3.1 \(\int x^m (A+B x) \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=45 \[ \frac{x^{m+3} (A c+b B)}{m+3}+\frac{A b x^{m+2}}{m+2}+\frac{B c x^{m+4}}{m+4} \]

[Out]

(A*b*x^(2 + m))/(2 + m) + ((b*B + A*c)*x^(3 + m))/(3 + m) + (B*c*x^(4 + m))/(4 +
m)

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Rubi [A]  time = 0.0708494, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x^{m+3} (A c+b B)}{m+3}+\frac{A b x^{m+2}}{m+2}+\frac{B c x^{m+4}}{m+4} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(A + B*x)*(b*x + c*x^2),x]

[Out]

(A*b*x^(2 + m))/(2 + m) + ((b*B + A*c)*x^(3 + m))/(3 + m) + (B*c*x^(4 + m))/(4 +
m)

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Rubi in Sympy [A]  time = 9.2961, size = 37, normalized size = 0.82 \[ \frac{A b x^{m + 2}}{m + 2} + \frac{B c x^{m + 4}}{m + 4} + \frac{x^{m + 3} \left (A c + B b\right )}{m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(B*x+A)*(c*x**2+b*x),x)

[Out]

A*b*x**(m + 2)/(m + 2) + B*c*x**(m + 4)/(m + 4) + x**(m + 3)*(A*c + B*b)/(m + 3)

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Mathematica [A]  time = 0.0647492, size = 40, normalized size = 0.89 \[ x^{m+2} \left (\frac{x (A c+b B)}{m+3}+\frac{A b}{m+2}+\frac{B c x^2}{m+4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(A + B*x)*(b*x + c*x^2),x]

[Out]

x^(2 + m)*((A*b)/(2 + m) + ((b*B + A*c)*x)/(3 + m) + (B*c*x^2)/(4 + m))

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Maple [B]  time = 0.005, size = 98, normalized size = 2.2 \[{\frac{{x}^{2+m} \left ( Bc{m}^{2}{x}^{2}+Ac{m}^{2}x+Bb{m}^{2}x+5\,Bcm{x}^{2}+Ab{m}^{2}+6\,Acmx+6\,Bbmx+6\,Bc{x}^{2}+7\,Abm+8\,Acx+8\,Bbx+12\,Ab \right ) }{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(B*x+A)*(c*x^2+b*x),x)

[Out]

x^(2+m)*(B*c*m^2*x^2+A*c*m^2*x+B*b*m^2*x+5*B*c*m*x^2+A*b*m^2+6*A*c*m*x+6*B*b*m*x
+6*B*c*x^2+7*A*b*m+8*A*c*x+8*B*b*x+12*A*b)/(4+m)/(3+m)/(2+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((c*x^2 + b*x)*(B*x + A)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286457, size = 127, normalized size = 2.82 \[ \frac{{\left ({\left (B c m^{2} + 5 \, B c m + 6 \, B c\right )} x^{4} +{\left ({\left (B b + A c\right )} m^{2} + 8 \, B b + 8 \, A c + 6 \,{\left (B b + A c\right )} m\right )} x^{3} +{\left (A b m^{2} + 7 \, A b m + 12 \, A b\right )} x^{2}\right )} x^{m}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)*x^m,x, algorithm="fricas")

[Out]

((B*c*m^2 + 5*B*c*m + 6*B*c)*x^4 + ((B*b + A*c)*m^2 + 8*B*b + 8*A*c + 6*(B*b + A
*c)*m)*x^3 + (A*b*m^2 + 7*A*b*m + 12*A*b)*x^2)*x^m/(m^3 + 9*m^2 + 26*m + 24)

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Sympy [A]  time = 2.24394, size = 394, normalized size = 8.76 \[ \begin{cases} - \frac{A b}{2 x^{2}} - \frac{A c}{x} - \frac{B b}{x} + B c \log{\left (x \right )} & \text{for}\: m = -4 \\- \frac{A b}{x} + A c \log{\left (x \right )} + B b \log{\left (x \right )} + B c x & \text{for}\: m = -3 \\A b \log{\left (x \right )} + A c x + B b x + \frac{B c x^{2}}{2} & \text{for}\: m = -2 \\\frac{A b m^{2} x^{2} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{7 A b m x^{2} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{12 A b x^{2} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{A c m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{6 A c m x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{8 A c x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{B b m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{6 B b m x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{8 B b x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{B c m^{2} x^{4} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{5 B c m x^{4} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac{6 B c x^{4} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(B*x+A)*(c*x**2+b*x),x)

[Out]

Piecewise((-A*b/(2*x**2) - A*c/x - B*b/x + B*c*log(x), Eq(m, -4)), (-A*b/x + A*c
*log(x) + B*b*log(x) + B*c*x, Eq(m, -3)), (A*b*log(x) + A*c*x + B*b*x + B*c*x**2
/2, Eq(m, -2)), (A*b*m**2*x**2*x**m/(m**3 + 9*m**2 + 26*m + 24) + 7*A*b*m*x**2*x
**m/(m**3 + 9*m**2 + 26*m + 24) + 12*A*b*x**2*x**m/(m**3 + 9*m**2 + 26*m + 24) +
A*c*m**2*x**3*x**m/(m**3 + 9*m**2 + 26*m + 24) + 6*A*c*m*x**3*x**m/(m**3 + 9*m*
*2 + 26*m + 24) + 8*A*c*x**3*x**m/(m**3 + 9*m**2 + 26*m + 24) + B*b*m**2*x**3*x*
*m/(m**3 + 9*m**2 + 26*m + 24) + 6*B*b*m*x**3*x**m/(m**3 + 9*m**2 + 26*m + 24) +
8*B*b*x**3*x**m/(m**3 + 9*m**2 + 26*m + 24) + B*c*m**2*x**4*x**m/(m**3 + 9*m**2
+ 26*m + 24) + 5*B*c*m*x**4*x**m/(m**3 + 9*m**2 + 26*m + 24) + 6*B*c*x**4*x**m/
(m**3 + 9*m**2 + 26*m + 24), True))

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GIAC/XCAS [A]  time = 0.272614, size = 234, normalized size = 5.2 \[ \frac{B c m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + B b m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + A c m^{2} x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, B c m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + A b m^{2} x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, B b m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, A c m x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, B c x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 7 \, A b m x^{2} e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, B b x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, A c x^{3} e^{\left (m{\rm ln}\left (x\right )\right )} + 12 \, A b x^{2} e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)*x^m,x, algorithm="giac")

[Out]

(B*c*m^2*x^4*e^(m*ln(x)) + B*b*m^2*x^3*e^(m*ln(x)) + A*c*m^2*x^3*e^(m*ln(x)) + 5
*B*c*m*x^4*e^(m*ln(x)) + A*b*m^2*x^2*e^(m*ln(x)) + 6*B*b*m*x^3*e^(m*ln(x)) + 6*A
*c*m*x^3*e^(m*ln(x)) + 6*B*c*x^4*e^(m*ln(x)) + 7*A*b*m*x^2*e^(m*ln(x)) + 8*B*b*x
^3*e^(m*ln(x)) + 8*A*c*x^3*e^(m*ln(x)) + 12*A*b*x^2*e^(m*ln(x)))/(m^3 + 9*m^2 +
26*m + 24)