*e + 3*a**2*b*d**4/2)

________________________________________________________________________________________