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

Optimal. Leaf size=240 \[ \frac {c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^6}-\frac {(d+e x)^7 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^6}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac {(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {5 c^2 (d+e x)^9 (2 c d-b e)}{9 e^6}+\frac {c^3 (d+e x)^{10}}{5 e^6} \]

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Rubi [A]  time = 0.42, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} \frac {c (d+e x)^8 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^6}-\frac {(d+e x)^7 (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 )}{7 e^6}+\frac {(d+e x)^6 \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 )}{3 e^6}-\frac {(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {5 c^2 (d+e x)^9 (2 c d-b e)}{9 e^6}+\frac {c^3 (d+e x)^{10}}{5 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

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

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 433, normalized size = 1.80 \begin {gather*} \frac {1}{3} x^6 \left (c e^2 \left (a^2 e^2+12 a b d e+12 b^2 d^2\right )+b^2 e^3 (a e+2 b d)+2 c^2 d^2 e (6 a e+5 b d)+c^3 d^4\right )+\frac {1}{5} x^5 \left (b \left (a^2 e^4+36 a c d^2 e^2+5 c^2 d^4\right )+8 b^2 \left (a d e^3+2 c d^3 e\right )+8 a c d e \left (a e^2+2 c d^2\right )+6 b^3 d^2 e^2\right )+\frac {1}{3} d^2 x^3 \left (8 a^2 c d e+8 a b^2 d e+6 a b \left (a e^2+c d^2\right )+b^3 d^2\right )+a^2 b d^4 x+\frac {1}{7} e x^7 \left (2 c^2 d e (8 a e+15 b d)+2 b c e^2 (3 a e+8 b d)+b^3 e^3+8 c^3 d^3\right )+\frac {1}{2} c e^2 x^8 \left (c e (a e+5 b d)+b^2 e^2+3 c^2 d^2\right )+a d^3 x^2 \left (2 a b e+a c d+b^2 d\right )+d x^4 \left (b^2 \left (3 a d e^2+c d^3\right )+a b e \left (a e^2+6 c d^2\right )+a c d \left (3 a e^2+c d^2\right )+b^3 d^2 e\right )+\frac {1}{9} c^2 e^3 x^9 (5 b e+8 c d)+\frac {1}{5} c^3 e^4 x^{10} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*b*d^4*x + a*d^3*(b^2*d + a*c*d + 2*a*b*e)*x^2 + (d^2*(b^3*d^2 + 8*a*b^2*d*e + 8*a^2*c*d*e + 6*a*b*(c*d^2 +
a*e^2))*x^3)/3 + d*(b^3*d^2*e + a*b*e*(6*c*d^2 + a*e^2) + a*c*d*(c*d^2 + 3*a*e^2) + b^2*(c*d^3 + 3*a*d*e^2))*
x^4 + ((6*b^3*d^2*e^2 + 8*a*c*d*e*(2*c*d^2 + a*e^2) + 8*b^2*(2*c*d^3*e + a*d*e^3) + b*(5*c^2*d^4 + 36*a*c*d^2*
e^2 + a^2*e^4))*x^5)/5 + ((c^3*d^4 + b^2*e^3*(2*b*d + a*e) + 2*c^2*d^2*e*(5*b*d + 6*a*e) + c*e^2*(12*b^2*d^2 +
12*a*b*d*e + a^2*e^2))*x^6)/3 + (e*(8*c^3*d^3 + b^3*e^3 + 2*b*c*e^2*(8*b*d + 3*a*e) + 2*c^2*d*e*(15*b*d + 8*a
*e))*x^7)/7 + (c*e^2*(3*c^2*d^2 + b^2*e^2 + c*e*(5*b*d + a*e))*x^8)/2 + (c^2*e^3*(8*c*d + 5*b*e)*x^9)/9 + (c^3
*e^4*x^10)/5

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.38, size = 561, normalized size = 2.34 \begin {gather*} \frac {1}{5} x^{10} e^{4} c^{3} + \frac {8}{9} x^{9} e^{3} d c^{3} + \frac {5}{9} x^{9} e^{4} c^{2} b + \frac {3}{2} x^{8} e^{2} d^{2} c^{3} + \frac {5}{2} x^{8} e^{3} d c^{2} b + \frac {1}{2} x^{8} e^{4} c b^{2} + \frac {1}{2} x^{8} e^{4} c^{2} a + \frac {8}{7} x^{7} e d^{3} c^{3} + \frac {30}{7} x^{7} e^{2} d^{2} c^{2} b + \frac {16}{7} x^{7} e^{3} d c b^{2} + \frac {1}{7} x^{7} e^{4} b^{3} + \frac {16}{7} x^{7} e^{3} d c^{2} a + \frac {6}{7} x^{7} e^{4} c b a + \frac {1}{3} x^{6} d^{4} c^{3} + \frac {10}{3} x^{6} e d^{3} c^{2} b + 4 x^{6} e^{2} d^{2} c b^{2} + \frac {2}{3} x^{6} e^{3} d b^{3} + 4 x^{6} e^{2} d^{2} c^{2} a + 4 x^{6} e^{3} d c b a + \frac {1}{3} x^{6} e^{4} b^{2} a + \frac {1}{3} x^{6} e^{4} c a^{2} + x^{5} d^{4} c^{2} b + \frac {16}{5} x^{5} e d^{3} c b^{2} + \frac {6}{5} x^{5} e^{2} d^{2} b^{3} + \frac {16}{5} x^{5} e d^{3} c^{2} a + \frac {36}{5} x^{5} e^{2} d^{2} c b a + \frac {8}{5} x^{5} e^{3} d b^{2} a + \frac {8}{5} x^{5} e^{3} d c a^{2} + \frac {1}{5} x^{5} e^{4} b a^{2} + x^{4} d^{4} c b^{2} + x^{4} e d^{3} b^{3} + x^{4} d^{4} c^{2} a + 6 x^{4} e d^{3} c b a + 3 x^{4} e^{2} d^{2} b^{2} a + 3 x^{4} e^{2} d^{2} c a^{2} + x^{4} e^{3} d b a^{2} + \frac {1}{3} x^{3} d^{4} b^{3} + 2 x^{3} d^{4} c b a + \frac {8}{3} x^{3} e d^{3} b^{2} a + \frac {8}{3} x^{3} e d^{3} c a^{2} + 2 x^{3} e^{2} d^{2} b a^{2} + x^{2} d^{4} b^{2} a + x^{2} d^{4} c a^{2} + 2 x^{2} e d^{3} b a^{2} + x d^{4} b a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^10*e^4*c^3 + 8/9*x^9*e^3*d*c^3 + 5/9*x^9*e^4*c^2*b + 3/2*x^8*e^2*d^2*c^3 + 5/2*x^8*e^3*d*c^2*b + 1/2*x^8
*e^4*c*b^2 + 1/2*x^8*e^4*c^2*a + 8/7*x^7*e*d^3*c^3 + 30/7*x^7*e^2*d^2*c^2*b + 16/7*x^7*e^3*d*c*b^2 + 1/7*x^7*e
^4*b^3 + 16/7*x^7*e^3*d*c^2*a + 6/7*x^7*e^4*c*b*a + 1/3*x^6*d^4*c^3 + 10/3*x^6*e*d^3*c^2*b + 4*x^6*e^2*d^2*c*b
^2 + 2/3*x^6*e^3*d*b^3 + 4*x^6*e^2*d^2*c^2*a + 4*x^6*e^3*d*c*b*a + 1/3*x^6*e^4*b^2*a + 1/3*x^6*e^4*c*a^2 + x^5
*d^4*c^2*b + 16/5*x^5*e*d^3*c*b^2 + 6/5*x^5*e^2*d^2*b^3 + 16/5*x^5*e*d^3*c^2*a + 36/5*x^5*e^2*d^2*c*b*a + 8/5*
x^5*e^3*d*b^2*a + 8/5*x^5*e^3*d*c*a^2 + 1/5*x^5*e^4*b*a^2 + x^4*d^4*c*b^2 + x^4*e*d^3*b^3 + x^4*d^4*c^2*a + 6*
x^4*e*d^3*c*b*a + 3*x^4*e^2*d^2*b^2*a + 3*x^4*e^2*d^2*c*a^2 + x^4*e^3*d*b*a^2 + 1/3*x^3*d^4*b^3 + 2*x^3*d^4*c*
b*a + 8/3*x^3*e*d^3*b^2*a + 8/3*x^3*e*d^3*c*a^2 + 2*x^3*e^2*d^2*b*a^2 + x^2*d^4*b^2*a + x^2*d^4*c*a^2 + 2*x^2*
e*d^3*b*a^2 + x*d^4*b*a^2

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giac [B]  time = 0.17, size = 543, normalized size = 2.26 \begin {gather*} \frac {1}{5} \, c^{3} x^{10} e^{4} + \frac {8}{9} \, c^{3} d x^{9} e^{3} + \frac {3}{2} \, c^{3} d^{2} x^{8} e^{2} + \frac {8}{7} \, c^{3} d^{3} x^{7} e + \frac {1}{3} \, c^{3} d^{4} x^{6} + \frac {5}{9} \, b c^{2} x^{9} e^{4} + \frac {5}{2} \, b c^{2} d x^{8} e^{3} + \frac {30}{7} \, b c^{2} d^{2} x^{7} e^{2} + \frac {10}{3} \, b c^{2} d^{3} x^{6} e + b c^{2} d^{4} x^{5} + \frac {1}{2} \, b^{2} c x^{8} e^{4} + \frac {1}{2} \, a c^{2} x^{8} e^{4} + \frac {16}{7} \, b^{2} c d x^{7} e^{3} + \frac {16}{7} \, a c^{2} d x^{7} e^{3} + 4 \, b^{2} c d^{2} x^{6} e^{2} + 4 \, a c^{2} d^{2} x^{6} e^{2} + \frac {16}{5} \, b^{2} c d^{3} x^{5} e + \frac {16}{5} \, a c^{2} d^{3} x^{5} e + b^{2} c d^{4} x^{4} + a c^{2} d^{4} x^{4} + \frac {1}{7} \, b^{3} x^{7} e^{4} + \frac {6}{7} \, a b c x^{7} e^{4} + \frac {2}{3} \, b^{3} d x^{6} e^{3} + 4 \, a b c d x^{6} e^{3} + \frac {6}{5} \, b^{3} d^{2} x^{5} e^{2} + \frac {36}{5} \, a b c d^{2} x^{5} e^{2} + b^{3} d^{3} x^{4} e + 6 \, a b c d^{3} x^{4} e + \frac {1}{3} \, b^{3} d^{4} x^{3} + 2 \, a b c d^{4} x^{3} + \frac {1}{3} \, a b^{2} x^{6} e^{4} + \frac {1}{3} \, a^{2} c x^{6} e^{4} + \frac {8}{5} \, a b^{2} d x^{5} e^{3} + \frac {8}{5} \, a^{2} c d x^{5} e^{3} + 3 \, a b^{2} d^{2} x^{4} e^{2} + 3 \, a^{2} c d^{2} x^{4} e^{2} + \frac {8}{3} \, a b^{2} d^{3} x^{3} e + \frac {8}{3} \, a^{2} c d^{3} x^{3} e + a b^{2} d^{4} x^{2} + a^{2} c d^{4} x^{2} + \frac {1}{5} \, a^{2} b x^{5} e^{4} + a^{2} b d x^{4} e^{3} + 2 \, a^{2} b d^{2} x^{3} e^{2} + 2 \, a^{2} b d^{3} x^{2} e + a^{2} b d^{4} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^3*x^10*e^4 + 8/9*c^3*d*x^9*e^3 + 3/2*c^3*d^2*x^8*e^2 + 8/7*c^3*d^3*x^7*e + 1/3*c^3*d^4*x^6 + 5/9*b*c^2*x
^9*e^4 + 5/2*b*c^2*d*x^8*e^3 + 30/7*b*c^2*d^2*x^7*e^2 + 10/3*b*c^2*d^3*x^6*e + b*c^2*d^4*x^5 + 1/2*b^2*c*x^8*e
^4 + 1/2*a*c^2*x^8*e^4 + 16/7*b^2*c*d*x^7*e^3 + 16/7*a*c^2*d*x^7*e^3 + 4*b^2*c*d^2*x^6*e^2 + 4*a*c^2*d^2*x^6*e
^2 + 16/5*b^2*c*d^3*x^5*e + 16/5*a*c^2*d^3*x^5*e + b^2*c*d^4*x^4 + a*c^2*d^4*x^4 + 1/7*b^3*x^7*e^4 + 6/7*a*b*c
*x^7*e^4 + 2/3*b^3*d*x^6*e^3 + 4*a*b*c*d*x^6*e^3 + 6/5*b^3*d^2*x^5*e^2 + 36/5*a*b*c*d^2*x^5*e^2 + b^3*d^3*x^4*
e + 6*a*b*c*d^3*x^4*e + 1/3*b^3*d^4*x^3 + 2*a*b*c*d^4*x^3 + 1/3*a*b^2*x^6*e^4 + 1/3*a^2*c*x^6*e^4 + 8/5*a*b^2*
d*x^5*e^3 + 8/5*a^2*c*d*x^5*e^3 + 3*a*b^2*d^2*x^4*e^2 + 3*a^2*c*d^2*x^4*e^2 + 8/3*a*b^2*d^3*x^3*e + 8/3*a^2*c*
d^3*x^3*e + a*b^2*d^4*x^2 + a^2*c*d^4*x^2 + 1/5*a^2*b*x^5*e^4 + a^2*b*d*x^4*e^3 + 2*a^2*b*d^2*x^3*e^2 + 2*a^2*
b*d^3*x^2*e + a^2*b*d^4*x

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maple [B]  time = 0.04, size = 554, normalized size = 2.31 \begin {gather*} \frac {c^{3} e^{4} x^{10}}{5}+\frac {\left (4 b \,c^{2} e^{4}+\left (b \,e^{4}+8 c d \,e^{3}\right ) c^{2}\right ) x^{9}}{9}+a^{2} b \,d^{4} x +\frac {\left (2 \left (2 a c +b^{2}\right ) c \,e^{4}+2 \left (b \,e^{4}+8 c d \,e^{3}\right ) b c +\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) c^{2}\right ) x^{8}}{8}+\frac {\left (4 a b c \,e^{4}+2 \left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) b c +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) c^{2}+\left (b \,e^{4}+8 c d \,e^{3}\right ) \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (2 a^{2} c \,e^{4}+2 \left (b \,e^{4}+8 c d \,e^{3}\right ) a b +2 \left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) b c +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) c^{2}+\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) \left (2 a c +b^{2}\right )\right ) x^{6}}{6}+\frac {\left (b \,c^{2} d^{4}+\left (b \,e^{4}+8 c d \,e^{3}\right ) a^{2}+2 \left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) a b +2 \left (4 b \,d^{3} e +2 c \,d^{4}\right ) b c +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 b^{2} c \,d^{4}+\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) a^{2}+2 \left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a b +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) b \,d^{4}+\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a^{2}+2 \left (4 b \,d^{3} e +2 c \,d^{4}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 a \,b^{2} d^{4}+\left (4 b \,d^{3} e +2 c \,d^{4}\right ) a^{2}\right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 430, normalized size = 1.79 \begin {gather*} \frac {1}{5} \, c^{3} e^{4} x^{10} + \frac {1}{9} \, {\left (8 \, c^{3} d e^{3} + 5 \, b c^{2} e^{4}\right )} x^{9} + \frac {1}{2} \, {\left (3 \, c^{3} d^{2} e^{2} + 5 \, b c^{2} d e^{3} + {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{8} + a^{2} b d^{4} x + \frac {1}{7} \, {\left (8 \, c^{3} d^{3} e + 30 \, b c^{2} d^{2} e^{2} + 16 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} + {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{7} + \frac {1}{3} \, {\left (c^{3} d^{4} + 10 \, b c^{2} d^{3} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, b c^{2} d^{4} + a^{2} b e^{4} + 16 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e + 6 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{2} + 8 \, {\left (a b^{2} + a^{2} c\right )} d e^{3}\right )} x^{5} + {\left (a^{2} b d e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{4} + {\left (b^{3} + 6 \, a b c\right )} d^{3} e + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b d^{2} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{4} + 8 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e\right )} x^{3} + {\left (2 \, a^{2} b d^{3} e + {\left (a b^{2} + a^{2} c\right )} d^{4}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^3*e^4*x^10 + 1/9*(8*c^3*d*e^3 + 5*b*c^2*e^4)*x^9 + 1/2*(3*c^3*d^2*e^2 + 5*b*c^2*d*e^3 + (b^2*c + a*c^2)*
e^4)*x^8 + a^2*b*d^4*x + 1/7*(8*c^3*d^3*e + 30*b*c^2*d^2*e^2 + 16*(b^2*c + a*c^2)*d*e^3 + (b^3 + 6*a*b*c)*e^4)
*x^7 + 1/3*(c^3*d^4 + 10*b*c^2*d^3*e + 12*(b^2*c + a*c^2)*d^2*e^2 + 2*(b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c)*
e^4)*x^6 + 1/5*(5*b*c^2*d^4 + a^2*b*e^4 + 16*(b^2*c + a*c^2)*d^3*e + 6*(b^3 + 6*a*b*c)*d^2*e^2 + 8*(a*b^2 + a^
2*c)*d*e^3)*x^5 + (a^2*b*d*e^3 + (b^2*c + a*c^2)*d^4 + (b^3 + 6*a*b*c)*d^3*e + 3*(a*b^2 + a^2*c)*d^2*e^2)*x^4
+ 1/3*(6*a^2*b*d^2*e^2 + (b^3 + 6*a*b*c)*d^4 + 8*(a*b^2 + a^2*c)*d^3*e)*x^3 + (2*a^2*b*d^3*e + (a*b^2 + a^2*c)
*d^4)*x^2

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mupad [B]  time = 1.88, size = 454, normalized size = 1.89 \begin {gather*} x^3\,\left (2\,a^2\,b\,d^2\,e^2+\frac {8\,c\,a^2\,d^3\,e}{3}+\frac {8\,a\,b^2\,d^3\,e}{3}+2\,c\,a\,b\,d^4+\frac {b^3\,d^4}{3}\right )+x^7\,\left (\frac {b^3\,e^4}{7}+\frac {16\,b^2\,c\,d\,e^3}{7}+\frac {30\,b\,c^2\,d^2\,e^2}{7}+\frac {6\,a\,b\,c\,e^4}{7}+\frac {8\,c^3\,d^3\,e}{7}+\frac {16\,a\,c^2\,d\,e^3}{7}\right )+x^5\,\left (\frac {a^2\,b\,e^4}{5}+\frac {8\,a^2\,c\,d\,e^3}{5}+\frac {8\,a\,b^2\,d\,e^3}{5}+\frac {36\,a\,b\,c\,d^2\,e^2}{5}+\frac {16\,a\,c^2\,d^3\,e}{5}+\frac {6\,b^3\,d^2\,e^2}{5}+\frac {16\,b^2\,c\,d^3\,e}{5}+b\,c^2\,d^4\right )+x^6\,\left (\frac {a^2\,c\,e^4}{3}+\frac {a\,b^2\,e^4}{3}+4\,a\,b\,c\,d\,e^3+4\,a\,c^2\,d^2\,e^2+\frac {2\,b^3\,d\,e^3}{3}+4\,b^2\,c\,d^2\,e^2+\frac {10\,b\,c^2\,d^3\,e}{3}+\frac {c^3\,d^4}{3}\right )+x^4\,\left (a^2\,b\,d\,e^3+3\,a^2\,c\,d^2\,e^2+3\,a\,b^2\,d^2\,e^2+6\,a\,b\,c\,d^3\,e+a\,c^2\,d^4+b^3\,d^3\,e+b^2\,c\,d^4\right )+\frac {c^3\,e^4\,x^{10}}{5}+a\,d^3\,x^2\,\left (d\,b^2+2\,a\,e\,b+a\,c\,d\right )+\frac {c\,e^2\,x^8\,\left (b^2\,e^2+5\,b\,c\,d\,e+3\,c^2\,d^2+a\,c\,e^2\right )}{2}+\frac {c^2\,e^3\,x^9\,\left (5\,b\,e+8\,c\,d\right )}{9}+a^2\,b\,d^4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((b^3*d^4)/3 + 2*a^2*b*d^2*e^2 + 2*a*b*c*d^4 + (8*a*b^2*d^3*e)/3 + (8*a^2*c*d^3*e)/3) + x^7*((b^3*e^4)/7 +
(8*c^3*d^3*e)/7 + (30*b*c^2*d^2*e^2)/7 + (6*a*b*c*e^4)/7 + (16*a*c^2*d*e^3)/7 + (16*b^2*c*d*e^3)/7) + x^5*((a
^2*b*e^4)/5 + b*c^2*d^4 + (6*b^3*d^2*e^2)/5 + (8*a*b^2*d*e^3)/5 + (16*a*c^2*d^3*e)/5 + (8*a^2*c*d*e^3)/5 + (16
*b^2*c*d^3*e)/5 + (36*a*b*c*d^2*e^2)/5) + x^6*((c^3*d^4)/3 + (a*b^2*e^4)/3 + (a^2*c*e^4)/3 + (2*b^3*d*e^3)/3 +
4*a*c^2*d^2*e^2 + 4*b^2*c*d^2*e^2 + (10*b*c^2*d^3*e)/3 + 4*a*b*c*d*e^3) + x^4*(a*c^2*d^4 + b^2*c*d^4 + b^3*d^
3*e + 3*a*b^2*d^2*e^2 + 3*a^2*c*d^2*e^2 + a^2*b*d*e^3 + 6*a*b*c*d^3*e) + (c^3*e^4*x^10)/5 + a*d^3*x^2*(b^2*d +
2*a*b*e + a*c*d) + (c*e^2*x^8*(b^2*e^2 + 3*c^2*d^2 + a*c*e^2 + 5*b*c*d*e))/2 + (c^2*e^3*x^9*(5*b*e + 8*c*d))/
9 + a^2*b*d^4*x

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sympy [B]  time = 0.16, size = 552, normalized size = 2.30 \begin {gather*} a^{2} b d^{4} x + \frac {c^{3} e^{4} x^{10}}{5} + x^{9} \left (\frac {5 b c^{2} e^{4}}{9} + \frac {8 c^{3} d e^{3}}{9}\right ) + x^{8} \left (\frac {a c^{2} e^{4}}{2} + \frac {b^{2} c e^{4}}{2} + \frac {5 b c^{2} d e^{3}}{2} + \frac {3 c^{3} d^{2} e^{2}}{2}\right ) + x^{7} \left (\frac {6 a b c e^{4}}{7} + \frac {16 a c^{2} d e^{3}}{7} + \frac {b^{3} e^{4}}{7} + \frac {16 b^{2} c d e^{3}}{7} + \frac {30 b c^{2} d^{2} e^{2}}{7} + \frac {8 c^{3} d^{3} e}{7}\right ) + x^{6} \left (\frac {a^{2} c e^{4}}{3} + \frac {a b^{2} e^{4}}{3} + 4 a b c d e^{3} + 4 a c^{2} d^{2} e^{2} + \frac {2 b^{3} d e^{3}}{3} + 4 b^{2} c d^{2} e^{2} + \frac {10 b c^{2} d^{3} e}{3} + \frac {c^{3} d^{4}}{3}\right ) + x^{5} \left (\frac {a^{2} b e^{4}}{5} + \frac {8 a^{2} c d e^{3}}{5} + \frac {8 a b^{2} d e^{3}}{5} + \frac {36 a b c d^{2} e^{2}}{5} + \frac {16 a c^{2} d^{3} e}{5} + \frac {6 b^{3} d^{2} e^{2}}{5} + \frac {16 b^{2} c d^{3} e}{5} + b c^{2} d^{4}\right ) + x^{4} \left (a^{2} b d e^{3} + 3 a^{2} c d^{2} e^{2} + 3 a b^{2} d^{2} e^{2} + 6 a b c d^{3} e + a c^{2} d^{4} + b^{3} d^{3} e + b^{2} c d^{4}\right ) + x^{3} \left (2 a^{2} b d^{2} e^{2} + \frac {8 a^{2} c d^{3} e}{3} + \frac {8 a b^{2} d^{3} e}{3} + 2 a b c d^{4} + \frac {b^{3} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} b d^{3} e + a^{2} c d^{4} + a b^{2} d^{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*b*d**4*x + c**3*e**4*x**10/5 + x**9*(5*b*c**2*e**4/9 + 8*c**3*d*e**3/9) + x**8*(a*c**2*e**4/2 + b**2*c*e*
*4/2 + 5*b*c**2*d*e**3/2 + 3*c**3*d**2*e**2/2) + x**7*(6*a*b*c*e**4/7 + 16*a*c**2*d*e**3/7 + b**3*e**4/7 + 16*
b**2*c*d*e**3/7 + 30*b*c**2*d**2*e**2/7 + 8*c**3*d**3*e/7) + x**6*(a**2*c*e**4/3 + a*b**2*e**4/3 + 4*a*b*c*d*e
**3 + 4*a*c**2*d**2*e**2 + 2*b**3*d*e**3/3 + 4*b**2*c*d**2*e**2 + 10*b*c**2*d**3*e/3 + c**3*d**4/3) + x**5*(a*
*2*b*e**4/5 + 8*a**2*c*d*e**3/5 + 8*a*b**2*d*e**3/5 + 36*a*b*c*d**2*e**2/5 + 16*a*c**2*d**3*e/5 + 6*b**3*d**2*
e**2/5 + 16*b**2*c*d**3*e/5 + b*c**2*d**4) + x**4*(a**2*b*d*e**3 + 3*a**2*c*d**2*e**2 + 3*a*b**2*d**2*e**2 + 6
*a*b*c*d**3*e + a*c**2*d**4 + b**3*d**3*e + b**2*c*d**4) + x**3*(2*a**2*b*d**2*e**2 + 8*a**2*c*d**3*e/3 + 8*a*
b**2*d**3*e/3 + 2*a*b*c*d**4 + b**3*d**4/3) + x**2*(2*a**2*b*d**3*e + a**2*c*d**4 + a*b**2*d**4)

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