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

Optimal. Leaf size=124 \[ \frac {(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^4}-\frac {(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac {3 c (d+e x)^7 (2 c d-b e)}{7 e^4}+\frac {c^2 (d+e x)^8}{4 e^4} \]

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Rubi [A]  time = 0.19, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {(d+e x)^6 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{6 e^4}-\frac {(d+e x)^5 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^4}-\frac {3 c (d+e x)^7 (2 c d-b e)}{7 e^4}+\frac {c^2 (d+e x)^8}{4 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 229, normalized size = 1.85 \begin {gather*} \frac {1}{6} e^2 x^6 \left (2 c e (a e+6 b d)+b^2 e^2+12 c^2 d^2\right )+\frac {1}{2} d^3 x^2 \left (4 a b e+2 a c d+b^2 d\right )+\frac {1}{3} d^2 x^3 \left (6 a b e^2+8 a c d e+4 b^2 d e+3 b c d^2\right )+\frac {1}{5} e x^5 \left (2 c d e (4 a e+9 b d)+b e^2 (a e+4 b d)+8 c^2 d^3\right )+\frac {1}{2} d x^4 \left (6 c d e (a e+b d)+b e^2 (2 a e+3 b d)+c^2 d^3\right )+a b d^4 x+\frac {1}{7} c e^3 x^7 (3 b e+8 c d)+\frac {1}{4} c^2 e^4 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*b*d^4*x + (d^3*(b^2*d + 2*a*c*d + 4*a*b*e)*x^2)/2 + (d^2*(3*b*c*d^2 + 4*b^2*d*e + 8*a*c*d*e + 6*a*b*e^2)*x^3
)/3 + (d*(c^2*d^3 + 6*c*d*e*(b*d + a*e) + b*e^2*(3*b*d + 2*a*e))*x^4)/2 + (e*(8*c^2*d^3 + b*e^2*(4*b*d + a*e)
+ 2*c*d*e*(9*b*d + 4*a*e))*x^5)/5 + (e^2*(12*c^2*d^2 + b^2*e^2 + 2*c*e*(6*b*d + a*e))*x^6)/6 + (c*e^3*(8*c*d +
3*b*e)*x^7)/7 + (c^2*e^4*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)^4 \left (a+b x+c x^2\right ) \, 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),x]

[Out]

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

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fricas [B]  time = 0.36, size = 280, normalized size = 2.26 \begin {gather*} \frac {1}{4} x^{8} e^{4} c^{2} + \frac {8}{7} x^{7} e^{3} d c^{2} + \frac {3}{7} x^{7} e^{4} c b + 2 x^{6} e^{2} d^{2} c^{2} + 2 x^{6} e^{3} d c b + \frac {1}{6} x^{6} e^{4} b^{2} + \frac {1}{3} x^{6} e^{4} c a + \frac {8}{5} x^{5} e d^{3} c^{2} + \frac {18}{5} x^{5} e^{2} d^{2} c b + \frac {4}{5} x^{5} e^{3} d b^{2} + \frac {8}{5} x^{5} e^{3} d c a + \frac {1}{5} x^{5} e^{4} b a + \frac {1}{2} x^{4} d^{4} c^{2} + 3 x^{4} e d^{3} c b + \frac {3}{2} x^{4} e^{2} d^{2} b^{2} + 3 x^{4} e^{2} d^{2} c a + x^{4} e^{3} d b a + x^{3} d^{4} c b + \frac {4}{3} x^{3} e d^{3} b^{2} + \frac {8}{3} x^{3} e d^{3} c a + 2 x^{3} e^{2} d^{2} b a + \frac {1}{2} x^{2} d^{4} b^{2} + x^{2} d^{4} c a + 2 x^{2} e d^{3} b a + x d^{4} b a \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),x, algorithm="fricas")

[Out]

1/4*x^8*e^4*c^2 + 8/7*x^7*e^3*d*c^2 + 3/7*x^7*e^4*c*b + 2*x^6*e^2*d^2*c^2 + 2*x^6*e^3*d*c*b + 1/6*x^6*e^4*b^2
+ 1/3*x^6*e^4*c*a + 8/5*x^5*e*d^3*c^2 + 18/5*x^5*e^2*d^2*c*b + 4/5*x^5*e^3*d*b^2 + 8/5*x^5*e^3*d*c*a + 1/5*x^5
*e^4*b*a + 1/2*x^4*d^4*c^2 + 3*x^4*e*d^3*c*b + 3/2*x^4*e^2*d^2*b^2 + 3*x^4*e^2*d^2*c*a + x^4*e^3*d*b*a + x^3*d
^4*c*b + 4/3*x^3*e*d^3*b^2 + 8/3*x^3*e*d^3*c*a + 2*x^3*e^2*d^2*b*a + 1/2*x^2*d^4*b^2 + x^2*d^4*c*a + 2*x^2*e*d
^3*b*a + x*d^4*b*a

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giac [B]  time = 0.16, size = 270, normalized size = 2.18 \begin {gather*} \frac {1}{4} \, c^{2} x^{8} e^{4} + \frac {8}{7} \, c^{2} d x^{7} e^{3} + 2 \, c^{2} d^{2} x^{6} e^{2} + \frac {8}{5} \, c^{2} d^{3} x^{5} e + \frac {1}{2} \, c^{2} d^{4} x^{4} + \frac {3}{7} \, b c x^{7} e^{4} + 2 \, b c d x^{6} e^{3} + \frac {18}{5} \, b c d^{2} x^{5} e^{2} + 3 \, b c d^{3} x^{4} e + b c d^{4} x^{3} + \frac {1}{6} \, b^{2} x^{6} e^{4} + \frac {1}{3} \, a c x^{6} e^{4} + \frac {4}{5} \, b^{2} d x^{5} e^{3} + \frac {8}{5} \, a c d x^{5} e^{3} + \frac {3}{2} \, b^{2} d^{2} x^{4} e^{2} + 3 \, a c d^{2} x^{4} e^{2} + \frac {4}{3} \, b^{2} d^{3} x^{3} e + \frac {8}{3} \, a c d^{3} x^{3} e + \frac {1}{2} \, b^{2} d^{4} x^{2} + a c d^{4} x^{2} + \frac {1}{5} \, a b x^{5} e^{4} + a b d x^{4} e^{3} + 2 \, a b d^{2} x^{3} e^{2} + 2 \, a b d^{3} x^{2} e + a 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),x, algorithm="giac")

[Out]

1/4*c^2*x^8*e^4 + 8/7*c^2*d*x^7*e^3 + 2*c^2*d^2*x^6*e^2 + 8/5*c^2*d^3*x^5*e + 1/2*c^2*d^4*x^4 + 3/7*b*c*x^7*e^
4 + 2*b*c*d*x^6*e^3 + 18/5*b*c*d^2*x^5*e^2 + 3*b*c*d^3*x^4*e + b*c*d^4*x^3 + 1/6*b^2*x^6*e^4 + 1/3*a*c*x^6*e^4
+ 4/5*b^2*d*x^5*e^3 + 8/5*a*c*d*x^5*e^3 + 3/2*b^2*d^2*x^4*e^2 + 3*a*c*d^2*x^4*e^2 + 4/3*b^2*d^3*x^3*e + 8/3*a
*c*d^3*x^3*e + 1/2*b^2*d^4*x^2 + a*c*d^4*x^2 + 1/5*a*b*x^5*e^4 + a*b*d*x^4*e^3 + 2*a*b*d^2*x^3*e^2 + 2*a*b*d^3
*x^2*e + a*b*d^4*x

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maple [B]  time = 0.04, size = 290, normalized size = 2.34 \begin {gather*} \frac {c^{2} e^{4} x^{8}}{4}+a b \,d^{4} x +\frac {\left (2 b c \,e^{4}+\left (b \,e^{4}+8 c d \,e^{3}\right ) c \right ) x^{7}}{7}+\frac {\left (2 a c \,e^{4}+\left (b \,e^{4}+8 c d \,e^{3}\right ) b +\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) c \right ) x^{6}}{6}+\frac {\left (\left (b \,e^{4}+8 c d \,e^{3}\right ) a +\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) b +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) c \right ) x^{5}}{5}+\frac {\left (\left (4 b d \,e^{3}+12 c \,d^{2} e^{2}\right ) a +\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) b +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) c \right ) x^{4}}{4}+\frac {\left (b c \,d^{4}+\left (6 b \,d^{2} e^{2}+8 c \,d^{3} e \right ) a +\left (4 b \,d^{3} e +2 c \,d^{4}\right ) b \right ) x^{3}}{3}+\frac {\left (b^{2} d^{4}+\left (4 b \,d^{3} e +2 c \,d^{4}\right ) a \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),x)

[Out]

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

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maxima [A]  time = 0.60, size = 231, normalized size = 1.86 \begin {gather*} \frac {1}{4} \, c^{2} e^{4} x^{8} + \frac {1}{7} \, {\left (8 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{7} + a b d^{4} x + \frac {1}{6} \, {\left (12 \, c^{2} d^{2} e^{2} + 12 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (8 \, c^{2} d^{3} e + 18 \, b c d^{2} e^{2} + a b e^{4} + 4 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{5} + \frac {1}{2} \, {\left (c^{2} d^{4} + 6 \, b c d^{3} e + 2 \, a b d e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, b c d^{4} + 6 \, a b d^{2} e^{2} + 4 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, a b d^{3} e + {\left (b^{2} + 2 \, a 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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.11, size = 238, normalized size = 1.92 \begin {gather*} x^5\,\left (\frac {4\,b^2\,d\,e^3}{5}+\frac {18\,b\,c\,d^2\,e^2}{5}+\frac {a\,b\,e^4}{5}+\frac {8\,c^2\,d^3\,e}{5}+\frac {8\,a\,c\,d\,e^3}{5}\right )+x^6\,\left (\frac {b^2\,e^4}{6}+2\,b\,c\,d\,e^3+2\,c^2\,d^2\,e^2+\frac {a\,c\,e^4}{3}\right )+x^4\,\left (\frac {3\,b^2\,d^2\,e^2}{2}+3\,b\,c\,d^3\,e+a\,b\,d\,e^3+\frac {c^2\,d^4}{2}+3\,a\,c\,d^2\,e^2\right )+x^2\,\left (\frac {b^2\,d^4}{2}+2\,a\,e\,b\,d^3+a\,c\,d^4\right )+x^3\,\left (\frac {4\,b^2\,d^3\,e}{3}+c\,b\,d^4+2\,a\,b\,d^2\,e^2+\frac {8\,a\,c\,d^3\,e}{3}\right )+\frac {c^2\,e^4\,x^8}{4}+\frac {c\,e^3\,x^7\,\left (3\,b\,e+8\,c\,d\right )}{7}+a\,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),x)

[Out]

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

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sympy [B]  time = 0.11, size = 279, normalized size = 2.25 \begin {gather*} a b d^{4} x + \frac {c^{2} e^{4} x^{8}}{4} + x^{7} \left (\frac {3 b c e^{4}}{7} + \frac {8 c^{2} d e^{3}}{7}\right ) + x^{6} \left (\frac {a c e^{4}}{3} + \frac {b^{2} e^{4}}{6} + 2 b c d e^{3} + 2 c^{2} d^{2} e^{2}\right ) + x^{5} \left (\frac {a b e^{4}}{5} + \frac {8 a c d e^{3}}{5} + \frac {4 b^{2} d e^{3}}{5} + \frac {18 b c d^{2} e^{2}}{5} + \frac {8 c^{2} d^{3} e}{5}\right ) + x^{4} \left (a b d e^{3} + 3 a c d^{2} e^{2} + \frac {3 b^{2} d^{2} e^{2}}{2} + 3 b c d^{3} e + \frac {c^{2} d^{4}}{2}\right ) + x^{3} \left (2 a b d^{2} e^{2} + \frac {8 a c d^{3} e}{3} + \frac {4 b^{2} d^{3} e}{3} + b c d^{4}\right ) + x^{2} \left (2 a b d^{3} e + a c d^{4} + \frac {b^{2} d^{4}}{2}\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),x)

[Out]

a*b*d**4*x + c**2*e**4*x**8/4 + x**7*(3*b*c*e**4/7 + 8*c**2*d*e**3/7) + x**6*(a*c*e**4/3 + b**2*e**4/6 + 2*b*c
*d*e**3 + 2*c**2*d**2*e**2) + x**5*(a*b*e**4/5 + 8*a*c*d*e**3/5 + 4*b**2*d*e**3/5 + 18*b*c*d**2*e**2/5 + 8*c**
2*d**3*e/5) + x**4*(a*b*d*e**3 + 3*a*c*d**2*e**2 + 3*b**2*d**2*e**2/2 + 3*b*c*d**3*e + c**2*d**4/2) + x**3*(2*
a*b*d**2*e**2 + 8*a*c*d**3*e/3 + 4*b**2*d**3*e/3 + b*c*d**4) + x**2*(2*a*b*d**3*e + a*c*d**4 + b**2*d**4/2)

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