3.113 \(\int (d x)^m \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=58 \[ \frac{b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac{c^2 (d x)^{m+5}}{d^5 (m+5)} \]

[Out]

(b^2*(d*x)^(3 + m))/(d^3*(3 + m)) + (2*b*c*(d*x)^(4 + m))/(d^4*(4 + m)) + (c^2*(
d*x)^(5 + m))/(d^5*(5 + m))

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Rubi [A]  time = 0.0967987, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{b^2 (d x)^{m+3}}{d^3 (m+3)}+\frac{2 b c (d x)^{m+4}}{d^4 (m+4)}+\frac{c^2 (d x)^{m+5}}{d^5 (m+5)} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(b*x + c*x^2)^2,x]

[Out]

(b^2*(d*x)^(3 + m))/(d^3*(3 + m)) + (2*b*c*(d*x)^(4 + m))/(d^4*(4 + m)) + (c^2*(
d*x)^(5 + m))/(d^5*(5 + m))

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Rubi in Sympy [A]  time = 17.5661, size = 51, normalized size = 0.88 \[ \frac{b^{2} \left (d x\right )^{m + 3}}{d^{3} \left (m + 3\right )} + \frac{2 b c \left (d x\right )^{m + 4}}{d^{4} \left (m + 4\right )} + \frac{c^{2} \left (d x\right )^{m + 5}}{d^{5} \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*(d*x)**(m + 3)/(d**3*(m + 3)) + 2*b*c*(d*x)**(m + 4)/(d**4*(m + 4)) + c**2*
(d*x)**(m + 5)/(d**5*(m + 5))

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Mathematica [A]  time = 0.0321996, size = 43, normalized size = 0.74 \[ (d x)^m \left (\frac{b^2 x^3}{m+3}+\frac{2 b c x^4}{m+4}+\frac{c^2 x^5}{m+5}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(d*x)^m*((b^2*x^3)/(3 + m) + (2*b*c*x^4)/(4 + m) + (c^2*x^5)/(5 + m))

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Maple [A]  time = 0.008, size = 90, normalized size = 1.6 \[{\frac{ \left ( dx \right ) ^{m} \left ({c}^{2}{m}^{2}{x}^{2}+2\,bc{m}^{2}x+7\,{c}^{2}m{x}^{2}+{b}^{2}{m}^{2}+16\,bcmx+12\,{c}^{2}{x}^{2}+9\,{b}^{2}m+30\,bcx+20\,{b}^{2} \right ){x}^{3}}{ \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^2+b*x)^2,x)

[Out]

(d*x)^m*(c^2*m^2*x^2+2*b*c*m^2*x+7*c^2*m*x^2+b^2*m^2+16*b*c*m*x+12*c^2*x^2+9*b^2
*m+30*b*c*x+20*b^2)*x^3/(5+m)/(4+m)/(3+m)

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Maxima [A]  time = 0.704428, size = 74, normalized size = 1.28 \[ \frac{c^{2} d^{m} x^{5} x^{m}}{m + 5} + \frac{2 \, b c d^{m} x^{4} x^{m}}{m + 4} + \frac{b^{2} d^{m} x^{3} x^{m}}{m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(d*x)^m,x, algorithm="maxima")

[Out]

c^2*d^m*x^5*x^m/(m + 5) + 2*b*c*d^m*x^4*x^m/(m + 4) + b^2*d^m*x^3*x^m/(m + 3)

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Fricas [A]  time = 0.228769, size = 120, normalized size = 2.07 \[ \frac{{\left ({\left (c^{2} m^{2} + 7 \, c^{2} m + 12 \, c^{2}\right )} x^{5} + 2 \,{\left (b c m^{2} + 8 \, b c m + 15 \, b c\right )} x^{4} +{\left (b^{2} m^{2} + 9 \, b^{2} m + 20 \, b^{2}\right )} x^{3}\right )} \left (d x\right )^{m}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((c^2*m^2 + 7*c^2*m + 12*c^2)*x^5 + 2*(b*c*m^2 + 8*b*c*m + 15*b*c)*x^4 + (b^2*m^
2 + 9*b^2*m + 20*b^2)*x^3)*(d*x)^m/(m^3 + 12*m^2 + 47*m + 60)

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Sympy [A]  time = 2.71435, size = 345, normalized size = 5.95 \[ \begin{cases} \frac{- \frac{b^{2}}{2 x^{2}} - \frac{2 b c}{x} + c^{2} \log{\left (x \right )}}{d^{5}} & \text{for}\: m = -5 \\\frac{- \frac{b^{2}}{x} + 2 b c \log{\left (x \right )} + c^{2} x}{d^{4}} & \text{for}\: m = -4 \\\frac{b^{2} \log{\left (x \right )} + 2 b c x + \frac{c^{2} x^{2}}{2}}{d^{3}} & \text{for}\: m = -3 \\\frac{b^{2} d^{m} m^{2} x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{9 b^{2} d^{m} m x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{20 b^{2} d^{m} x^{3} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{2 b c d^{m} m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{16 b c d^{m} m x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{30 b c d^{m} x^{4} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{c^{2} d^{m} m^{2} x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{7 c^{2} d^{m} m x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} + \frac{12 c^{2} d^{m} x^{5} x^{m}}{m^{3} + 12 m^{2} + 47 m + 60} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((-b**2/(2*x**2) - 2*b*c/x + c**2*log(x))/d**5, Eq(m, -5)), ((-b**2/x
+ 2*b*c*log(x) + c**2*x)/d**4, Eq(m, -4)), ((b**2*log(x) + 2*b*c*x + c**2*x**2/2
)/d**3, Eq(m, -3)), (b**2*d**m*m**2*x**3*x**m/(m**3 + 12*m**2 + 47*m + 60) + 9*b
**2*d**m*m*x**3*x**m/(m**3 + 12*m**2 + 47*m + 60) + 20*b**2*d**m*x**3*x**m/(m**3
+ 12*m**2 + 47*m + 60) + 2*b*c*d**m*m**2*x**4*x**m/(m**3 + 12*m**2 + 47*m + 60)
+ 16*b*c*d**m*m*x**4*x**m/(m**3 + 12*m**2 + 47*m + 60) + 30*b*c*d**m*x**4*x**m/
(m**3 + 12*m**2 + 47*m + 60) + c**2*d**m*m**2*x**5*x**m/(m**3 + 12*m**2 + 47*m +
60) + 7*c**2*d**m*m*x**5*x**m/(m**3 + 12*m**2 + 47*m + 60) + 12*c**2*d**m*x**5*
x**m/(m**3 + 12*m**2 + 47*m + 60), True))

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GIAC/XCAS [A]  time = 0.210075, size = 215, normalized size = 3.71 \[ \frac{c^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 2 \, b c m^{2} x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + 7 \, c^{2} m x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + b^{2} m^{2} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 16 \, b c m x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + 12 \, c^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 9 \, b^{2} m x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 30 \, b c x^{4} e^{\left (m{\rm ln}\left (d x\right )\right )} + 20 \, b^{2} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )}}{m^{3} + 12 \, m^{2} + 47 \, m + 60} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^2*m^2*x^5*e^(m*ln(d*x)) + 2*b*c*m^2*x^4*e^(m*ln(d*x)) + 7*c^2*m*x^5*e^(m*ln(d
*x)) + b^2*m^2*x^3*e^(m*ln(d*x)) + 16*b*c*m*x^4*e^(m*ln(d*x)) + 12*c^2*x^5*e^(m*
ln(d*x)) + 9*b^2*m*x^3*e^(m*ln(d*x)) + 30*b*c*x^4*e^(m*ln(d*x)) + 20*b^2*x^3*e^(
m*ln(d*x)))/(m^3 + 12*m^2 + 47*m + 60)