3.11.28 \(\int (a+b x) (a c-b c x)^3 \, dx\) [1028]

Optimal. Leaf size=38 \[ -\frac {a c^3 (a-b x)^4}{2 b}+\frac {c^3 (a-b x)^5}{5 b} \]

[Out]

-1/2*a*c^3*(-b*x+a)^4/b+1/5*c^3*(-b*x+a)^5/b

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \begin {gather*} \frac {c^3 (a-b x)^5}{5 b}-\frac {a c^3 (a-b x)^4}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

-1/2*(a*c^3*(a - b*x)^4)/b + (c^3*(a - b*x)^5)/(5*b)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 40, normalized size = 1.05 \begin {gather*} c^3 \left (a^4 x-a^3 b x^2+\frac {1}{2} a b^3 x^4-\frac {b^4 x^5}{5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 45, normalized size = 1.18

method	result	size
gosper	\(\frac {x \left (-2 b^{4} x^{4}+5 a \,b^{3} x^{3}-10 a^{3} b x +10 a^{4}\right ) c^{3}}{10}\)	\(37\)
default	\(-\frac {1}{5} b^{4} c^{3} x^{5}+\frac {1}{2} a \,b^{3} c^{3} x^{4}-a^{3} b \,c^{3} x^{2}+a^{4} c^{3} x\)	\(45\)
norman	\(-\frac {1}{5} b^{4} c^{3} x^{5}+\frac {1}{2} a \,b^{3} c^{3} x^{4}-a^{3} b \,c^{3} x^{2}+a^{4} c^{3} x\)	\(45\)
risch	\(-\frac {1}{5} b^{4} c^{3} x^{5}+\frac {1}{2} a \,b^{3} c^{3} x^{4}-a^{3} b \,c^{3} x^{2}+a^{4} c^{3} x\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 44, normalized size = 1.16 \begin {gather*} -\frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.90, size = 44, normalized size = 1.16 \begin {gather*} -\frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.01, size = 44, normalized size = 1.16 \begin {gather*} a^{4} c^{3} x - a^{3} b c^{3} x^{2} + \frac {a b^{3} c^{3} x^{4}}{2} - \frac {b^{4} c^{3} x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*c**3*x - a**3*b*c**3*x**2 + a*b**3*c**3*x**4/2 - b**4*c**3*x**5/5

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Giac [A]
time = 2.59, size = 44, normalized size = 1.16 \begin {gather*} -\frac {1}{5} \, b^{4} c^{3} x^{5} + \frac {1}{2} \, a b^{3} c^{3} x^{4} - a^{3} b c^{3} x^{2} + a^{4} c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.16, size = 44, normalized size = 1.16 \begin {gather*} a^4\,c^3\,x-a^3\,b\,c^3\,x^2+\frac {a\,b^3\,c^3\,x^4}{2}-\frac {b^4\,c^3\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^3*(a + b*x),x)

[Out]

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

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