3.20.40 \(\int (a+b x) (a c+b c x)^m (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=24 \[ \frac {(a c+b c x)^{m+8}}{b c^8 (m+8)} \]

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 27, 32} \begin {gather*} \frac {(a c+b c x)^{m+8}}{b c^8 (m+8)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c + b*c*x)^(8 + m)/(b*c^8*(8 + m))

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 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) (a c+b c x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\frac {\int (a c+b c x)^{1+m} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx}{c}\\ &=\frac {\int (a+b x)^6 (a c+b c x)^{1+m} \, dx}{c}\\ &=\frac {\int (a c+b c x)^{7+m} \, dx}{c^7}\\ &=\frac {(a c+b c x)^{8+m}}{b c^8 (8+m)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.04 \begin {gather*} \frac {(a+b x)^8 (c (a+b x))^m}{b (m+8)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^8*(c*(a + b*x))^m)/(b*(8 + m))

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IntegrateAlgebraic [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (a c+b c x)^m \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)*(a*c + b*c*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

Defer[IntegrateAlgebraic][(a + b*x)*(a*c + b*c*x)^m*(a^2 + 2*a*b*x + b^2*x^2)^3, x]

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fricas [B]  time = 0.42, size = 102, normalized size = 4.25 \begin {gather*} \frac {{\left (b^{8} x^{8} + 8 \, a b^{7} x^{7} + 28 \, a^{2} b^{6} x^{6} + 56 \, a^{3} b^{5} x^{5} + 70 \, a^{4} b^{4} x^{4} + 56 \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2} x^{2} + 8 \, a^{7} b x + a^{8}\right )} {\left (b c x + a c\right )}^{m}}{b m + 8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^8*x^8 + 8*a*b^7*x^7 + 28*a^2*b^6*x^6 + 56*a^3*b^5*x^5 + 70*a^4*b^4*x^4 + 56*a^5*b^3*x^3 + 28*a^6*b^2*x^2 +
8*a^7*b*x + a^8)*(b*c*x + a*c)^m/(b*m + 8*b)

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giac [B]  time = 0.21, size = 183, normalized size = 7.62 \begin {gather*} \frac {{\left (b c x + a c\right )}^{m} b^{8} x^{8} + 8 \, {\left (b c x + a c\right )}^{m} a b^{7} x^{7} + 28 \, {\left (b c x + a c\right )}^{m} a^{2} b^{6} x^{6} + 56 \, {\left (b c x + a c\right )}^{m} a^{3} b^{5} x^{5} + 70 \, {\left (b c x + a c\right )}^{m} a^{4} b^{4} x^{4} + 56 \, {\left (b c x + a c\right )}^{m} a^{5} b^{3} x^{3} + 28 \, {\left (b c x + a c\right )}^{m} a^{6} b^{2} x^{2} + 8 \, {\left (b c x + a c\right )}^{m} a^{7} b x + {\left (b c x + a c\right )}^{m} a^{8}}{b m + 8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*c*x + a*c)^m*b^8*x^8 + 8*(b*c*x + a*c)^m*a*b^7*x^7 + 28*(b*c*x + a*c)^m*a^2*b^6*x^6 + 56*(b*c*x + a*c)^m*a
^3*b^5*x^5 + 70*(b*c*x + a*c)^m*a^4*b^4*x^4 + 56*(b*c*x + a*c)^m*a^5*b^3*x^3 + 28*(b*c*x + a*c)^m*a^6*b^2*x^2
+ 8*(b*c*x + a*c)^m*a^7*b*x + (b*c*x + a*c)^m*a^8)/(b*m + 8*b)

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maple [A]  time = 0.05, size = 45, normalized size = 1.88 \begin {gather*} \frac {\left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3} \left (b c x +a c \right )^{m}}{\left (m +8\right ) b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^2/b/(m+8)*(b*c*x+a*c)^m*(b^2*x^2+2*a*b*x+a^2)^3

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maxima [B]  time = 0.91, size = 1221, normalized size = 50.88

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7*(b^2*c^m*(m + 1)*x^2 + a*b*c^m*m*x - a^2*c^m)*(b*x + a)^m*a^6/((m^2 + 3*m + 2)*b) + 21*((m^2 + 3*m + 2)*b^3*
c^m*x^3 + (m^2 + m)*a*b^2*c^m*x^2 - 2*a^2*b*c^m*m*x + 2*a^3*c^m)*(b*x + a)^m*a^5/((m^3 + 6*m^2 + 11*m + 6)*b)
+ (b*c*x + a*c)^(m + 1)*a^7/(b*c*(m + 1)) + 35*((m^3 + 6*m^2 + 11*m + 6)*b^4*c^m*x^4 + (m^3 + 3*m^2 + 2*m)*a*b
^3*c^m*x^3 - 3*(m^2 + m)*a^2*b^2*c^m*x^2 + 6*a^3*b*c^m*m*x - 6*a^4*c^m)*(b*x + a)^m*a^4/((m^4 + 10*m^3 + 35*m^
2 + 50*m + 24)*b) + 35*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*b^5*c^m*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a*b^4*c
^m*x^4 - 4*(m^3 + 3*m^2 + 2*m)*a^2*b^3*c^m*x^3 + 12*(m^2 + m)*a^3*b^2*c^m*x^2 - 24*a^4*b*c^m*m*x + 24*a^5*c^m)
*(b*x + a)^m*a^3/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b) + 21*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 +
274*m + 120)*b^6*c^m*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a*b^5*c^m*x^5 - 5*(m^4 + 6*m^3 + 11*m^2 + 6
*m)*a^2*b^4*c^m*x^4 + 20*(m^3 + 3*m^2 + 2*m)*a^3*b^3*c^m*x^3 - 60*(m^2 + m)*a^4*b^2*c^m*x^2 + 120*a^5*b*c^m*m*
x - 120*a^6*c^m)*(b*x + a)^m*a^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b) + 7*((m^6 +
21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*b^7*c^m*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2
+ 120*m)*a*b^6*c^m*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^2*b^5*c^m*x^5 + 30*(m^4 + 6*m^3 + 11*m^2
+ 6*m)*a^3*b^4*c^m*x^4 - 120*(m^3 + 3*m^2 + 2*m)*a^4*b^3*c^m*x^3 + 360*(m^2 + m)*a^5*b^2*c^m*x^2 - 720*a^6*b*
c^m*m*x + 720*a^7*c^m)*(b*x + a)^m*a/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 50
40)*b) + ((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*b^8*c^m*x^8 + (m^7 + 21*
m^6 + 175*m^5 + 735*m^4 + 1624*m^3 + 1764*m^2 + 720*m)*a*b^7*c^m*x^7 - 7*(m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 27
4*m^2 + 120*m)*a^2*b^6*c^m*x^6 + 42*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^3*b^5*c^m*x^5 - 210*(m^4 + 6*m^3
+ 11*m^2 + 6*m)*a^4*b^4*c^m*x^4 + 840*(m^3 + 3*m^2 + 2*m)*a^5*b^3*c^m*x^3 - 2520*(m^2 + m)*a^6*b^2*c^m*x^2 +
5040*a^7*b*c^m*m*x - 5040*a^8*c^m)*(b*x + a)^m/((m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 1
18124*m^2 + 109584*m + 40320)*b)

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mupad [B]  time = 2.18, size = 139, normalized size = 5.79 \begin {gather*} {\left (a\,c+b\,c\,x\right )}^m\,\left (\frac {a^8}{b\,\left (m+8\right )}+\frac {b^7\,x^8}{m+8}+\frac {8\,a^7\,x}{m+8}+\frac {28\,a^6\,b\,x^2}{m+8}+\frac {8\,a\,b^6\,x^7}{m+8}+\frac {56\,a^5\,b^2\,x^3}{m+8}+\frac {70\,a^4\,b^3\,x^4}{m+8}+\frac {56\,a^3\,b^4\,x^5}{m+8}+\frac {28\,a^2\,b^5\,x^6}{m+8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*c + b*c*x)^m*(a^8/(b*(m + 8)) + (b^7*x^8)/(m + 8) + (8*a^7*x)/(m + 8) + (28*a^6*b*x^2)/(m + 8) + (8*a*b^6*x
^7)/(m + 8) + (56*a^5*b^2*x^3)/(m + 8) + (70*a^4*b^3*x^4)/(m + 8) + (56*a^3*b^4*x^5)/(m + 8) + (28*a^2*b^5*x^6
)/(m + 8))

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sympy [A]  time = 3.92, size = 270, normalized size = 11.25 \begin {gather*} \begin {cases} \frac {x}{a c^{8}} & \text {for}\: b = 0 \wedge m = -8 \\a^{7} x \left (a c\right )^{m} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + x \right )}}{b c^{8}} & \text {for}\: m = -8 \\\frac {a^{8} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac {8 a^{7} b x \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac {28 a^{6} b^{2} x^{2} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac {56 a^{5} b^{3} x^{3} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac {70 a^{4} b^{4} x^{4} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac {56 a^{3} b^{5} x^{5} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac {28 a^{2} b^{6} x^{6} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac {8 a b^{7} x^{7} \left (a c + b c x\right )^{m}}{b m + 8 b} + \frac {b^{8} x^{8} \left (a c + b c x\right )^{m}}{b m + 8 b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/(a*c**8), Eq(b, 0) & Eq(m, -8)), (a**7*x*(a*c)**m, Eq(b, 0)), (log(a/b + x)/(b*c**8), Eq(m, -8)),
(a**8*(a*c + b*c*x)**m/(b*m + 8*b) + 8*a**7*b*x*(a*c + b*c*x)**m/(b*m + 8*b) + 28*a**6*b**2*x**2*(a*c + b*c*x
)**m/(b*m + 8*b) + 56*a**5*b**3*x**3*(a*c + b*c*x)**m/(b*m + 8*b) + 70*a**4*b**4*x**4*(a*c + b*c*x)**m/(b*m +
8*b) + 56*a**3*b**5*x**5*(a*c + b*c*x)**m/(b*m + 8*b) + 28*a**2*b**6*x**6*(a*c + b*c*x)**m/(b*m + 8*b) + 8*a*b
**7*x**7*(a*c + b*c*x)**m/(b*m + 8*b) + b**8*x**8*(a*c + b*c*x)**m/(b*m + 8*b), True))

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