3.1922 \(\int (a+b x) (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=14 \[ \frac{(a+b x)^8}{8 b} \]

[Out]

(a + b*x)^8/(8*b)

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Rubi [A]  time = 0.0025781, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {27, 32} \[ \frac{(a+b x)^8}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a + b*x)^8/(8*b)

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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 \, dx\\ &=\frac{(a+b x)^8}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0012208, size = 14, normalized size = 1. \[ \frac{(a+b x)^8}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(a + b*x)^8/(8*b)

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Maple [B]  time = 0., size = 76, normalized size = 5.4 \begin{align*}{\frac{{b}^{7}{x}^{8}}{8}}+a{b}^{6}{x}^{7}+{\frac{7\,{a}^{2}{b}^{5}{x}^{6}}{2}}+7\,{a}^{3}{b}^{4}{x}^{5}+{\frac{35\,{a}^{4}{b}^{3}{x}^{4}}{4}}+7\,{a}^{5}{b}^{2}{x}^{3}+{\frac{7\,{a}^{6}b{x}^{2}}{2}}+{a}^{7}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/8*b^7*x^8+a*b^6*x^7+7/2*a^2*b^5*x^6+7*a^3*b^4*x^5+35/4*a^4*b^3*x^4+7*a^5*b^2*x^3+7/2*a^6*b*x^2+a^7*x

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Maxima [A]  time = 0.957142, size = 31, normalized size = 2.21 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{4}}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/8*(b^2*x^2 + 2*a*b*x + a^2)^4/b

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Fricas [B]  time = 1.32962, size = 159, normalized size = 11.36 \begin{align*} \frac{1}{8} x^{8} b^{7} + x^{7} b^{6} a + \frac{7}{2} x^{6} b^{5} a^{2} + 7 x^{5} b^{4} a^{3} + \frac{35}{4} x^{4} b^{3} a^{4} + 7 x^{3} b^{2} a^{5} + \frac{7}{2} x^{2} b a^{6} + x a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/8*x^8*b^7 + x^7*b^6*a + 7/2*x^6*b^5*a^2 + 7*x^5*b^4*a^3 + 35/4*x^4*b^3*a^4 + 7*x^3*b^2*a^5 + 7/2*x^2*b*a^6 +
x*a^7

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Sympy [B]  time = 0.078726, size = 83, normalized size = 5.93 \begin{align*} a^{7} x + \frac{7 a^{6} b x^{2}}{2} + 7 a^{5} b^{2} x^{3} + \frac{35 a^{4} b^{3} x^{4}}{4} + 7 a^{3} b^{4} x^{5} + \frac{7 a^{2} b^{5} x^{6}}{2} + a b^{6} x^{7} + \frac{b^{7} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**7*x + 7*a**6*b*x**2/2 + 7*a**5*b**2*x**3 + 35*a**4*b**3*x**4/4 + 7*a**3*b**4*x**5 + 7*a**2*b**5*x**6/2 + a*
b**6*x**7 + b**7*x**8/8

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Giac [B]  time = 1.1127, size = 101, normalized size = 7.21 \begin{align*} \frac{1}{8} \, b^{7} x^{8} + a b^{6} x^{7} + \frac{7}{2} \, a^{2} b^{5} x^{6} + 7 \, a^{3} b^{4} x^{5} + \frac{35}{4} \, a^{4} b^{3} x^{4} + 7 \, a^{5} b^{2} x^{3} + \frac{7}{2} \, a^{6} b x^{2} + a^{7} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/8*b^7*x^8 + a*b^6*x^7 + 7/2*a^2*b^5*x^6 + 7*a^3*b^4*x^5 + 35/4*a^4*b^3*x^4 + 7*a^5*b^2*x^3 + 7/2*a^6*b*x^2 +
a^7*x