3.11.18 \(\int (a+b x)^5 (a c+b c x) \, dx\) [1018]

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

[Out]

1/7*c*(b*x+a)^7/b

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {c (a+b x)^7}{7 b} \end {gather*}

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 = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {c (a+b x)^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(a + b*x)^7)/(7*b)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(13)=26\).
time = 0.13, size = 72, normalized size = 4.80

method	result	size
gosper	\(\frac {c x \left (x^{6} b^{6}+7 a \,x^{5} b^{5}+21 a^{2} x^{4} b^{4}+35 a^{3} b^{3} x^{3}+35 a^{4} x^{2} b^{2}+21 a^{5} x b +7 a^{6}\right )}{7}\)	\(67\)
default	\(\frac {1}{7} b^{6} c \,x^{7}+a \,b^{5} c \,x^{6}+3 a^{2} b^{4} c \,x^{5}+5 a^{3} b^{3} c \,x^{4}+5 a^{4} b^{2} c \,x^{3}+3 a^{5} b c \,x^{2}+a^{6} c x\)	\(72\)
norman	\(\frac {1}{7} b^{6} c \,x^{7}+a \,b^{5} c \,x^{6}+3 a^{2} b^{4} c \,x^{5}+5 a^{3} b^{3} c \,x^{4}+5 a^{4} b^{2} c \,x^{3}+3 a^{5} b c \,x^{2}+a^{6} c x\)	\(72\)
risch	\(\frac {b^{6} c \,x^{7}}{7}+a \,b^{5} c \,x^{6}+3 a^{2} b^{4} c \,x^{5}+5 a^{3} b^{3} c \,x^{4}+5 a^{4} b^{2} c \,x^{3}+3 a^{5} b c \,x^{2}+a^{6} c x +\frac {c \,a^{7}}{7 b}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (13) = 26\).
time = 0.29, size = 71, normalized size = 4.73 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (13) = 26\).
time = 0.67, size = 71, normalized size = 4.73 \begin {gather*} \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 {gather*}

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*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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (10) = 20\).
time = 0.02, size = 78, normalized size = 5.20 \begin {gather*} 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 {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (13) = 26\).
time = 1.58, size = 71, normalized size = 4.73 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 71, normalized size = 4.73 \begin {gather*} c\,a^6\,x+3\,c\,a^5\,b\,x^2+5\,c\,a^4\,b^2\,x^3+5\,c\,a^3\,b^3\,x^4+3\,c\,a^2\,b^4\,x^5+c\,a\,b^5\,x^6+\frac {c\,b^6\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c + b*c*x)*(a + b*x)^5,x)

[Out]

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

________________________________________________________________________________________