3.1451 \(\int (d+e x)^4 (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=65 \[ -\frac{b (d+e x)^6 (b d-a e)}{3 e^3}+\frac{(d+e x)^5 (b d-a e)^2}{5 e^3}+\frac{b^2 (d+e x)^7}{7 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0977775, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 43} \[ -\frac{b (d+e x)^6 (b d-a e)}{3 e^3}+\frac{(d+e x)^5 (b d-a e)^2}{5 e^3}+\frac{b^2 (d+e x)^7}{7 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [B]  time = 0.0261923, size = 148, normalized size = 2.28 \[ \frac{1}{5} e^2 x^5 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+d e x^4 \left (a^2 e^2+3 a b d e+b^2 d^2\right )+\frac{1}{3} d^2 x^3 \left (6 a^2 e^2+8 a b d e+b^2 d^2\right )+a^2 d^4 x+a d^3 x^2 (2 a e+b d)+\frac{1}{3} b e^3 x^6 (a e+2 b d)+\frac{1}{7} b^2 e^4 x^7 \]

Antiderivative was successfully verified.

[In]

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

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Maple [B]  time = 0.04, size = 163, normalized size = 2.5 \begin{align*}{\frac{{e}^{4}{b}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{4}ab+4\,d{e}^{3}{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}{e}^{4}+8\,d{e}^{3}ab+6\,{d}^{2}{e}^{2}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{2}+12\,{d}^{2}{e}^{2}ab+4\,{d}^{3}e{b}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{2}+8\,{d}^{3}eab+{d}^{4}{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{d}^{3}e{a}^{2}+2\,{d}^{4}ab \right ){x}^{2}}{2}}+{d}^{4}{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*e^4*b^2*x^7+1/6*(2*a*b*e^4+4*b^2*d*e^3)*x^6+1/5*(a^2*e^4+8*a*b*d*e^3+6*b^2*d^2*e^2)*x^5+1/4*(4*a^2*d*e^3+1
2*a*b*d^2*e^2+4*b^2*d^3*e)*x^4+1/3*(6*a^2*d^2*e^2+8*a*b*d^3*e+b^2*d^4)*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 [B]  time = 1.14135, size = 211, normalized size = 3.25 \begin{align*} \frac{1}{7} \, b^{2} e^{4} x^{7} + a^{2} d^{4} x + \frac{1}{3} \,{\left (2 \, b^{2} d e^{3} + a b e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (6 \, b^{2} d^{2} e^{2} + 8 \, a b d e^{3} + a^{2} e^{4}\right )} x^{5} +{\left (b^{2} d^{3} e + 3 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{4} + 8 \, a b d^{3} e + 6 \, a^{2} d^{2} e^{2}\right )} x^{3} +{\left (a b d^{4} + 2 \, a^{2} d^{3} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.70062, size = 363, normalized size = 5.58 \begin{align*} \frac{1}{7} x^{7} e^{4} b^{2} + \frac{2}{3} x^{6} e^{3} d b^{2} + \frac{1}{3} x^{6} e^{4} b a + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} + \frac{8}{5} x^{5} e^{3} d b a + \frac{1}{5} x^{5} e^{4} a^{2} + x^{4} e d^{3} b^{2} + 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{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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e^4*b^2 + 2/3*x^6*e^3*d*b^2 + 1/3*x^6*e^4*b*a + 6/5*x^5*e^2*d^2*b^2 + 8/5*x^5*e^3*d*b*a + 1/5*x^5*e^4*
a^2 + x^4*e*d^3*b^2 + 3*x^4*e^2*d^2*b*a + x^4*e^3*d*a^2 + 1/3*x^3*d^4*b^2 + 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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Sympy [B]  time = 0.100004, size = 168, normalized size = 2.58 \begin{align*} a^{2} d^{4} x + \frac{b^{2} e^{4} x^{7}}{7} + x^{6} \left (\frac{a b e^{4}}{3} + \frac{2 b^{2} d e^{3}}{3}\right ) + x^{5} \left (\frac{a^{2} e^{4}}{5} + \frac{8 a b d e^{3}}{5} + \frac{6 b^{2} d^{2} e^{2}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 3 a b d^{2} e^{2} + b^{2} d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac{8 a b d^{3} e}{3} + \frac{b^{2} d^{4}}{3}\right ) + x^{2} \left (2 a^{2} d^{3} e + a b d^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.18998, size = 221, normalized size = 3.4 \begin{align*} \frac{1}{7} \, b^{2} x^{7} e^{4} + \frac{2}{3} \, b^{2} d x^{6} e^{3} + \frac{6}{5} \, b^{2} d^{2} x^{5} e^{2} + b^{2} d^{3} x^{4} e + \frac{1}{3} \, b^{2} 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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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