3.2107 \(\int (d+e x)^4 (a+b x+c x^2) \, dx\)

Optimal. Leaf size=69 \[ \frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^3}-\frac{(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac{c (d+e x)^7}{7 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0982364, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {698} \[ \frac{(d+e x)^5 \left (a e^2-b d e+c d^2\right )}{5 e^3}-\frac{(d+e x)^6 (2 c d-b e)}{6 e^3}+\frac{c (d+e x)^7}{7 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a + b*x + c*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0388166, size = 135, normalized size = 1.96 \[ \frac{1}{5} e^2 x^5 \left (a e^2+4 b d e+6 c d^2\right )+\frac{1}{2} d e x^4 \left (2 a e^2+3 b d e+2 c d^2\right )+\frac{1}{3} d^2 x^3 \left (6 a e^2+4 b d e+c d^2\right )+\frac{1}{2} d^3 x^2 (4 a e+b d)+a d^4 x+\frac{1}{6} e^3 x^6 (b e+4 c d)+\frac{1}{7} c e^4 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.039, size = 136, normalized size = 2. \begin{align*}{\frac{{e}^{4}c{x}^{7}}{7}}+{\frac{ \left ({e}^{4}b+4\,d{e}^{3}c \right ){x}^{6}}{6}}+{\frac{ \left ({e}^{4}a+4\,d{e}^{3}b+6\,{d}^{2}{e}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,ad{e}^{3}+6\,{d}^{2}{e}^{2}b+4\,c{d}^{3}e \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,a{d}^{2}{e}^{2}+4\,{d}^{3}eb+c{d}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{d}^{3}ea+b{d}^{4} \right ){x}^{2}}{2}}+{d}^{4}ax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.978365, size = 182, normalized size = 2.64 \begin{align*} \frac{1}{7} \, c e^{4} x^{7} + \frac{1}{6} \,{\left (4 \, c d e^{3} + b e^{4}\right )} x^{6} + a d^{4} x + \frac{1}{5} \,{\left (6 \, c d^{2} e^{2} + 4 \, b d e^{3} + a e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, c d^{3} e + 3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{4} + 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{4} + 4 \, a d^{3} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.77728, size = 331, normalized size = 4.8 \begin{align*} \frac{1}{7} x^{7} e^{4} c + \frac{2}{3} x^{6} e^{3} d c + \frac{1}{6} x^{6} e^{4} b + \frac{6}{5} x^{5} e^{2} d^{2} c + \frac{4}{5} x^{5} e^{3} d b + \frac{1}{5} x^{5} e^{4} a + x^{4} e d^{3} c + \frac{3}{2} x^{4} e^{2} d^{2} b + x^{4} e^{3} d a + \frac{1}{3} x^{3} d^{4} c + \frac{4}{3} x^{3} e d^{3} b + 2 x^{3} e^{2} d^{2} a + \frac{1}{2} x^{2} d^{4} b + 2 x^{2} e d^{3} a + x d^{4} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.090638, size = 146, normalized size = 2.12 \begin{align*} a d^{4} x + \frac{c e^{4} x^{7}}{7} + x^{6} \left (\frac{b e^{4}}{6} + \frac{2 c d e^{3}}{3}\right ) + x^{5} \left (\frac{a e^{4}}{5} + \frac{4 b d e^{3}}{5} + \frac{6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (a d e^{3} + \frac{3 b d^{2} e^{2}}{2} + c d^{3} e\right ) + x^{3} \left (2 a d^{2} e^{2} + \frac{4 b d^{3} e}{3} + \frac{c d^{4}}{3}\right ) + x^{2} \left (2 a d^{3} e + \frac{b d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.10771, size = 189, normalized size = 2.74 \begin{align*} \frac{1}{7} \, c x^{7} e^{4} + \frac{2}{3} \, c d x^{6} e^{3} + \frac{6}{5} \, c d^{2} x^{5} e^{2} + c d^{3} x^{4} e + \frac{1}{3} \, c d^{4} x^{3} + \frac{1}{6} \, b x^{6} e^{4} + \frac{4}{5} \, b d x^{5} e^{3} + \frac{3}{2} \, b d^{2} x^{4} e^{2} + \frac{4}{3} \, b d^{3} x^{3} e + \frac{1}{2} \, b d^{4} x^{2} + \frac{1}{5} \, a x^{5} e^{4} + a d x^{4} e^{3} + 2 \, a d^{2} x^{3} e^{2} + 2 \, a d^{3} x^{2} e + a d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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