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

Optimal. Leaf size=240 \[ \frac {2 c (d+e x)^6 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac {(d+e x)^5 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^6}-\frac {(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {5 c^2 (d+e x)^7 (2 c d-b e)}{7 e^6}+\frac {c^3 (d+e x)^8}{4 e^6} \]

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Rubi [A]  time = 0.23, 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 {2 c (d+e x)^6 \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 (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^6}+\frac {(d+e x)^4 \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 )}{2 e^6}-\frac {(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {5 c^2 (d+e x)^7 (2 c d-b e)}{7 e^6}+\frac {c^3 (d+e x)^8}{4 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^2*(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)^3)/(3*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)^4)/(2*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)^5)/(5*e^6) + (2*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^6)/(3*e^6) - (5*c^2*(2*c*d - b*e)
*(d + e*x)^7)/(7*e^6) + (c^3*(d + e*x)^8)/(4*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)^2 \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)^2}{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)^3}{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)^4}{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)^5}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^6}{e^5}+\frac {2 c^3 (d+e x)^7}{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)^3}{3 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)^4}{2 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)^5}{5 e^6}+\frac {2 c \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 {5 c^2 (2 c d-b e) (d+e x)^7}{7 e^6}+\frac {c^3 (d+e x)^8}{4 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 244, normalized size = 1.02 \begin {gather*} \frac {1}{3} x^3 \left (4 a^2 c d e+4 a b^2 d e+a b \left (a e^2+6 c d^2\right )+b^3 d^2\right )+a^2 b d^2 x+\frac {1}{3} c x^6 \left (c e (2 a e+5 b d)+2 b^2 e^2+c^2 d^2\right )+a d x^2 \left (a b e+a c d+b^2 d\right )+\frac {1}{5} x^5 \left (b c \left (6 a e^2+5 c d^2\right )+8 a c^2 d e+b^3 e^2+8 b^2 c d e\right )+\frac {1}{2} x^4 \left (b^2 \left (a e^2+2 c d^2\right )+6 a b c d e+a c \left (a e^2+2 c d^2\right )+b^3 d e\right )+\frac {1}{7} c^2 e x^7 (5 b e+4 c d)+\frac {1}{4} c^3 e^2 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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)^2 \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)^2*(a + b*x + c*x^2)^2,x]

[Out]

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

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fricas [A]  time = 0.40, size = 298, normalized size = 1.24 \begin {gather*} \frac {1}{4} x^{8} e^{2} c^{3} + \frac {4}{7} x^{7} e d c^{3} + \frac {5}{7} x^{7} e^{2} c^{2} b + \frac {1}{3} x^{6} d^{2} c^{3} + \frac {5}{3} x^{6} e d c^{2} b + \frac {2}{3} x^{6} e^{2} c b^{2} + \frac {2}{3} x^{6} e^{2} c^{2} a + x^{5} d^{2} c^{2} b + \frac {8}{5} x^{5} e d c b^{2} + \frac {1}{5} x^{5} e^{2} b^{3} + \frac {8}{5} x^{5} e d c^{2} a + \frac {6}{5} x^{5} e^{2} c b a + x^{4} d^{2} c b^{2} + \frac {1}{2} x^{4} e d b^{3} + x^{4} d^{2} c^{2} a + 3 x^{4} e d c b a + \frac {1}{2} x^{4} e^{2} b^{2} a + \frac {1}{2} x^{4} e^{2} c a^{2} + \frac {1}{3} x^{3} d^{2} b^{3} + 2 x^{3} d^{2} c b a + \frac {4}{3} x^{3} e d b^{2} a + \frac {4}{3} x^{3} e d c a^{2} + \frac {1}{3} x^{3} e^{2} b a^{2} + x^{2} d^{2} b^{2} a + x^{2} d^{2} c a^{2} + x^{2} e d b a^{2} + x d^{2} b a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^8*e^2*c^3 + 4/7*x^7*e*d*c^3 + 5/7*x^7*e^2*c^2*b + 1/3*x^6*d^2*c^3 + 5/3*x^6*e*d*c^2*b + 2/3*x^6*e^2*c*b^
2 + 2/3*x^6*e^2*c^2*a + x^5*d^2*c^2*b + 8/5*x^5*e*d*c*b^2 + 1/5*x^5*e^2*b^3 + 8/5*x^5*e*d*c^2*a + 6/5*x^5*e^2*
c*b*a + x^4*d^2*c*b^2 + 1/2*x^4*e*d*b^3 + x^4*d^2*c^2*a + 3*x^4*e*d*c*b*a + 1/2*x^4*e^2*b^2*a + 1/2*x^4*e^2*c*
a^2 + 1/3*x^3*d^2*b^3 + 2*x^3*d^2*c*b*a + 4/3*x^3*e*d*b^2*a + 4/3*x^3*e*d*c*a^2 + 1/3*x^3*e^2*b*a^2 + x^2*d^2*
b^2*a + x^2*d^2*c*a^2 + x^2*e*d*b*a^2 + x*d^2*b*a^2

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giac [A]  time = 0.17, size = 298, normalized size = 1.24 \begin {gather*} \frac {1}{4} \, c^{3} x^{8} e^{2} + \frac {4}{7} \, c^{3} d x^{7} e + \frac {1}{3} \, c^{3} d^{2} x^{6} + \frac {5}{7} \, b c^{2} x^{7} e^{2} + \frac {5}{3} \, b c^{2} d x^{6} e + b c^{2} d^{2} x^{5} + \frac {2}{3} \, b^{2} c x^{6} e^{2} + \frac {2}{3} \, a c^{2} x^{6} e^{2} + \frac {8}{5} \, b^{2} c d x^{5} e + \frac {8}{5} \, a c^{2} d x^{5} e + b^{2} c d^{2} x^{4} + a c^{2} d^{2} x^{4} + \frac {1}{5} \, b^{3} x^{5} e^{2} + \frac {6}{5} \, a b c x^{5} e^{2} + \frac {1}{2} \, b^{3} d x^{4} e + 3 \, a b c d x^{4} e + \frac {1}{3} \, b^{3} d^{2} x^{3} + 2 \, a b c d^{2} x^{3} + \frac {1}{2} \, a b^{2} x^{4} e^{2} + \frac {1}{2} \, a^{2} c x^{4} e^{2} + \frac {4}{3} \, a b^{2} d x^{3} e + \frac {4}{3} \, a^{2} c d x^{3} e + a b^{2} d^{2} x^{2} + a^{2} c d^{2} x^{2} + \frac {1}{3} \, a^{2} b x^{3} e^{2} + a^{2} b d x^{2} e + a^{2} b d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c^3*x^8*e^2 + 4/7*c^3*d*x^7*e + 1/3*c^3*d^2*x^6 + 5/7*b*c^2*x^7*e^2 + 5/3*b*c^2*d*x^6*e + b*c^2*d^2*x^5 +
2/3*b^2*c*x^6*e^2 + 2/3*a*c^2*x^6*e^2 + 8/5*b^2*c*d*x^5*e + 8/5*a*c^2*d*x^5*e + b^2*c*d^2*x^4 + a*c^2*d^2*x^4
+ 1/5*b^3*x^5*e^2 + 6/5*a*b*c*x^5*e^2 + 1/2*b^3*d*x^4*e + 3*a*b*c*d*x^4*e + 1/3*b^3*d^2*x^3 + 2*a*b*c*d^2*x^3
+ 1/2*a*b^2*x^4*e^2 + 1/2*a^2*c*x^4*e^2 + 4/3*a*b^2*d*x^3*e + 4/3*a^2*c*d*x^3*e + a*b^2*d^2*x^2 + a^2*c*d^2*x^
2 + 1/3*a^2*b*x^3*e^2 + a^2*b*d*x^2*e + a^2*b*d^2*x

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maple [A]  time = 0.04, size = 302, normalized size = 1.26 \begin {gather*} \frac {c^{3} e^{2} x^{8}}{4}+\frac {\left (4 b \,c^{2} e^{2}+\left (b \,e^{2}+4 c d e \right ) c^{2}\right ) x^{7}}{7}+a^{2} b \,d^{2} x +\frac {\left (2 \left (2 a c +b^{2}\right ) c \,e^{2}+2 \left (b \,e^{2}+4 c d e \right ) b c +\left (2 b d e +2 c \,d^{2}\right ) c^{2}\right ) x^{6}}{6}+\frac {\left (4 a b c \,e^{2}+b \,c^{2} d^{2}+2 \left (2 b d e +2 c \,d^{2}\right ) b c +\left (b \,e^{2}+4 c d e \right ) \left (2 a c +b^{2}\right )\right ) x^{5}}{5}+\frac {\left (2 a^{2} c \,e^{2}+2 b^{2} c \,d^{2}+2 \left (b \,e^{2}+4 c d e \right ) a b +\left (2 b d e +2 c \,d^{2}\right ) \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) b \,d^{2}+\left (b \,e^{2}+4 c d e \right ) a^{2}+2 \left (2 b d e +2 c \,d^{2}\right ) a b \right ) x^{3}}{3}+\frac {\left (2 a \,b^{2} d^{2}+\left (2 b d e +2 c \,d^{2}\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)^2*(c*x^2+b*x+a)^2,x)

[Out]

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

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maxima [A]  time = 0.60, size = 241, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, c^{3} e^{2} x^{8} + \frac {1}{7} \, {\left (4 \, c^{3} d e + 5 \, b c^{2} e^{2}\right )} x^{7} + \frac {1}{3} \, {\left (c^{3} d^{2} + 5 \, b c^{2} d e + 2 \, {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x^{6} + a^{2} b d^{2} x + \frac {1}{5} \, {\left (5 \, b c^{2} d^{2} + 8 \, {\left (b^{2} c + a c^{2}\right )} d e + {\left (b^{3} + 6 \, a b c\right )} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (2 \, {\left (b^{2} c + a c^{2}\right )} d^{2} + {\left (b^{3} + 6 \, a b c\right )} d e + {\left (a b^{2} + a^{2} c\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (a^{2} b e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{2} + 4 \, {\left (a b^{2} + a^{2} c\right )} d e\right )} x^{3} + {\left (a^{2} b d e + {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.83, size = 242, normalized size = 1.01 \begin {gather*} x^6\,\left (\frac {2\,b^2\,c\,e^2}{3}+\frac {5\,b\,c^2\,d\,e}{3}+\frac {c^3\,d^2}{3}+\frac {2\,a\,c^2\,e^2}{3}\right )+x^3\,\left (\frac {a^2\,b\,e^2}{3}+\frac {4\,c\,a^2\,d\,e}{3}+\frac {4\,a\,b^2\,d\,e}{3}+2\,c\,a\,b\,d^2+\frac {b^3\,d^2}{3}\right )+x^5\,\left (\frac {b^3\,e^2}{5}+\frac {8\,b^2\,c\,d\,e}{5}+b\,c^2\,d^2+\frac {6\,a\,b\,c\,e^2}{5}+\frac {8\,a\,c^2\,d\,e}{5}\right )+x^4\,\left (\frac {a^2\,c\,e^2}{2}+\frac {a\,b^2\,e^2}{2}+3\,a\,b\,c\,d\,e+a\,c^2\,d^2+\frac {b^3\,d\,e}{2}+b^2\,c\,d^2\right )+\frac {c^3\,e^2\,x^8}{4}+\frac {c^2\,e\,x^7\,\left (5\,b\,e+4\,c\,d\right )}{7}+a^2\,b\,d^2\,x+a\,d\,x^2\,\left (d\,b^2+a\,e\,b+a\,c\,d\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.12, size = 294, normalized size = 1.22 \begin {gather*} a^{2} b d^{2} x + \frac {c^{3} e^{2} x^{8}}{4} + x^{7} \left (\frac {5 b c^{2} e^{2}}{7} + \frac {4 c^{3} d e}{7}\right ) + x^{6} \left (\frac {2 a c^{2} e^{2}}{3} + \frac {2 b^{2} c e^{2}}{3} + \frac {5 b c^{2} d e}{3} + \frac {c^{3} d^{2}}{3}\right ) + x^{5} \left (\frac {6 a b c e^{2}}{5} + \frac {8 a c^{2} d e}{5} + \frac {b^{3} e^{2}}{5} + \frac {8 b^{2} c d e}{5} + b c^{2} d^{2}\right ) + x^{4} \left (\frac {a^{2} c e^{2}}{2} + \frac {a b^{2} e^{2}}{2} + 3 a b c d e + a c^{2} d^{2} + \frac {b^{3} d e}{2} + b^{2} c d^{2}\right ) + x^{3} \left (\frac {a^{2} b e^{2}}{3} + \frac {4 a^{2} c d e}{3} + \frac {4 a b^{2} d e}{3} + 2 a b c d^{2} + \frac {b^{3} d^{2}}{3}\right ) + x^{2} \left (a^{2} b d e + a^{2} c d^{2} + a b^{2} d^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*b*d**2*x + c**3*e**2*x**8/4 + x**7*(5*b*c**2*e**2/7 + 4*c**3*d*e/7) + x**6*(2*a*c**2*e**2/3 + 2*b**2*c*e*
*2/3 + 5*b*c**2*d*e/3 + c**3*d**2/3) + x**5*(6*a*b*c*e**2/5 + 8*a*c**2*d*e/5 + b**3*e**2/5 + 8*b**2*c*d*e/5 +
b*c**2*d**2) + x**4*(a**2*c*e**2/2 + a*b**2*e**2/2 + 3*a*b*c*d*e + a*c**2*d**2 + b**3*d*e/2 + b**2*c*d**2) + x
**3*(a**2*b*e**2/3 + 4*a**2*c*d*e/3 + 4*a*b**2*d*e/3 + 2*a*b*c*d**2 + b**3*d**2/3) + x**2*(a**2*b*d*e + a**2*c
*d**2 + a*b**2*d**2)

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