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

Optimal. Leaf size=156 \[ \frac {(d+e x)^7 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac {(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac {c (d+e x)^8 (2 c d-b e)}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \]

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Rubi [A]  time = 0.25, antiderivative size = 156, 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 {(d+e x)^7 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{7 e^5}-\frac {(d+e x)^6 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right )^2}{5 e^5}-\frac {c (d+e x)^8 (2 c d-b e)}{4 e^5}+\frac {c^2 (d+e x)^9}{9 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 283, normalized size = 1.81 \begin {gather*} \frac {1}{5} x^5 \left (e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+4 c d^2 e (3 a e+2 b d)+c^2 d^4\right )+\frac {1}{2} d x^4 \left (2 a^2 e^3+6 a b d e^2+4 a c d^2 e+2 b^2 d^2 e+b c d^3\right )+a^2 d^4 x+\frac {1}{7} e^2 x^7 \left (2 c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )+\frac {1}{3} d^2 x^3 \left (8 a b d e+2 a \left (3 a e^2+c d^2\right )+b^2 d^2\right )+\frac {1}{3} e x^6 \left (2 c d e (2 a e+3 b d)+b e^2 (a e+2 b d)+2 c^2 d^3\right )+a d^3 x^2 (2 a e+b d)+\frac {1}{4} c e^3 x^8 (b e+2 c d)+\frac {1}{9} c^2 e^4 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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 )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.35, size = 340, normalized size = 2.18 \begin {gather*} \frac {1}{9} x^{9} e^{4} c^{2} + \frac {1}{2} x^{8} e^{3} d c^{2} + \frac {1}{4} x^{8} e^{4} c b + \frac {6}{7} x^{7} e^{2} d^{2} c^{2} + \frac {8}{7} x^{7} e^{3} d c b + \frac {1}{7} x^{7} e^{4} b^{2} + \frac {2}{7} x^{7} e^{4} c a + \frac {2}{3} x^{6} e d^{3} c^{2} + 2 x^{6} e^{2} d^{2} c b + \frac {2}{3} x^{6} e^{3} d b^{2} + \frac {4}{3} x^{6} e^{3} d c a + \frac {1}{3} x^{6} e^{4} b a + \frac {1}{5} x^{5} d^{4} c^{2} + \frac {8}{5} x^{5} e d^{3} c b + \frac {6}{5} x^{5} e^{2} d^{2} b^{2} + \frac {12}{5} x^{5} e^{2} d^{2} c a + \frac {8}{5} x^{5} e^{3} d b a + \frac {1}{5} x^{5} e^{4} a^{2} + \frac {1}{2} x^{4} d^{4} c b + x^{4} e d^{3} b^{2} + 2 x^{4} e d^{3} c a + 3 x^{4} e^{2} d^{2} b a + x^{4} e^{3} d a^{2} + \frac {1}{3} x^{3} d^{4} b^{2} + \frac {2}{3} x^{3} d^{4} c a + \frac {8}{3} x^{3} e d^{3} b a + 2 x^{3} e^{2} d^{2} a^{2} + x^{2} d^{4} b a + 2 x^{2} e d^{3} a^{2} + x d^{4} a^{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)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.16, size = 328, normalized size = 2.10 \begin {gather*} \frac {1}{9} \, c^{2} x^{9} e^{4} + \frac {1}{2} \, c^{2} d x^{8} e^{3} + \frac {6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac {2}{3} \, c^{2} d^{3} x^{6} e + \frac {1}{5} \, c^{2} d^{4} x^{5} + \frac {1}{4} \, b c x^{8} e^{4} + \frac {8}{7} \, b c d x^{7} e^{3} + 2 \, b c d^{2} x^{6} e^{2} + \frac {8}{5} \, b c d^{3} x^{5} e + \frac {1}{2} \, b c d^{4} x^{4} + \frac {1}{7} \, b^{2} x^{7} e^{4} + \frac {2}{7} \, a c x^{7} e^{4} + \frac {2}{3} \, b^{2} d x^{6} e^{3} + \frac {4}{3} \, a c d x^{6} e^{3} + \frac {6}{5} \, b^{2} d^{2} x^{5} e^{2} + \frac {12}{5} \, a c d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + 2 \, a c d^{3} x^{4} e + \frac {1}{3} \, b^{2} d^{4} x^{3} + \frac {2}{3} \, a c d^{4} x^{3} + \frac {1}{3} \, a b x^{6} e^{4} + \frac {8}{5} \, a b d x^{5} e^{3} + 3 \, a b d^{2} x^{4} e^{2} + \frac {8}{3} \, a b d^{3} x^{3} e + a b d^{4} x^{2} + \frac {1}{5} \, a^{2} x^{5} e^{4} + a^{2} d x^{4} e^{3} + 2 \, a^{2} d^{2} x^{3} e^{2} + 2 \, a^{2} d^{3} x^{2} e + a^{2} 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)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 283, normalized size = 1.81 \begin {gather*} \frac {c^{2} e^{4} x^{9}}{9}+\frac {\left (2 e^{4} b c +4 d \,e^{3} c^{2}\right ) x^{8}}{8}+a^{2} d^{4} x +\frac {\left (8 b c d \,e^{3}+6 c^{2} d^{2} e^{2}+\left (2 a c +b^{2}\right ) e^{4}\right ) x^{7}}{7}+\frac {\left (2 a b \,e^{4}+12 b c \,d^{2} e^{2}+4 c^{2} d^{3} e +4 \left (2 a c +b^{2}\right ) d \,e^{3}\right ) x^{6}}{6}+\frac {\left (a^{2} e^{4}+8 a b d \,e^{3}+8 b c \,d^{3} e +c^{2} d^{4}+6 \left (2 a c +b^{2}\right ) d^{2} e^{2}\right ) x^{5}}{5}+\frac {\left (4 a^{2} d \,e^{3}+12 a b \,d^{2} e^{2}+2 b c \,d^{4}+4 \left (2 a c +b^{2}\right ) d^{3} e \right ) x^{4}}{4}+\frac {\left (6 a^{2} d^{2} e^{2}+8 a b \,d^{3} e +\left (2 a c +b^{2}\right ) d^{4}\right ) x^{3}}{3}+\frac {\left (4 d^{3} e \,a^{2}+2 d^{4} a 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)^2,x)

[Out]

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

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maxima [A]  time = 1.10, size = 277, normalized size = 1.78 \begin {gather*} \frac {1}{9} \, c^{2} e^{4} x^{9} + \frac {1}{4} \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{8} + \frac {1}{7} \, {\left (6 \, c^{2} d^{2} e^{2} + 8 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac {1}{3} \, {\left (2 \, c^{2} d^{3} e + 6 \, b c d^{2} e^{2} + a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (c^{2} d^{4} + 8 \, b c d^{3} e + 8 \, a b d e^{3} + a^{2} e^{4} + 6 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} x^{5} + \frac {1}{2} \, {\left (b c d^{4} + 6 \, a b d^{2} e^{2} + 2 \, a^{2} d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e\right )} x^{4} + \frac {1}{3} \, {\left (8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{4}\right )} x^{3} + {\left (a b d^{4} + 2 \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.11, size = 283, normalized size = 1.81 \begin {gather*} x^4\,\left (a^2\,d\,e^3+3\,a\,b\,d^2\,e^2+2\,c\,a\,d^3\,e+b^2\,d^3\,e+\frac {c\,b\,d^4}{2}\right )+x^6\,\left (\frac {2\,b^2\,d\,e^3}{3}+2\,b\,c\,d^2\,e^2+\frac {a\,b\,e^4}{3}+\frac {2\,c^2\,d^3\,e}{3}+\frac {4\,a\,c\,d\,e^3}{3}\right )+x^5\,\left (\frac {a^2\,e^4}{5}+\frac {8\,a\,b\,d\,e^3}{5}+\frac {12\,a\,c\,d^2\,e^2}{5}+\frac {6\,b^2\,d^2\,e^2}{5}+\frac {8\,b\,c\,d^3\,e}{5}+\frac {c^2\,d^4}{5}\right )+x^3\,\left (2\,a^2\,d^2\,e^2+\frac {8\,a\,b\,d^3\,e}{3}+\frac {2\,c\,a\,d^4}{3}+\frac {b^2\,d^4}{3}\right )+x^7\,\left (\frac {b^2\,e^4}{7}+\frac {8\,b\,c\,d\,e^3}{7}+\frac {6\,c^2\,d^2\,e^2}{7}+\frac {2\,a\,c\,e^4}{7}\right )+a^2\,d^4\,x+\frac {c^2\,e^4\,x^9}{9}+a\,d^3\,x^2\,\left (2\,a\,e+b\,d\right )+\frac {c\,e^3\,x^8\,\left (b\,e+2\,c\,d\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.13, size = 337, normalized size = 2.16 \begin {gather*} a^{2} d^{4} x + \frac {c^{2} e^{4} x^{9}}{9} + x^{8} \left (\frac {b c e^{4}}{4} + \frac {c^{2} d e^{3}}{2}\right ) + x^{7} \left (\frac {2 a c e^{4}}{7} + \frac {b^{2} e^{4}}{7} + \frac {8 b c d e^{3}}{7} + \frac {6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac {a b e^{4}}{3} + \frac {4 a c d e^{3}}{3} + \frac {2 b^{2} d e^{3}}{3} + 2 b c d^{2} e^{2} + \frac {2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac {a^{2} e^{4}}{5} + \frac {8 a b d e^{3}}{5} + \frac {12 a c d^{2} e^{2}}{5} + \frac {6 b^{2} d^{2} e^{2}}{5} + \frac {8 b c d^{3} e}{5} + \frac {c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + 2 a c d^{3} e + b^{2} d^{3} e + \frac {b c d^{4}}{2}\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac {8 a b d^{3} e}{3} + \frac {2 a c d^{4}}{3} + \frac {b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\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)**2,x)

[Out]

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

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