### 3.1468 $$\int (d+e x) (a^2+2 a b x+b^2 x^2)^2 \, dx$$

Optimal. Leaf size=38 $\frac{(a+b x)^5 (b d-a e)}{5 b^2}+\frac{e (a+b x)^6}{6 b^2}$

[Out]

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

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Rubi [A]  time = 0.0158858, antiderivative size = 38, 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} $\frac{(a+b x)^5 (b d-a e)}{5 b^2}+\frac{e (a+b x)^6}{6 b^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [B]  time = 0.0161652, size = 84, normalized size = 2.21 $\frac{1}{30} x \left (15 a^2 b^2 x^2 (4 d+3 e x)+20 a^3 b x (3 d+2 e x)+15 a^4 (2 d+e x)+6 a b^3 x^3 (5 d+4 e x)+b^4 x^4 (6 d+5 e x)\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x*(15*a^4*(2*d + e*x) + 20*a^3*b*x*(3*d + 2*e*x) + 15*a^2*b^2*x^2*(4*d + 3*e*x) + 6*a*b^3*x^3*(5*d + 4*e*x) +
b^4*x^4*(6*d + 5*e*x)))/30

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Maple [B]  time = 0.039, size = 97, normalized size = 2.6 \begin{align*}{\frac{e{b}^{4}{x}^{6}}{6}}+{\frac{ \left ( 4\,ea{b}^{3}+d{b}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,{a}^{2}e{b}^{2}+4\,ad{b}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 4\,e{a}^{3}b+6\,d{b}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{4}+4\,d{a}^{3}b \right ){x}^{2}}{2}}+{a}^{4}dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.1702, size = 130, normalized size = 3.42 \begin{align*} \frac{1}{6} \, b^{4} e x^{6} + a^{4} d x + \frac{1}{5} \,{\left (b^{4} d + 4 \, a b^{3} e\right )} x^{5} + \frac{1}{2} \,{\left (2 \, a b^{3} d + 3 \, a^{2} b^{2} e\right )} x^{4} + \frac{2}{3} \,{\left (3 \, a^{2} b^{2} d + 2 \, a^{3} b e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b d + a^{4} e\right )} x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.52879, size = 217, normalized size = 5.71 \begin{align*} \frac{1}{6} x^{6} e b^{4} + \frac{1}{5} x^{5} d b^{4} + \frac{4}{5} x^{5} e b^{3} a + x^{4} d b^{3} a + \frac{3}{2} x^{4} e b^{2} a^{2} + 2 x^{3} d b^{2} a^{2} + \frac{4}{3} x^{3} e b a^{3} + 2 x^{2} d b a^{3} + \frac{1}{2} x^{2} e a^{4} + x d a^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.076732, size = 100, normalized size = 2.63 \begin{align*} a^{4} d x + \frac{b^{4} e x^{6}}{6} + x^{5} \left (\frac{4 a b^{3} e}{5} + \frac{b^{4} d}{5}\right ) + x^{4} \left (\frac{3 a^{2} b^{2} e}{2} + a b^{3} d\right ) + x^{3} \left (\frac{4 a^{3} b e}{3} + 2 a^{2} b^{2} d\right ) + x^{2} \left (\frac{a^{4} e}{2} + 2 a^{3} b d\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.15537, size = 138, normalized size = 3.63 \begin{align*} \frac{1}{6} \, b^{4} x^{6} e + \frac{1}{5} \, b^{4} d x^{5} + \frac{4}{5} \, a b^{3} x^{5} e + a b^{3} d x^{4} + \frac{3}{2} \, a^{2} b^{2} x^{4} e + 2 \, a^{2} b^{2} d x^{3} + \frac{4}{3} \, a^{3} b x^{3} e + 2 \, a^{3} b d x^{2} + \frac{1}{2} \, a^{4} x^{2} e + a^{4} d x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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