3.70 \(\int (e x)^m (a+b x) (a c-b c x)^2 \, dx\)

Optimal. Leaf size=93 \[ -\frac{a^2 b c^2 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^3 c^2 (e x)^{m+1}}{e (m+1)}-\frac{a b^2 c^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{b^3 c^2 (e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.0468879, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {75} \[ -\frac{a^2 b c^2 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^3 c^2 (e x)^{m+1}}{e (m+1)}-\frac{a b^2 c^2 (e x)^{m+3}}{e^3 (m+3)}+\frac{b^3 c^2 (e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x)*(a*c - b*c*x)^2,x]

[Out]

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

Rule 75

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

Mathematica [A]  time = 0.06659, size = 88, normalized size = 0.95 \[ \frac{c^2 x (e x)^m \left (\frac{a (2 m+5) \left (a^2 \left (m^2+5 m+6\right )-2 a b \left (m^2+4 m+3\right ) x+b^2 \left (m^2+3 m+2\right ) x^2\right )}{(m+1) (m+2) (m+3)}+(b x-a)^3\right )}{m+4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x*(e*x)^m*((-a + b*x)^3 + (a*(5 + 2*m)*(a^2*(6 + 5*m + m^2) - 2*a*b*(3 + 4*m + m^2)*x + b^2*(2 + 3*m + m^
2)*x^2))/((1 + m)*(2 + m)*(3 + m))))/(4 + m)

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Maple [A]  time = 0.005, size = 174, normalized size = 1.9 \begin{align*}{\frac{{c}^{2} \left ( ex \right ) ^{m} \left ({b}^{3}{m}^{3}{x}^{3}-a{b}^{2}{m}^{3}{x}^{2}+6\,{b}^{3}{m}^{2}{x}^{3}-{a}^{2}b{m}^{3}x-7\,a{b}^{2}{m}^{2}{x}^{2}+11\,{b}^{3}m{x}^{3}+{a}^{3}{m}^{3}-8\,{a}^{2}b{m}^{2}x-14\,a{b}^{2}m{x}^{2}+6\,{b}^{3}{x}^{3}+9\,{a}^{3}{m}^{2}-19\,{a}^{2}bmx-8\,a{b}^{2}{x}^{2}+26\,{a}^{3}m-12\,{a}^{2}bx+24\,{a}^{3} \right ) x}{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x+a)*(-b*c*x+a*c)^2,x)

[Out]

c^2*(e*x)^m*(b^3*m^3*x^3-a*b^2*m^3*x^2+6*b^3*m^2*x^3-a^2*b*m^3*x-7*a*b^2*m^2*x^2+11*b^3*m*x^3+a^3*m^3-8*a^2*b*
m^2*x-14*a*b^2*m*x^2+6*b^3*x^3+9*a^3*m^2-19*a^2*b*m*x-8*a*b^2*x^2+26*a^3*m-12*a^2*b*x+24*a^3)*x/(4+m)/(3+m)/(2
+m)/(1+m)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.81561, size = 423, normalized size = 4.55 \begin{align*} \frac{{\left ({\left (b^{3} c^{2} m^{3} + 6 \, b^{3} c^{2} m^{2} + 11 \, b^{3} c^{2} m + 6 \, b^{3} c^{2}\right )} x^{4} -{\left (a b^{2} c^{2} m^{3} + 7 \, a b^{2} c^{2} m^{2} + 14 \, a b^{2} c^{2} m + 8 \, a b^{2} c^{2}\right )} x^{3} -{\left (a^{2} b c^{2} m^{3} + 8 \, a^{2} b c^{2} m^{2} + 19 \, a^{2} b c^{2} m + 12 \, a^{2} b c^{2}\right )} x^{2} +{\left (a^{3} c^{2} m^{3} + 9 \, a^{3} c^{2} m^{2} + 26 \, a^{3} c^{2} m + 24 \, a^{3} c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^3*c^2*m^3 + 6*b^3*c^2*m^2 + 11*b^3*c^2*m + 6*b^3*c^2)*x^4 - (a*b^2*c^2*m^3 + 7*a*b^2*c^2*m^2 + 14*a*b^2*c^
2*m + 8*a*b^2*c^2)*x^3 - (a^2*b*c^2*m^3 + 8*a^2*b*c^2*m^2 + 19*a^2*b*c^2*m + 12*a^2*b*c^2)*x^2 + (a^3*c^2*m^3
+ 9*a^3*c^2*m^2 + 26*a^3*c^2*m + 24*a^3*c^2)*x)*(e*x)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

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Sympy [A]  time = 1.14576, size = 821, normalized size = 8.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-a**3*c**2/(3*x**3) + a**2*b*c**2/(2*x**2) + a*b**2*c**2/x + b**3*c**2*log(x))/e**4, Eq(m, -4)), (
(-a**3*c**2/(2*x**2) + a**2*b*c**2/x - a*b**2*c**2*log(x) + b**3*c**2*x)/e**3, Eq(m, -3)), ((-a**3*c**2/x - a*
*2*b*c**2*log(x) - a*b**2*c**2*x + b**3*c**2*x**2/2)/e**2, Eq(m, -2)), ((a**3*c**2*log(x) - a**2*b*c**2*x - a*
b**2*c**2*x**2/2 + b**3*c**2*x**3/3)/e, Eq(m, -1)), (a**3*c**2*e**m*m**3*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50
*m + 24) + 9*a**3*c**2*e**m*m**2*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*a**3*c**2*e**m*m*x*x**m/(m
**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 24*a**3*c**2*e**m*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - a**2*
b*c**2*e**m*m**3*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 8*a**2*b*c**2*e**m*m**2*x**2*x**m/(m**4 +
10*m**3 + 35*m**2 + 50*m + 24) - 19*a**2*b*c**2*e**m*m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 12*a
**2*b*c**2*e**m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - a*b**2*c**2*e**m*m**3*x**3*x**m/(m**4 + 10*
m**3 + 35*m**2 + 50*m + 24) - 7*a*b**2*c**2*e**m*m**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 14*a*
b**2*c**2*e**m*m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 8*a*b**2*c**2*e**m*x**3*x**m/(m**4 + 10*m*
*3 + 35*m**2 + 50*m + 24) + b**3*c**2*e**m*m**3*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 6*b**3*c**2
*e**m*m**2*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 11*b**3*c**2*e**m*m*x**4*x**m/(m**4 + 10*m**3 +
35*m**2 + 50*m + 24) + 6*b**3*c**2*e**m*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24), True))

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Giac [B]  time = 1.19902, size = 410, normalized size = 4.41 \begin{align*} \frac{b^{3} c^{2} m^{3} x^{4} x^{m} e^{m} - a b^{2} c^{2} m^{3} x^{3} x^{m} e^{m} + 6 \, b^{3} c^{2} m^{2} x^{4} x^{m} e^{m} - a^{2} b c^{2} m^{3} x^{2} x^{m} e^{m} - 7 \, a b^{2} c^{2} m^{2} x^{3} x^{m} e^{m} + 11 \, b^{3} c^{2} m x^{4} x^{m} e^{m} + a^{3} c^{2} m^{3} x x^{m} e^{m} - 8 \, a^{2} b c^{2} m^{2} x^{2} x^{m} e^{m} - 14 \, a b^{2} c^{2} m x^{3} x^{m} e^{m} + 6 \, b^{3} c^{2} x^{4} x^{m} e^{m} + 9 \, a^{3} c^{2} m^{2} x x^{m} e^{m} - 19 \, a^{2} b c^{2} m x^{2} x^{m} e^{m} - 8 \, a b^{2} c^{2} x^{3} x^{m} e^{m} + 26 \, a^{3} c^{2} m x x^{m} e^{m} - 12 \, a^{2} b c^{2} x^{2} x^{m} e^{m} + 24 \, a^{3} c^{2} x x^{m} e^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*c^2*m^3*x^4*x^m*e^m - a*b^2*c^2*m^3*x^3*x^m*e^m + 6*b^3*c^2*m^2*x^4*x^m*e^m - a^2*b*c^2*m^3*x^2*x^m*e^m -
7*a*b^2*c^2*m^2*x^3*x^m*e^m + 11*b^3*c^2*m*x^4*x^m*e^m + a^3*c^2*m^3*x*x^m*e^m - 8*a^2*b*c^2*m^2*x^2*x^m*e^m
- 14*a*b^2*c^2*m*x^3*x^m*e^m + 6*b^3*c^2*x^4*x^m*e^m + 9*a^3*c^2*m^2*x*x^m*e^m - 19*a^2*b*c^2*m*x^2*x^m*e^m -
8*a*b^2*c^2*x^3*x^m*e^m + 26*a^3*c^2*m*x*x^m*e^m - 12*a^2*b*c^2*x^2*x^m*e^m + 24*a^3*c^2*x*x^m*e^m)/(m^4 + 10*
m^3 + 35*m^2 + 50*m + 24)