3.1016 \(\int (a+b x)^5 (a c+b c x)^3 \, dx\)

Optimal. Leaf size=17 \[ \frac{c^3 (a+b x)^9}{9 b} \]

[Out]

(c^3*(a + b*x)^9)/(9*b)

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Rubi [A]  time = 0.0038523, 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} \[ \frac{c^3 (a+b x)^9}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*(a*c + b*c*x)^3,x]

[Out]

(c^3*(a + b*x)^9)/(9*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)^3 \, dx &=c^3 \int (a+b x)^8 \, dx\\ &=\frac{c^3 (a+b x)^9}{9 b}\\ \end{align*}

Mathematica [A]  time = 0.0020879, size = 17, normalized size = 1. \[ \frac{c^3 (a+b x)^9}{9 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5*(a*c + b*c*x)^3,x]

[Out]

(c^3*(a + b*x)^9)/(9*b)

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Maple [B]  time = 0.001, size = 114, normalized size = 6.7 \begin{align*}{\frac{{b}^{8}{c}^{3}{x}^{9}}{9}}+a{b}^{7}{c}^{3}{x}^{8}+4\,{a}^{2}{b}^{6}{c}^{3}{x}^{7}+{\frac{28\,{a}^{3}{b}^{5}{c}^{3}{x}^{6}}{3}}+14\,{a}^{4}{b}^{4}{c}^{3}{x}^{5}+14\,{a}^{5}{b}^{3}{c}^{3}{x}^{4}+{\frac{28\,{a}^{6}{c}^{3}{b}^{2}{x}^{3}}{3}}+4\,{a}^{7}{c}^{3}b{x}^{2}+{a}^{8}{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(b*c*x+a*c)^3,x)

[Out]

1/9*b^8*c^3*x^9+a*b^7*c^3*x^8+4*a^2*b^6*c^3*x^7+28/3*a^3*b^5*c^3*x^6+14*a^4*b^4*c^3*x^5+14*a^5*b^3*c^3*x^4+28/
3*a^6*c^3*b^2*x^3+4*a^7*c^3*b*x^2+a^8*c^3*x

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Maxima [B]  time = 1.03238, size = 153, normalized size = 9. \begin{align*} \frac{1}{9} \, b^{8} c^{3} x^{9} + a b^{7} c^{3} x^{8} + 4 \, a^{2} b^{6} c^{3} x^{7} + \frac{28}{3} \, a^{3} b^{5} c^{3} x^{6} + 14 \, a^{4} b^{4} c^{3} x^{5} + 14 \, a^{5} b^{3} c^{3} x^{4} + \frac{28}{3} \, a^{6} b^{2} c^{3} x^{3} + 4 \, a^{7} b c^{3} x^{2} + a^{8} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

1/9*b^8*c^3*x^9 + a*b^7*c^3*x^8 + 4*a^2*b^6*c^3*x^7 + 28/3*a^3*b^5*c^3*x^6 + 14*a^4*b^4*c^3*x^5 + 14*a^5*b^3*c
^3*x^4 + 28/3*a^6*b^2*c^3*x^3 + 4*a^7*b*c^3*x^2 + a^8*c^3*x

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Fricas [B]  time = 1.32902, size = 231, normalized size = 13.59 \begin{align*} \frac{1}{9} x^{9} c^{3} b^{8} + x^{8} c^{3} b^{7} a + 4 x^{7} c^{3} b^{6} a^{2} + \frac{28}{3} x^{6} c^{3} b^{5} a^{3} + 14 x^{5} c^{3} b^{4} a^{4} + 14 x^{4} c^{3} b^{3} a^{5} + \frac{28}{3} x^{3} c^{3} b^{2} a^{6} + 4 x^{2} c^{3} b a^{7} + x c^{3} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

1/9*x^9*c^3*b^8 + x^8*c^3*b^7*a + 4*x^7*c^3*b^6*a^2 + 28/3*x^6*c^3*b^5*a^3 + 14*x^5*c^3*b^4*a^4 + 14*x^4*c^3*b
^3*a^5 + 28/3*x^3*c^3*b^2*a^6 + 4*x^2*c^3*b*a^7 + x*c^3*a^8

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Sympy [B]  time = 0.088348, size = 124, normalized size = 7.29 \begin{align*} a^{8} c^{3} x + 4 a^{7} b c^{3} x^{2} + \frac{28 a^{6} b^{2} c^{3} x^{3}}{3} + 14 a^{5} b^{3} c^{3} x^{4} + 14 a^{4} b^{4} c^{3} x^{5} + \frac{28 a^{3} b^{5} c^{3} x^{6}}{3} + 4 a^{2} b^{6} c^{3} x^{7} + a b^{7} c^{3} x^{8} + \frac{b^{8} c^{3} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(b*c*x+a*c)**3,x)

[Out]

a**8*c**3*x + 4*a**7*b*c**3*x**2 + 28*a**6*b**2*c**3*x**3/3 + 14*a**5*b**3*c**3*x**4 + 14*a**4*b**4*c**3*x**5
+ 28*a**3*b**5*c**3*x**6/3 + 4*a**2*b**6*c**3*x**7 + a*b**7*c**3*x**8 + b**8*c**3*x**9/9

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Giac [B]  time = 1.07406, size = 153, normalized size = 9. \begin{align*} \frac{1}{9} \, b^{8} c^{3} x^{9} + a b^{7} c^{3} x^{8} + 4 \, a^{2} b^{6} c^{3} x^{7} + \frac{28}{3} \, a^{3} b^{5} c^{3} x^{6} + 14 \, a^{4} b^{4} c^{3} x^{5} + 14 \, a^{5} b^{3} c^{3} x^{4} + \frac{28}{3} \, a^{6} b^{2} c^{3} x^{3} + 4 \, a^{7} b c^{3} x^{2} + a^{8} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(b*c*x+a*c)^3,x, algorithm="giac")

[Out]

1/9*b^8*c^3*x^9 + a*b^7*c^3*x^8 + 4*a^2*b^6*c^3*x^7 + 28/3*a^3*b^5*c^3*x^6 + 14*a^4*b^4*c^3*x^5 + 14*a^5*b^3*c
^3*x^4 + 28/3*a^6*b^2*c^3*x^3 + 4*a^7*b*c^3*x^2 + a^8*c^3*x