3.1018 \(\int (a+b x)^5 (a c+b c x) \, dx\)

Optimal. Leaf size=15 \[ \frac{c (a+b x)^7}{7 b} \]

[Out]

(c*(a + b*x)^7)/(7*b)

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Rubi [A]  time = 0.0029926, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {21, 32} \[ \frac{c (a+b x)^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*(a*c + b*c*x),x]

[Out]

(c*(a + b*x)^7)/(7*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) \, dx &=c \int (a+b x)^6 \, dx\\ &=\frac{c (a+b x)^7}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0017526, size = 15, normalized size = 1. \[ \frac{c (a+b x)^7}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(a + b*x)^7)/(7*b)

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Maple [B]  time = 0.001, size = 72, normalized size = 4.8 \begin{align*}{\frac{{b}^{6}c{x}^{7}}{7}}+a{b}^{5}c{x}^{6}+3\,{a}^{2}{b}^{4}c{x}^{5}+5\,{a}^{3}{b}^{3}c{x}^{4}+5\,{a}^{4}{b}^{2}c{x}^{3}+3\,{a}^{5}bc{x}^{2}+{a}^{6}cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.993697, size = 96, normalized size = 6.4 \begin{align*} \frac{1}{7} \, b^{6} c x^{7} + a b^{5} c x^{6} + 3 \, a^{2} b^{4} c x^{5} + 5 \, a^{3} b^{3} c x^{4} + 5 \, a^{4} b^{2} c x^{3} + 3 \, a^{5} b c x^{2} + a^{6} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.3624, size = 147, normalized size = 9.8 \begin{align*} \frac{1}{7} x^{7} c b^{6} + x^{6} c b^{5} a + 3 x^{5} c b^{4} a^{2} + 5 x^{4} c b^{3} a^{3} + 5 x^{3} c b^{2} a^{4} + 3 x^{2} c b a^{5} + x c a^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.075771, size = 78, normalized size = 5.2 \begin{align*} a^{6} c x + 3 a^{5} b c x^{2} + 5 a^{4} b^{2} c x^{3} + 5 a^{3} b^{3} c x^{4} + 3 a^{2} b^{4} c x^{5} + a b^{5} c x^{6} + \frac{b^{6} c x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.05803, size = 96, normalized size = 6.4 \begin{align*} \frac{1}{7} \, b^{6} c x^{7} + a b^{5} c x^{6} + 3 \, a^{2} b^{4} c x^{5} + 5 \, a^{3} b^{3} c x^{4} + 5 \, a^{4} b^{2} c x^{3} + 3 \, a^{5} b c x^{2} + a^{6} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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