3.1.70 \(\int (e x)^m (a+b x) (a c-b c x)^2 \, dx\) [70]

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

[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 = 0.03, 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 = {76} \begin {gather*} \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)} \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 76

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 = 0.08, size = 88, normalized size = 0.95 \begin {gather*} \frac {c^2 x (e x)^m \left ((-a+b x)^3+\frac {a (5+2 m) \left (a^2 \left (6+5 m+m^2\right )-2 a b \left (3+4 m+m^2\right ) x+b^2 \left (2+3 m+m^2\right ) x^2\right )}{(1+m) (2+m) (3+m)}\right )}{4+m} \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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Maple [A]
time = 0.09, size = 92, normalized size = 0.99

method	result	size
norman	\(\frac {a^{3} c^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b^{3} c^{2} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {a \,b^{2} c^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}-\frac {a^{2} b \,c^{2} x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}\)	\(92\)
gosper	\(\frac {c^{2} \left (e x \right )^{m} \left (b^{3} m^{3} x^{3}-a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}-a^{2} b \,m^{3} x -7 a \,b^{2} m^{2} x^{2}+11 m \,x^{3} b^{3}+a^{3} m^{3}-8 a^{2} b \,m^{2} x -14 a \,b^{2} m \,x^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}-19 a^{2} b m x -8 a \,b^{2} x^{2}+26 a^{3} m -12 a^{2} b x +24 a^{3}\right ) x}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(174\)
risch	\(\frac {c^{2} \left (e x \right )^{m} \left (b^{3} m^{3} x^{3}-a \,b^{2} m^{3} x^{2}+6 b^{3} m^{2} x^{3}-a^{2} b \,m^{3} x -7 a \,b^{2} m^{2} x^{2}+11 m \,x^{3} b^{3}+a^{3} m^{3}-8 a^{2} b \,m^{2} x -14 a \,b^{2} m \,x^{2}+6 b^{3} x^{3}+9 a^{3} m^{2}-19 a^{2} b m x -8 a \,b^{2} x^{2}+26 a^{3} m -12 a^{2} b x +24 a^{3}\right ) x}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(174\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 93, normalized size = 1.00 \begin {gather*} \frac {b^{3} c^{2} x^{4} e^{\left (m \log \left (x\right ) + m\right )}}{m + 4} - \frac {a b^{2} c^{2} x^{3} e^{\left (m \log \left (x\right ) + m\right )}}{m + 3} - \frac {a^{2} b c^{2} x^{2} e^{\left (m \log \left (x\right ) + m\right )}}{m + 2} + \frac {\left (x e\right )^{m + 1} a^{3} c^{2} e^{\left (-1\right )}}{m + 1} \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*x^4*e^(m*log(x) + m)/(m + 4) - a*b^2*c^2*x^3*e^(m*log(x) + m)/(m + 3) - a^2*b*c^2*x^2*e^(m*log(x) + m)
/(m + 2) + (x*e)^(m + 1)*a^3*c^2*e^(-1)/(m + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (93) = 186\).
time = 1.13, size = 208, normalized size = 2.24 \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 (x e\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)*(x*e)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (82) = 164\).
time = 0.27, size = 794, normalized size = 8.54 \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 {\left (x \right )}}{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 {\left (x \right )} + b^{3} c^{2} x}{e^{3}} & \text {for}\: m = -3 \\\frac {- \frac {a^{3} c^{2}}{x} - a^{2} b c^{2} \log {\left (x \right )} - 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 {\left (x \right )} - 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} m^{3} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 a^{3} c^{2} m^{2} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 a^{3} c^{2} m x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 a^{3} c^{2} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {a^{2} b c^{2} m^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {8 a^{2} b c^{2} m^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {19 a^{2} b c^{2} m x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {12 a^{2} b c^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {a b^{2} c^{2} m^{3} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {7 a b^{2} c^{2} m^{2} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {14 a b^{2} c^{2} m x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} - \frac {8 a b^{2} c^{2} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {b^{3} c^{2} m^{3} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} c^{2} m^{2} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {11 b^{3} c^{2} m x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 b^{3} c^{2} x^{4} \left (e x\right )^{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*m**3*x*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*
m + 24) + 9*a**3*c**2*m**2*x*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*a**3*c**2*m*x*(e*x)**m/(m**4
+ 10*m**3 + 35*m**2 + 50*m + 24) + 24*a**3*c**2*x*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - a**2*b*c*
*2*m**3*x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 8*a**2*b*c**2*m**2*x**2*(e*x)**m/(m**4 + 10*m**
3 + 35*m**2 + 50*m + 24) - 19*a**2*b*c**2*m*x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 12*a**2*b*c
**2*x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - a*b**2*c**2*m**3*x**3*(e*x)**m/(m**4 + 10*m**3 + 35
*m**2 + 50*m + 24) - 7*a*b**2*c**2*m**2*x**3*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 14*a*b**2*c**2*
m*x**3*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) - 8*a*b**2*c**2*x**3*(e*x)**m/(m**4 + 10*m**3 + 35*m**2
+ 50*m + 24) + b**3*c**2*m**3*x**4*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 6*b**3*c**2*m**2*x**4*(e
*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 11*b**3*c**2*m*x**4*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m
+ 24) + 6*b**3*c**2*x**4*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (93) = 186\).
time = 1.11, 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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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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