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

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

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Rubi [A]  time = 0.05, antiderivative size = 93, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.09, size = 88, normalized size = 0.95 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m (a+b x) (a c-b c x)^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.22, size = 207, normalized size = 2.23 \begin {gather*} \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 {gather*}

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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giac [B]  time = 1.27, size = 304, normalized size = 3.27 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 174, normalized size = 1.87 \begin {gather*} \frac {\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} b m x -8 a \,b^{2} x^{2}+26 a^{3} m -12 a^{2} b x +24 a^{3}\right ) c^{2} x \left (e x \right )^{m}}{\left (m +4\right ) \left (m +3\right ) \left (m +2\right ) \left (m +1\right )} \end {gather*}

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/(m+4)/(m+3)/(m
+2)/(m+1)

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maxima [A]  time = 1.15, size = 90, normalized size = 0.97 \begin {gather*} \frac {b^{3} c^{2} e^{m} x^{4} x^{m}}{m + 4} - \frac {a b^{2} c^{2} e^{m} x^{3} x^{m}}{m + 3} - \frac {a^{2} b c^{2} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{3} c^{2}}{e {\left (m + 1\right )}} \end {gather*}

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]

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

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mupad [B]  time = 0.43, size = 181, normalized size = 1.95 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {a^3\,c^2\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b^3\,c^2\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a\,b^2\,c^2\,x^3\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}-\frac {a^2\,b\,c^2\,x^2\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.23, size = 821, normalized size = 8.83 \begin {gather*} \begin {cases} \frac {- \frac {a^{3} c^{2}}{3 x^{3}} + \frac {a^{2} b c^{2}}{2 x^{2}} + \frac {a b^{2} c^{2}}{x} + b^{3} c^{2} \log {\relax (x )}}{e^{4}} & \text {for}\: m = -4 \\\frac {- \frac {a^{3} c^{2}}{2 x^{2}} + \frac {a^{2} b c^{2}}{x} - a b^{2} c^{2} \log {\relax (x )} + b^{3} c^{2} x}{e^{3}} & \text {for}\: m = -3 \\\frac {- \frac {a^{3} c^{2}}{x} - a^{2} b c^{2} \log {\relax (x )} - a b^{2} c^{2} x + \frac {b^{3} c^{2} x^{2}}{2}}{e^{2}} & \text {for}\: m = -2 \\\frac {a^{3} c^{2} \log {\relax (x )} - a^{2} b c^{2} x - \frac {a b^{2} c^{2} x^{2}}{2} + \frac {b^{3} c^{2} x^{3}}{3}}{e} & \text {for}\: m = -1 \\\frac {a^{3} c^{2} e^{m} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 a^{3} c^{2} e^{m} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 a^{3} c^{2} e^{m} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a^{3} c^{2} e^{m} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {a^{2} b c^{2} e^{m} m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {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} - \frac {19 a^{2} b c^{2} e^{m} m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {12 a^{2} b c^{2} e^{m} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {a b^{2} c^{2} e^{m} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {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} - \frac {14 a b^{2} c^{2} e^{m} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {8 a b^{2} c^{2} e^{m} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {b^{3} c^{2} e^{m} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} c^{2} e^{m} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {11 b^{3} c^{2} e^{m} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} c^{2} e^{m} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text {otherwise} \end {cases} \end {gather*}

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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