3.1.69 \(\int (e x)^m (a+b x) (a c-b c x)^3 \, dx\) [69]

Optimal. Leaf size=94 \[ \frac {a^4 c^3 (e x)^{1+m}}{e (1+m)}-\frac {2 a^3 b c^3 (e x)^{2+m}}{e^2 (2+m)}+\frac {2 a b^3 c^3 (e x)^{4+m}}{e^4 (4+m)}-\frac {b^4 c^3 (e x)^{5+m}}{e^5 (5+m)} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 94, 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^4 c^3 (e x)^{m+1}}{e (m+1)}-\frac {2 a^3 b c^3 (e x)^{m+2}}{e^2 (m+2)}+\frac {2 a b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}-\frac {b^4 c^3 (e x)^{m+5}}{e^5 (m+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

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

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Mathematica [A]
time = 0.08, size = 112, normalized size = 1.19 \begin {gather*} -\frac {c^3 x (e x)^m \left (-a^4 \left (40+38 m+11 m^2+m^3\right )+2 a^3 b \left (20+29 m+10 m^2+m^3\right ) x-2 a b^3 \left (10+17 m+8 m^2+m^3\right ) x^3+b^4 \left (8+14 m+7 m^2+m^3\right ) x^4\right )}{(1+m) (2+m) (4+m) (5+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x)*(a*c - b*c*x)^3,x]

[Out]

-((c^3*x*(e*x)^m*(-(a^4*(40 + 38*m + 11*m^2 + m^3)) + 2*a^3*b*(20 + 29*m + 10*m^2 + m^3)*x - 2*a*b^3*(10 + 17*
m + 8*m^2 + m^3)*x^3 + b^4*(8 + 14*m + 7*m^2 + m^3)*x^4))/((1 + m)*(2 + m)*(4 + m)*(5 + m)))

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Maple [A]
time = 0.06, size = 93, normalized size = 0.99

method	result	size
norman	\(\frac {a^{4} c^{3} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}-\frac {b^{4} c^{3} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {2 a \,b^{3} c^{3} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {2 a^{3} c^{3} b \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}\)	\(93\)
gosper	\(\frac {c^{3} \left (e x \right )^{m} \left (-b^{4} m^{3} x^{4}+2 a \,b^{3} m^{3} x^{3}-7 b^{4} m^{2} x^{4}+16 a \,b^{3} m^{2} x^{3}-14 m \,x^{4} b^{4}-2 a^{3} b \,m^{3} x +34 a \,b^{3} m \,x^{3}-8 b^{4} x^{4}+a^{4} m^{3}-20 a^{3} b \,m^{2} x +20 a \,b^{3} x^{3}+11 a^{4} m^{2}-58 a^{3} b m x +38 a^{4} m -40 a^{3} b x +40 a^{4}\right ) x}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(175\)
risch	\(\frac {c^{3} \left (e x \right )^{m} \left (-b^{4} m^{3} x^{4}+2 a \,b^{3} m^{3} x^{3}-7 b^{4} m^{2} x^{4}+16 a \,b^{3} m^{2} x^{3}-14 m \,x^{4} b^{4}-2 a^{3} b \,m^{3} x +34 a \,b^{3} m \,x^{3}-8 b^{4} x^{4}+a^{4} m^{3}-20 a^{3} b \,m^{2} x +20 a \,b^{3} x^{3}+11 a^{4} m^{2}-58 a^{3} b m x +38 a^{4} m -40 a^{3} b x +40 a^{4}\right ) x}{\left (5+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x+a)*(-b*c*x+a*c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 94, normalized size = 1.00 \begin {gather*} -\frac {b^{4} c^{3} x^{5} e^{\left (m \log \left (x\right ) + m\right )}}{m + 5} + \frac {2 \, a b^{3} c^{3} x^{4} e^{\left (m \log \left (x\right ) + m\right )}}{m + 4} - \frac {2 \, a^{3} b c^{3} x^{2} e^{\left (m \log \left (x\right ) + m\right )}}{m + 2} + \frac {\left (x e\right )^{m + 1} a^{4} c^{3} 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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (94) = 188\).
time = 1.86, size = 210, normalized size = 2.23 \begin {gather*} -\frac {{\left ({\left (b^{4} c^{3} m^{3} + 7 \, b^{4} c^{3} m^{2} + 14 \, b^{4} c^{3} m + 8 \, b^{4} c^{3}\right )} x^{5} - 2 \, {\left (a b^{3} c^{3} m^{3} + 8 \, a b^{3} c^{3} m^{2} + 17 \, a b^{3} c^{3} m + 10 \, a b^{3} c^{3}\right )} x^{4} + 2 \, {\left (a^{3} b c^{3} m^{3} + 10 \, a^{3} b c^{3} m^{2} + 29 \, a^{3} b c^{3} m + 20 \, a^{3} b c^{3}\right )} x^{2} - {\left (a^{4} c^{3} m^{3} + 11 \, a^{4} c^{3} m^{2} + 38 \, a^{4} c^{3} m + 40 \, a^{4} c^{3}\right )} x\right )} \left (x e\right )^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \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)^3,x, algorithm="fricas")

[Out]

-((b^4*c^3*m^3 + 7*b^4*c^3*m^2 + 14*b^4*c^3*m + 8*b^4*c^3)*x^5 - 2*(a*b^3*c^3*m^3 + 8*a*b^3*c^3*m^2 + 17*a*b^3
*c^3*m + 10*a*b^3*c^3)*x^4 + 2*(a^3*b*c^3*m^3 + 10*a^3*b*c^3*m^2 + 29*a^3*b*c^3*m + 20*a^3*b*c^3)*x^2 - (a^4*c
^3*m^3 + 11*a^4*c^3*m^2 + 38*a^4*c^3*m + 40*a^4*c^3)*x)*(x*e)^m/(m^4 + 12*m^3 + 49*m^2 + 78*m + 40)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 811 vs. \(2 (85) = 170\).
time = 0.31, size = 811, normalized size = 8.63 \begin {gather*} \begin {cases} \frac {- \frac {a^{4} c^{3}}{4 x^{4}} + \frac {2 a^{3} b c^{3}}{3 x^{3}} - \frac {2 a b^{3} c^{3}}{x} - b^{4} c^{3} \log {\left (x \right )}}{e^{5}} & \text {for}\: m = -5 \\\frac {- \frac {a^{4} c^{3}}{3 x^{3}} + \frac {a^{3} b c^{3}}{x^{2}} + 2 a b^{3} c^{3} \log {\left (x \right )} - b^{4} c^{3} x}{e^{4}} & \text {for}\: m = -4 \\\frac {- \frac {a^{4} c^{3}}{x} - 2 a^{3} b c^{3} \log {\left (x \right )} + a b^{3} c^{3} x^{2} - \frac {b^{4} c^{3} x^{3}}{3}}{e^{2}} & \text {for}\: m = -2 \\\frac {a^{4} c^{3} \log {\left (x \right )} - 2 a^{3} b c^{3} x + \frac {2 a b^{3} c^{3} x^{3}}{3} - \frac {b^{4} c^{3} x^{4}}{4}}{e} & \text {for}\: m = -1 \\\frac {a^{4} c^{3} m^{3} x \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {11 a^{4} c^{3} m^{2} x \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {38 a^{4} c^{3} m x \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {40 a^{4} c^{3} x \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {2 a^{3} b c^{3} m^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {20 a^{3} b c^{3} m^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {58 a^{3} b c^{3} m x^{2} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {40 a^{3} b c^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {2 a b^{3} c^{3} m^{3} x^{4} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {16 a b^{3} c^{3} m^{2} x^{4} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {34 a b^{3} c^{3} m x^{4} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} + \frac {20 a b^{3} c^{3} x^{4} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {b^{4} c^{3} m^{3} x^{5} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {7 b^{4} c^{3} m^{2} x^{5} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {14 b^{4} c^{3} m x^{5} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} - \frac {8 b^{4} c^{3} x^{5} \left (e x\right )^{m}}{m^{4} + 12 m^{3} + 49 m^{2} + 78 m + 40} & \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)**3,x)

[Out]

Piecewise(((-a**4*c**3/(4*x**4) + 2*a**3*b*c**3/(3*x**3) - 2*a*b**3*c**3/x - b**4*c**3*log(x))/e**5, Eq(m, -5)
), ((-a**4*c**3/(3*x**3) + a**3*b*c**3/x**2 + 2*a*b**3*c**3*log(x) - b**4*c**3*x)/e**4, Eq(m, -4)), ((-a**4*c*
*3/x - 2*a**3*b*c**3*log(x) + a*b**3*c**3*x**2 - b**4*c**3*x**3/3)/e**2, Eq(m, -2)), ((a**4*c**3*log(x) - 2*a*
*3*b*c**3*x + 2*a*b**3*c**3*x**3/3 - b**4*c**3*x**4/4)/e, Eq(m, -1)), (a**4*c**3*m**3*x*(e*x)**m/(m**4 + 12*m*
*3 + 49*m**2 + 78*m + 40) + 11*a**4*c**3*m**2*x*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 38*a**4*c**3
*m*x*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 40*a**4*c**3*x*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*
m + 40) - 2*a**3*b*c**3*m**3*x**2*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 20*a**3*b*c**3*m**2*x**2*(
e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 58*a**3*b*c**3*m*x**2*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78
*m + 40) - 40*a**3*b*c**3*x**2*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 2*a*b**3*c**3*m**3*x**4*(e*x)
**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 16*a*b**3*c**3*m**2*x**4*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*
m + 40) + 34*a*b**3*c**3*m*x**4*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) + 20*a*b**3*c**3*x**4*(e*x)**m
/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - b**4*c**3*m**3*x**5*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40)
- 7*b**4*c**3*m**2*x**5*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40) - 14*b**4*c**3*m*x**5*(e*x)**m/(m**4 +
12*m**3 + 49*m**2 + 78*m + 40) - 8*b**4*c**3*x**5*(e*x)**m/(m**4 + 12*m**3 + 49*m**2 + 78*m + 40), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 306 vs. \(2 (94) = 188\).
time = 1.32, size = 306, normalized size = 3.26 \begin {gather*} -\frac {b^{4} c^{3} m^{3} x^{5} x^{m} e^{m} - 2 \, a b^{3} c^{3} m^{3} x^{4} x^{m} e^{m} + 7 \, b^{4} c^{3} m^{2} x^{5} x^{m} e^{m} - 16 \, a b^{3} c^{3} m^{2} x^{4} x^{m} e^{m} + 14 \, b^{4} c^{3} m x^{5} x^{m} e^{m} + 2 \, a^{3} b c^{3} m^{3} x^{2} x^{m} e^{m} - 34 \, a b^{3} c^{3} m x^{4} x^{m} e^{m} + 8 \, b^{4} c^{3} x^{5} x^{m} e^{m} - a^{4} c^{3} m^{3} x x^{m} e^{m} + 20 \, a^{3} b c^{3} m^{2} x^{2} x^{m} e^{m} - 20 \, a b^{3} c^{3} x^{4} x^{m} e^{m} - 11 \, a^{4} c^{3} m^{2} x x^{m} e^{m} + 58 \, a^{3} b c^{3} m x^{2} x^{m} e^{m} - 38 \, a^{4} c^{3} m x x^{m} e^{m} + 40 \, a^{3} b c^{3} x^{2} x^{m} e^{m} - 40 \, a^{4} c^{3} x x^{m} e^{m}}{m^{4} + 12 \, m^{3} + 49 \, m^{2} + 78 \, m + 40} \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)^3,x, algorithm="giac")

[Out]

-(b^4*c^3*m^3*x^5*x^m*e^m - 2*a*b^3*c^3*m^3*x^4*x^m*e^m + 7*b^4*c^3*m^2*x^5*x^m*e^m - 16*a*b^3*c^3*m^2*x^4*x^m
*e^m + 14*b^4*c^3*m*x^5*x^m*e^m + 2*a^3*b*c^3*m^3*x^2*x^m*e^m - 34*a*b^3*c^3*m*x^4*x^m*e^m + 8*b^4*c^3*x^5*x^m
*e^m - a^4*c^3*m^3*x*x^m*e^m + 20*a^3*b*c^3*m^2*x^2*x^m*e^m - 20*a*b^3*c^3*x^4*x^m*e^m - 11*a^4*c^3*m^2*x*x^m*
e^m + 58*a^3*b*c^3*m*x^2*x^m*e^m - 38*a^4*c^3*m*x*x^m*e^m + 40*a^3*b*c^3*x^2*x^m*e^m - 40*a^4*c^3*x*x^m*e^m)/(
m^4 + 12*m^3 + 49*m^2 + 78*m + 40)

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Mupad [B]
time = 0.46, size = 182, normalized size = 1.94 \begin {gather*} {\left (e\,x\right )}^m\,\left (\frac {a^4\,c^3\,x\,\left (m^3+11\,m^2+38\,m+40\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {b^4\,c^3\,x^5\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}+\frac {2\,a\,b^3\,c^3\,x^4\,\left (m^3+8\,m^2+17\,m+10\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}-\frac {2\,a^3\,b\,c^3\,x^2\,\left (m^3+10\,m^2+29\,m+20\right )}{m^4+12\,m^3+49\,m^2+78\,m+40}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^3*(e*x)^m*(a + b*x),x)

[Out]

(e*x)^m*((a^4*c^3*x*(38*m + 11*m^2 + m^3 + 40))/(78*m + 49*m^2 + 12*m^3 + m^4 + 40) - (b^4*c^3*x^5*(14*m + 7*m
^2 + m^3 + 8))/(78*m + 49*m^2 + 12*m^3 + m^4 + 40) + (2*a*b^3*c^3*x^4*(17*m + 8*m^2 + m^3 + 10))/(78*m + 49*m^
2 + 12*m^3 + m^4 + 40) - (2*a^3*b*c^3*x^2*(29*m + 10*m^2 + m^3 + 20))/(78*m + 49*m^2 + 12*m^3 + m^4 + 40))

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