3.11.39 \(\int (a+b x)^2 (a c-b c x)^2 \, dx\) [1039]

Optimal. Leaf size=38 \[ a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {41, 200} \begin {gather*} a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(a*c - b*c*x)^2,x]

[Out]

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 38, normalized size = 1.00 \begin {gather*} a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 35, normalized size = 0.92

method	result	size
gosper	\(\frac {x \left (3 b^{4} x^{4}-10 a^{2} b^{2} x^{2}+15 a^{4}\right ) c^{2}}{15}\)	\(32\)
default	\(a^{4} c^{2} x -\frac {2}{3} a^{2} b^{2} c^{2} x^{3}+\frac {1}{5} b^{4} c^{2} x^{5}\)	\(35\)
norman	\(a^{4} c^{2} x -\frac {2}{3} a^{2} b^{2} c^{2} x^{3}+\frac {1}{5} b^{4} c^{2} x^{5}\)	\(35\)
risch	\(a^{4} c^{2} x -\frac {2}{3} a^{2} b^{2} c^{2} x^{3}+\frac {1}{5} b^{4} c^{2} x^{5}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(-b*c*x+a*c)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 34, normalized size = 0.89 \begin {gather*} \frac {1}{5} \, b^{4} c^{2} x^{5} - \frac {2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.53, size = 34, normalized size = 0.89 \begin {gather*} \frac {1}{5} \, b^{4} c^{2} x^{5} - \frac {2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 36, normalized size = 0.95 \begin {gather*} a^{4} c^{2} x - \frac {2 a^{2} b^{2} c^{2} x^{3}}{3} + \frac {b^{4} c^{2} x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(-b*c*x+a*c)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 3.11, size = 34, normalized size = 0.89 \begin {gather*} \frac {1}{5} \, b^{4} c^{2} x^{5} - \frac {2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 31, normalized size = 0.82 \begin {gather*} \frac {c^2\,x\,\left (15\,a^4-10\,a^2\,b^2\,x^2+3\,b^4\,x^4\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^2*(a + b*x)^2,x)

[Out]

(c^2*x*(15*a^4 + 3*b^4*x^4 - 10*a^2*b^2*x^2))/15

________________________________________________________________________________________