3.1017 \(\int (a+b x)^5 (a c+b c x)^2 \, dx\)

Optimal. Leaf size=17 \[ \frac{c^2 (a+b x)^8}{8 b} \]

[Out]

(c^2*(a + b*x)^8)/(8*b)

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Rubi [A]  time = 0.0037584, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {21, 32} \[ \frac{c^2 (a+b x)^8}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*(a*c + b*c*x)^2,x]

[Out]

(c^2*(a + b*x)^8)/(8*b)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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)^5 (a c+b c x)^2 \, dx &=c^2 \int (a+b x)^7 \, dx\\ &=\frac{c^2 (a+b x)^8}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0020709, size = 17, normalized size = 1. \[ \frac{c^2 (a+b x)^8}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5*(a*c + b*c*x)^2,x]

[Out]

(c^2*(a + b*x)^8)/(8*b)

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Maple [B]  time = 0.002, size = 100, normalized size = 5.9 \begin{align*}{\frac{{b}^{7}{c}^{2}{x}^{8}}{8}}+a{b}^{6}{c}^{2}{x}^{7}+{\frac{7\,{a}^{2}{b}^{5}{c}^{2}{x}^{6}}{2}}+7\,{a}^{3}{b}^{4}{c}^{2}{x}^{5}+{\frac{35\,{a}^{4}{b}^{3}{c}^{2}{x}^{4}}{4}}+7\,{a}^{5}{b}^{2}{c}^{2}{x}^{3}+{\frac{7\,{a}^{6}{c}^{2}b{x}^{2}}{2}}+{a}^{7}{c}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(b*c*x+a*c)^2,x)

[Out]

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

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Maxima [B]  time = 1.00701, size = 134, normalized size = 7.88 \begin{align*} \frac{1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac{7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac{35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac{7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.37589, size = 203, normalized size = 11.94 \begin{align*} \frac{1}{8} x^{8} c^{2} b^{7} + x^{7} c^{2} b^{6} a + \frac{7}{2} x^{6} c^{2} b^{5} a^{2} + 7 x^{5} c^{2} b^{4} a^{3} + \frac{35}{4} x^{4} c^{2} b^{3} a^{4} + 7 x^{3} c^{2} b^{2} a^{5} + \frac{7}{2} x^{2} c^{2} b a^{6} + x c^{2} a^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.085406, size = 110, normalized size = 6.47 \begin{align*} a^{7} c^{2} x + \frac{7 a^{6} b c^{2} x^{2}}{2} + 7 a^{5} b^{2} c^{2} x^{3} + \frac{35 a^{4} b^{3} c^{2} x^{4}}{4} + 7 a^{3} b^{4} c^{2} x^{5} + \frac{7 a^{2} b^{5} c^{2} x^{6}}{2} + a b^{6} c^{2} x^{7} + \frac{b^{7} c^{2} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(b*c*x+a*c)**2,x)

[Out]

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

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Giac [B]  time = 1.07167, size = 134, normalized size = 7.88 \begin{align*} \frac{1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac{7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac{35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac{7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^2,x, algorithm="giac")

[Out]

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