3.344 \(\int x^m (a+b x) (A+B x) \, dx\)

Optimal. Leaf size=45 \[ \frac{x^{m+2} (a B+A b)}{m+2}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+3}}{m+3} \]

[Out]

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

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Rubi [A]  time = 0.0549507, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{x^{m+2} (a B+A b)}{m+2}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+3}}{m+3} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x)*(A + B*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.9168, size = 37, normalized size = 0.82 \[ \frac{A a x^{m + 1}}{m + 1} + \frac{B b x^{m + 3}}{m + 3} + \frac{x^{m + 2} \left (A b + B a\right )}{m + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0432342, size = 41, normalized size = 0.91 \[ x^m \left (\frac{x^2 (a B+A b)}{m+2}+\frac{a A x}{m+1}+\frac{b B x^3}{m+3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x)*(A + B*x),x]

[Out]

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

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Maple [B]  time = 0.004, size = 98, normalized size = 2.2 \[{\frac{{x}^{1+m} \left ( Bb{m}^{2}{x}^{2}+Ab{m}^{2}x+Ba{m}^{2}x+3\,Bbm{x}^{2}+Aa{m}^{2}+4\,Abmx+4\,Bamx+2\,bB{x}^{2}+5\,Aam+3\,Abx+3\,Bax+6\,Aa \right ) }{ \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x+a)*(B*x+A),x)

[Out]

x^(1+m)*(B*b*m^2*x^2+A*b*m^2*x+B*a*m^2*x+3*B*b*m*x^2+A*a*m^2+4*A*b*m*x+4*B*a*m*x
+2*B*b*x^2+5*A*a*m+3*A*b*x+3*B*a*x+6*A*a)/(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)*(b*x + a)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219533, size = 124, normalized size = 2.76 \[ \frac{{\left ({\left (B b m^{2} + 3 \, B b m + 2 \, B b\right )} x^{3} +{\left ({\left (B a + A b\right )} m^{2} + 3 \, B a + 3 \, A b + 4 \,{\left (B a + A b\right )} m\right )} x^{2} +{\left (A a m^{2} + 5 \, A a m + 6 \, A a\right )} x\right )} x^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((B*b*m^2 + 3*B*b*m + 2*B*b)*x^3 + ((B*a + A*b)*m^2 + 3*B*a + 3*A*b + 4*(B*a + A
*b)*m)*x^2 + (A*a*m^2 + 5*A*a*m + 6*A*a)*x)*x^m/(m^3 + 6*m^2 + 11*m + 6)

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Sympy [A]  time = 1.64109, size = 389, normalized size = 8.64 \[ \begin{cases} - \frac{A a}{2 x^{2}} - \frac{A b}{x} - \frac{B a}{x} + B b \log{\left (x \right )} & \text{for}\: m = -3 \\- \frac{A a}{x} + A b \log{\left (x \right )} + B a \log{\left (x \right )} + B b x & \text{for}\: m = -2 \\A a \log{\left (x \right )} + A b x + B a x + \frac{B b x^{2}}{2} & \text{for}\: m = -1 \\\frac{A a m^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{5 A a m x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 A a x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{A b m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{4 A b m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 A b x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{B a m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{4 B a m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 B a x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{B b m^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 B b m x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 B b x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x+a)*(B*x+A),x)

[Out]

Piecewise((-A*a/(2*x**2) - A*b/x - B*a/x + B*b*log(x), Eq(m, -3)), (-A*a/x + A*b
*log(x) + B*a*log(x) + B*b*x, Eq(m, -2)), (A*a*log(x) + A*b*x + B*a*x + B*b*x**2
/2, Eq(m, -1)), (A*a*m**2*x*x**m/(m**3 + 6*m**2 + 11*m + 6) + 5*A*a*m*x*x**m/(m*
*3 + 6*m**2 + 11*m + 6) + 6*A*a*x*x**m/(m**3 + 6*m**2 + 11*m + 6) + A*b*m**2*x**
2*x**m/(m**3 + 6*m**2 + 11*m + 6) + 4*A*b*m*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6)
+ 3*A*b*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + B*a*m**2*x**2*x**m/(m**3 + 6*m**
2 + 11*m + 6) + 4*B*a*m*x**2*x**m/(m**3 + 6*m**2 + 11*m + 6) + 3*B*a*x**2*x**m/(
m**3 + 6*m**2 + 11*m + 6) + B*b*m**2*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6) + 3*B*
b*m*x**3*x**m/(m**3 + 6*m**2 + 11*m + 6) + 2*B*b*x**3*x**m/(m**3 + 6*m**2 + 11*m
+ 6), True))

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GIAC/XCAS [A]  time = 0.213786, size = 225, normalized size = 5. \[ \frac{B b m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + B a m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + A b m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, B b m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + A a m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 4 \, B a m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 4 \, A b m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, B b x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, A a m x e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, B a x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, A b x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, A a x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b*m^2*x^3*e^(m*ln(x)) + B*a*m^2*x^2*e^(m*ln(x)) + A*b*m^2*x^2*e^(m*ln(x)) + 3
*B*b*m*x^3*e^(m*ln(x)) + A*a*m^2*x*e^(m*ln(x)) + 4*B*a*m*x^2*e^(m*ln(x)) + 4*A*b
*m*x^2*e^(m*ln(x)) + 2*B*b*x^3*e^(m*ln(x)) + 5*A*a*m*x*e^(m*ln(x)) + 3*B*a*x^2*e
^(m*ln(x)) + 3*A*b*x^2*e^(m*ln(x)) + 6*A*a*x*e^(m*ln(x)))/(m^3 + 6*m^2 + 11*m +
6)