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

Optimal result

Integrand size = 19, antiderivative size = 38 \[ \int (a+b x)^2 (a c-b c x)^2 \, dx=a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5 \]

[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] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {41, 200} \[ \int (a+b x)^2 (a c-b c x)^2 \, dx=a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5 \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int (a+b x)^2 (a c-b c x)^2 \, dx=a^4 c^2 x-\frac {2}{3} a^2 b^2 c^2 x^3+\frac {1}{5} b^4 c^2 x^5 \]

[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] (verified)

Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84

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	\(a^{4} c^{2} x -\frac {2}{3} a^{2} b^{2} c^{2} x^{3}+\frac {1}{5} b^{4} c^{2} x^{5}\)	\(35\)
parallelrisch	\(a^{4} c^{2} x -\frac {2}{3} a^{2} b^{2} c^{2} x^{3}+\frac {1}{5} b^{4} c^{2} x^{5}\)	\(35\)

[In]

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

[Out]

1/15*x*(3*b^4*x^4-10*a^2*b^2*x^2+15*a^4)*c^2

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int (a+b x)^2 (a c-b c x)^2 \, dx=\frac {1}{5} \, b^{4} c^{2} x^{5} - \frac {2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \]

[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] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int (a+b x)^2 (a c-b c x)^2 \, dx=a^{4} c^{2} x - \frac {2 a^{2} b^{2} c^{2} x^{3}}{3} + \frac {b^{4} c^{2} x^{5}}{5} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int (a+b x)^2 (a c-b c x)^2 \, dx=\frac {1}{5} \, b^{4} c^{2} x^{5} - \frac {2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int (a+b x)^2 (a c-b c x)^2 \, dx=\frac {1}{5} \, b^{4} c^{2} x^{5} - \frac {2}{3} \, a^{2} b^{2} c^{2} x^{3} + a^{4} c^{2} x \]

[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] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int (a+b x)^2 (a c-b c x)^2 \, dx=\frac {c^2\,x\,\left (15\,a^4-10\,a^2\,b^2\,x^2+3\,b^4\,x^4\right )}{15} \]

[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