3.1.4 \(\int (a+b x) (a c-b c x)^3 \, dx\)

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

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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 = {43} \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]

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

Rule 43

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 {1}{5} b^4 x^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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (a c-b c x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.13, size = 44, normalized size = 1.16 \begin {gather*} -\frac {1}{5} x^{5} c^{3} b^{4} + \frac {1}{2} x^{4} c^{3} b^{3} a - x^{2} c^{3} b a^{3} + x c^{3} a^{4} \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*x^5*c^3*b^4 + 1/2*x^4*c^3*b^3*a - x^2*c^3*b*a^3 + x*c^3*a^4

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giac [A]  time = 1.19, 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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maple [A]  time = 0.00, size = 45, normalized size = 1.18 \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]

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

[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 = 1.02, 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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mupad [B]  time = 0.05, 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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sympy [A]  time = 0.08, 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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