3.787 \(\int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx\)

Optimal. Leaf size=58 \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{b^2 (d x)^{m+5}}{d^5 (m+5)} \]

[Out]

(a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (b^2*(d*
x)^(5 + m))/(d^5*(5 + m))

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Rubi [A]  time = 0.062182, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{2 a b (d x)^{m+3}}{d^3 (m+3)}+\frac{b^2 (d x)^{m+5}}{d^5 (m+5)} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(a^2*(d*x)^(1 + m))/(d*(1 + m)) + (2*a*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (b^2*(d*
x)^(5 + m))/(d^5*(5 + m))

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Rubi in Sympy [A]  time = 22.5281, size = 49, normalized size = 0.84 \[ \frac{a^{2} \left (d x\right )^{m + 1}}{d \left (m + 1\right )} + \frac{2 a b \left (d x\right )^{m + 3}}{d^{3} \left (m + 3\right )} + \frac{b^{2} \left (d x\right )^{m + 5}}{d^{5} \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

a**2*(d*x)**(m + 1)/(d*(m + 1)) + 2*a*b*(d*x)**(m + 3)/(d**3*(m + 3)) + b**2*(d*
x)**(m + 5)/(d**5*(m + 5))

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Mathematica [A]  time = 0.0304457, size = 41, normalized size = 0.71 \[ (d x)^m \left (\frac{a^2 x}{m+1}+\frac{2 a b x^3}{m+3}+\frac{b^2 x^5}{m+5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(d*x)^m*((a^2*x)/(1 + m) + (2*a*b*x^3)/(3 + m) + (b^2*x^5)/(5 + m))

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Maple [A]  time = 0.01, size = 94, normalized size = 1.6 \[{\frac{ \left ( dx \right ) ^{m} \left ({b}^{2}{m}^{2}{x}^{4}+4\,{b}^{2}m{x}^{4}+2\,ab{m}^{2}{x}^{2}+3\,{b}^{2}{x}^{4}+12\,abm{x}^{2}+{a}^{2}{m}^{2}+10\,ab{x}^{2}+8\,{a}^{2}m+15\,{a}^{2} \right ) x}{ \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

(d*x)^m*(b^2*m^2*x^4+4*b^2*m*x^4+2*a*b*m^2*x^2+3*b^2*x^4+12*a*b*m*x^2+a^2*m^2+10
*a*b*x^2+8*a^2*m+15*a^2)*x/(5+m)/(3+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^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285003, size = 117, normalized size = 2.02 \[ \frac{{\left ({\left (b^{2} m^{2} + 4 \, b^{2} m + 3 \, b^{2}\right )} x^{5} + 2 \,{\left (a b m^{2} + 6 \, a b m + 5 \, a b\right )} x^{3} +{\left (a^{2} m^{2} + 8 \, a^{2} m + 15 \, a^{2}\right )} x\right )} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^m,x, algorithm="fricas")

[Out]

((b^2*m^2 + 4*b^2*m + 3*b^2)*x^5 + 2*(a*b*m^2 + 6*a*b*m + 5*a*b)*x^3 + (a^2*m^2
+ 8*a^2*m + 15*a^2)*x)*(d*x)^m/(m^3 + 9*m^2 + 23*m + 15)

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Sympy [A]  time = 2.53314, size = 345, normalized size = 5.95 \[ \begin{cases} \frac{- \frac{a^{2}}{4 x^{4}} - \frac{a b}{x^{2}} + b^{2} \log{\left (x \right )}}{d^{5}} & \text{for}\: m = -5 \\\frac{- \frac{a^{2}}{2 x^{2}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{2}}{2}}{d^{3}} & \text{for}\: m = -3 \\\frac{a^{2} \log{\left (x \right )} + a b x^{2} + \frac{b^{2} x^{4}}{4}}{d} & \text{for}\: m = -1 \\\frac{a^{2} d^{m} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{8 a^{2} d^{m} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{15 a^{2} d^{m} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{2 a b d^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{12 a b d^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{10 a b d^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{b^{2} d^{m} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{4 b^{2} d^{m} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{3 b^{2} d^{m} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

Piecewise(((-a**2/(4*x**4) - a*b/x**2 + b**2*log(x))/d**5, Eq(m, -5)), ((-a**2/(
2*x**2) + 2*a*b*log(x) + b**2*x**2/2)/d**3, Eq(m, -3)), ((a**2*log(x) + a*b*x**2
+ b**2*x**4/4)/d, Eq(m, -1)), (a**2*d**m*m**2*x*x**m/(m**3 + 9*m**2 + 23*m + 15
) + 8*a**2*d**m*m*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + 15*a**2*d**m*x*x**m/(m**3
+ 9*m**2 + 23*m + 15) + 2*a*b*d**m*m**2*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) +
12*a*b*d**m*m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 10*a*b*d**m*x**3*x**m/(m*
*3 + 9*m**2 + 23*m + 15) + b**2*d**m*m**2*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15)
+ 4*b**2*d**m*m*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15) + 3*b**2*d**m*x**5*x**m/(m
**3 + 9*m**2 + 23*m + 15), True))

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GIAC/XCAS [A]  time = 0.266189, size = 207, normalized size = 3.57 \[ \frac{b^{2} m^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 4 \, b^{2} m x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 2 \, a b m^{2} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 3 \, b^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 12 \, a b m x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (d x\right )\right )} + 10 \, a b x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 8 \, a^{2} m x e^{\left (m{\rm ln}\left (d x\right )\right )} + 15 \, a^{2} x e^{\left (m{\rm ln}\left (d x\right )\right )}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^m,x, algorithm="giac")

[Out]

(b^2*m^2*x^5*e^(m*ln(d*x)) + 4*b^2*m*x^5*e^(m*ln(d*x)) + 2*a*b*m^2*x^3*e^(m*ln(d
*x)) + 3*b^2*x^5*e^(m*ln(d*x)) + 12*a*b*m*x^3*e^(m*ln(d*x)) + a^2*m^2*x*e^(m*ln(
d*x)) + 10*a*b*x^3*e^(m*ln(d*x)) + 8*a^2*m*x*e^(m*ln(d*x)) + 15*a^2*x*e^(m*ln(d*
x)))/(m^3 + 9*m^2 + 23*m + 15)