3.1.14 \(\int x^3 (a+b x) (a c-b c x)^4 \, dx\) [14]

Optimal. Leaf size=87 \[ \frac {1}{4} a^5 c^4 x^4-\frac {3}{5} a^4 b c^4 x^5+\frac {1}{3} a^3 b^2 c^4 x^6+\frac {2}{7} a^2 b^3 c^4 x^7-\frac {3}{8} a b^4 c^4 x^8+\frac {1}{9} b^5 c^4 x^9 \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \begin {gather*} \frac {1}{4} a^5 c^4 x^4-\frac {3}{5} a^4 b c^4 x^5+\frac {1}{3} a^3 b^2 c^4 x^6+\frac {2}{7} a^2 b^3 c^4 x^7-\frac {3}{8} a b^4 c^4 x^8+\frac {1}{9} b^5 c^4 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(a^5*c^4*x^4)/4 - (3*a^4*b*c^4*x^5)/5 + (a^3*b^2*c^4*x^6)/3 + (2*a^2*b^3*c^4*x^7)/7 - (3*a*b^4*c^4*x^8)/8 + (b
^5*c^4*x^9)/9

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

\begin {align*} \int x^3 (a+b x) (a c-b c x)^4 \, dx &=\int \left (a^5 c^4 x^3-3 a^4 b c^4 x^4+2 a^3 b^2 c^4 x^5+2 a^2 b^3 c^4 x^6-3 a b^4 c^4 x^7+b^5 c^4 x^8\right ) \, dx\\ &=\frac {1}{4} a^5 c^4 x^4-\frac {3}{5} a^4 b c^4 x^5+\frac {1}{3} a^3 b^2 c^4 x^6+\frac {2}{7} a^2 b^3 c^4 x^7-\frac {3}{8} a b^4 c^4 x^8+\frac {1}{9} b^5 c^4 x^9\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 87, normalized size = 1.00 \begin {gather*} \frac {1}{4} a^5 c^4 x^4-\frac {3}{5} a^4 b c^4 x^5+\frac {1}{3} a^3 b^2 c^4 x^6+\frac {2}{7} a^2 b^3 c^4 x^7-\frac {3}{8} a b^4 c^4 x^8+\frac {1}{9} b^5 c^4 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(a^5*c^4*x^4)/4 - (3*a^4*b*c^4*x^5)/5 + (a^3*b^2*c^4*x^6)/3 + (2*a^2*b^3*c^4*x^7)/7 - (3*a*b^4*c^4*x^8)/8 + (b
^5*c^4*x^9)/9

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Maple [A]
time = 0.06, size = 76, normalized size = 0.87

method	result	size
gosper	\(\frac {x^{4} \left (280 b^{5} x^{5}-945 a \,b^{4} x^{4}+720 a^{2} b^{3} x^{3}+840 a^{3} b^{2} x^{2}-1512 a^{4} b x +630 a^{5}\right ) c^{4}}{2520}\)	\(61\)
default	\(\frac {1}{4} a^{5} c^{4} x^{4}-\frac {3}{5} a^{4} b \,c^{4} x^{5}+\frac {1}{3} a^{3} b^{2} c^{4} x^{6}+\frac {2}{7} a^{2} b^{3} c^{4} x^{7}-\frac {3}{8} a \,b^{4} c^{4} x^{8}+\frac {1}{9} b^{5} c^{4} x^{9}\)	\(76\)
norman	\(\frac {1}{4} a^{5} c^{4} x^{4}-\frac {3}{5} a^{4} b \,c^{4} x^{5}+\frac {1}{3} a^{3} b^{2} c^{4} x^{6}+\frac {2}{7} a^{2} b^{3} c^{4} x^{7}-\frac {3}{8} a \,b^{4} c^{4} x^{8}+\frac {1}{9} b^{5} c^{4} x^{9}\)	\(76\)
risch	\(\frac {1}{4} a^{5} c^{4} x^{4}-\frac {3}{5} a^{4} b \,c^{4} x^{5}+\frac {1}{3} a^{3} b^{2} c^{4} x^{6}+\frac {2}{7} a^{2} b^{3} c^{4} x^{7}-\frac {3}{8} a \,b^{4} c^{4} x^{8}+\frac {1}{9} b^{5} c^{4} x^{9}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)*(-b*c*x+a*c)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 75, normalized size = 0.86 \begin {gather*} \frac {1}{9} \, b^{5} c^{4} x^{9} - \frac {3}{8} \, a b^{4} c^{4} x^{8} + \frac {2}{7} \, a^{2} b^{3} c^{4} x^{7} + \frac {1}{3} \, a^{3} b^{2} c^{4} x^{6} - \frac {3}{5} \, a^{4} b c^{4} x^{5} + \frac {1}{4} \, a^{5} c^{4} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.22, size = 75, normalized size = 0.86 \begin {gather*} \frac {1}{9} \, b^{5} c^{4} x^{9} - \frac {3}{8} \, a b^{4} c^{4} x^{8} + \frac {2}{7} \, a^{2} b^{3} c^{4} x^{7} + \frac {1}{3} \, a^{3} b^{2} c^{4} x^{6} - \frac {3}{5} \, a^{4} b c^{4} x^{5} + \frac {1}{4} \, a^{5} c^{4} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.01, size = 85, normalized size = 0.98 \begin {gather*} \frac {a^{5} c^{4} x^{4}}{4} - \frac {3 a^{4} b c^{4} x^{5}}{5} + \frac {a^{3} b^{2} c^{4} x^{6}}{3} + \frac {2 a^{2} b^{3} c^{4} x^{7}}{7} - \frac {3 a b^{4} c^{4} x^{8}}{8} + \frac {b^{5} c^{4} x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

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

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Giac [A]
time = 1.83, size = 75, normalized size = 0.86 \begin {gather*} \frac {1}{9} \, b^{5} c^{4} x^{9} - \frac {3}{8} \, a b^{4} c^{4} x^{8} + \frac {2}{7} \, a^{2} b^{3} c^{4} x^{7} + \frac {1}{3} \, a^{3} b^{2} c^{4} x^{6} - \frac {3}{5} \, a^{4} b c^{4} x^{5} + \frac {1}{4} \, a^{5} c^{4} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 75, normalized size = 0.86 \begin {gather*} \frac {a^5\,c^4\,x^4}{4}-\frac {3\,a^4\,b\,c^4\,x^5}{5}+\frac {a^3\,b^2\,c^4\,x^6}{3}+\frac {2\,a^2\,b^3\,c^4\,x^7}{7}-\frac {3\,a\,b^4\,c^4\,x^8}{8}+\frac {b^5\,c^4\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a*c - b*c*x)^4*(a + b*x),x)

[Out]

(a^5*c^4*x^4)/4 + (b^5*c^4*x^9)/9 - (3*a^4*b*c^4*x^5)/5 - (3*a*b^4*c^4*x^8)/8 + (a^3*b^2*c^4*x^6)/3 + (2*a^2*b
^3*c^4*x^7)/7

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