3.442 \(\int x^m (a+b x)^3 \, dx\)

Optimal. Leaf size=61 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{3 a^2 b x^{m+2}}{m+2}+\frac{3 a b^2 x^{m+3}}{m+3}+\frac{b^3 x^{m+4}}{m+4} \]

[Out]

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

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Rubi [A]  time = 0.051404, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^3 x^{m+1}}{m+1}+\frac{3 a^2 b x^{m+2}}{m+2}+\frac{3 a b^2 x^{m+3}}{m+3}+\frac{b^3 x^{m+4}}{m+4} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.31098, size = 53, normalized size = 0.87 \[ \frac{a^{3} x^{m + 1}}{m + 1} + \frac{3 a^{2} b x^{m + 2}}{m + 2} + \frac{3 a b^{2} x^{m + 3}}{m + 3} + \frac{b^{3} x^{m + 4}}{m + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x+a)**3,x)

[Out]

a**3*x**(m + 1)/(m + 1) + 3*a**2*b*x**(m + 2)/(m + 2) + 3*a*b**2*x**(m + 3)/(m +
3) + b**3*x**(m + 4)/(m + 4)

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Mathematica [A]  time = 0.0388325, size = 55, normalized size = 0.9 \[ x^m \left (\frac{a^3 x}{m+1}+\frac{3 a^2 b x^2}{m+2}+\frac{3 a b^2 x^3}{m+3}+\frac{b^3 x^4}{m+4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x)^3,x]

[Out]

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

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Maple [B]  time = 0.007, size = 170, normalized size = 2.8 \[{\frac{{x}^{1+m} \left ({b}^{3}{m}^{3}{x}^{3}+3\,a{b}^{2}{m}^{3}{x}^{2}+6\,{b}^{3}{m}^{2}{x}^{3}+3\,{a}^{2}b{m}^{3}x+21\,a{b}^{2}{m}^{2}{x}^{2}+11\,{b}^{3}m{x}^{3}+{a}^{3}{m}^{3}+24\,{a}^{2}b{m}^{2}x+42\,a{b}^{2}m{x}^{2}+6\,{b}^{3}{x}^{3}+9\,{a}^{3}{m}^{2}+57\,{a}^{2}bmx+24\,a{b}^{2}{x}^{2}+26\,{a}^{3}m+36\,{a}^{2}bx+24\,{a}^{3} \right ) }{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a)^3,x)

[Out]

x^(1+m)*(b^3*m^3*x^3+3*a*b^2*m^3*x^2+6*b^3*m^2*x^3+3*a^2*b*m^3*x+21*a*b^2*m^2*x^
2+11*b^3*m*x^3+a^3*m^3+24*a^2*b*m^2*x+42*a*b^2*m*x^2+6*b^3*x^3+9*a^3*m^2+57*a^2*
b*m*x+24*a*b^2*x^2+26*a^3*m+36*a^2*b*x+24*a^3)/(4+m)/(3+m)/(2+m)/(1+m)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221384, size = 212, normalized size = 3.48 \[ \frac{{\left ({\left (b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}\right )} x^{4} + 3 \,{\left (a b^{2} m^{3} + 7 \, a b^{2} m^{2} + 14 \, a b^{2} m + 8 \, a b^{2}\right )} x^{3} + 3 \,{\left (a^{2} b m^{3} + 8 \, a^{2} b m^{2} + 19 \, a^{2} b m + 12 \, a^{2} b\right )} x^{2} +{\left (a^{3} m^{3} + 9 \, a^{3} m^{2} + 26 \, a^{3} m + 24 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*x^m,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.31849, size = 663, normalized size = 10.87 \[ \begin{cases} - \frac{a^{3}}{3 x^{3}} - \frac{3 a^{2} b}{2 x^{2}} - \frac{3 a b^{2}}{x} + b^{3} \log{\left (x \right )} & \text{for}\: m = -4 \\- \frac{a^{3}}{2 x^{2}} - \frac{3 a^{2} b}{x} + 3 a b^{2} \log{\left (x \right )} + b^{3} x & \text{for}\: m = -3 \\- \frac{a^{3}}{x} + 3 a^{2} b \log{\left (x \right )} + 3 a b^{2} x + \frac{b^{3} x^{2}}{2} & \text{for}\: m = -2 \\a^{3} \log{\left (x \right )} + 3 a^{2} b x + \frac{3 a b^{2} x^{2}}{2} + \frac{b^{3} x^{3}}{3} & \text{for}\: m = -1 \\\frac{a^{3} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{9 a^{3} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{26 a^{3} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{3 a^{2} b m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a^{2} b m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{57 a^{2} b m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{36 a^{2} b x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{3 a b^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{21 a b^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{42 a b^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a b^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{b^{3} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 b^{3} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{11 b^{3} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 b^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x+a)**3,x)

[Out]

Piecewise((-a**3/(3*x**3) - 3*a**2*b/(2*x**2) - 3*a*b**2/x + b**3*log(x), Eq(m,
-4)), (-a**3/(2*x**2) - 3*a**2*b/x + 3*a*b**2*log(x) + b**3*x, Eq(m, -3)), (-a**
3/x + 3*a**2*b*log(x) + 3*a*b**2*x + b**3*x**2/2, Eq(m, -2)), (a**3*log(x) + 3*a
**2*b*x + 3*a*b**2*x**2/2 + b**3*x**3/3, Eq(m, -1)), (a**3*m**3*x*x**m/(m**4 + 1
0*m**3 + 35*m**2 + 50*m + 24) + 9*a**3*m**2*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 5
0*m + 24) + 26*a**3*m*x*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 24*a**3*x*
x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 3*a**2*b*m**3*x**2*x**m/(m**4 + 10
*m**3 + 35*m**2 + 50*m + 24) + 24*a**2*b*m**2*x**2*x**m/(m**4 + 10*m**3 + 35*m**
2 + 50*m + 24) + 57*a**2*b*m*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) +
36*a**2*b*x**2*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 3*a*b**2*m**3*x**3*
x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 21*a*b**2*m**2*x**3*x**m/(m**4 + 1
0*m**3 + 35*m**2 + 50*m + 24) + 42*a*b**2*m*x**3*x**m/(m**4 + 10*m**3 + 35*m**2
+ 50*m + 24) + 24*a*b**2*x**3*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + b**3
*m**3*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 6*b**3*m**2*x**4*x**m/(
m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 11*b**3*m*x**4*x**m/(m**4 + 10*m**3 + 35
*m**2 + 50*m + 24) + 6*b**3*x**4*x**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24), Tr
ue))

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GIAC/XCAS [A]  time = 0.207705, size = 346, normalized size = 5.67 \[ \frac{b^{3} m^{3} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, a b^{2} m^{3} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, b^{3} m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, a^{2} b m^{3} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 21 \, a b^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 11 \, b^{3} m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{3} m^{3} x e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, a^{2} b m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 42 \, a b^{2} m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, b^{3} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 9 \, a^{3} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 57 \, a^{2} b m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, a b^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 26 \, a^{3} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 36 \, a^{2} b x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 24 \, a^{3} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*x^m,x, algorithm="giac")

[Out]

(b^3*m^3*x^4*e^(m*ln(x)) + 3*a*b^2*m^3*x^3*e^(m*ln(x)) + 6*b^3*m^2*x^4*e^(m*ln(x
)) + 3*a^2*b*m^3*x^2*e^(m*ln(x)) + 21*a*b^2*m^2*x^3*e^(m*ln(x)) + 11*b^3*m*x^4*e
^(m*ln(x)) + a^3*m^3*x*e^(m*ln(x)) + 24*a^2*b*m^2*x^2*e^(m*ln(x)) + 42*a*b^2*m*x
^3*e^(m*ln(x)) + 6*b^3*x^4*e^(m*ln(x)) + 9*a^3*m^2*x*e^(m*ln(x)) + 57*a^2*b*m*x^
2*e^(m*ln(x)) + 24*a*b^2*x^3*e^(m*ln(x)) + 26*a^3*m*x*e^(m*ln(x)) + 36*a^2*b*x^2
*e^(m*ln(x)) + 24*a^3*x*e^(m*ln(x)))/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)