3.1107 \(\int (d x)^m \left (a+b x^2+c x^4\right ) \, dx\)

Optimal. Leaf size=52 \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+3}}{d^3 (m+3)}+\frac{c (d x)^{m+5}}{d^5 (m+5)} \]

[Out]

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

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Rubi [A]  time = 0.0435398, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a (d x)^{m+1}}{d (m+1)}+\frac{b (d x)^{m+3}}{d^3 (m+3)}+\frac{c (d x)^{m+5}}{d^5 (m+5)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.7902, size = 42, normalized size = 0.81 \[ \frac{a \left (d x\right )^{m + 1}}{d \left (m + 1\right )} + \frac{b \left (d x\right )^{m + 3}}{d^{3} \left (m + 3\right )} + \frac{c \left (d x\right )^{m + 5}}{d^{5} \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0336494, size = 35, normalized size = 0.67 \[ (d x)^m \left (\frac{a x}{m+1}+\frac{b x^3}{m+3}+\frac{c x^5}{m+5}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 78, normalized size = 1.5 \[{\frac{ \left ( c{m}^{2}{x}^{4}+4\,cm{x}^{4}+b{m}^{2}{x}^{2}+3\,c{x}^{4}+6\,bm{x}^{2}+a{m}^{2}+5\,b{x}^{2}+8\,am+15\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^4+b*x^2+a),x)

[Out]

x*(c*m^2*x^4+4*c*m*x^4+b*m^2*x^2+3*c*x^4+6*b*m*x^2+a*m^2+5*b*x^2+8*a*m+15*a)*(d*
x)^m/(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((c*x^4 + b*x^2 + a)*(d*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302217, size = 96, normalized size = 1.85 \[ \frac{{\left ({\left (c m^{2} + 4 \, c m + 3 \, c\right )} x^{5} +{\left (b m^{2} + 6 \, b m + 5 \, b\right )} x^{3} +{\left (a m^{2} + 8 \, a m + 15 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.44005, size = 314, normalized size = 6.04 \[ \begin{cases} \frac{- \frac{a}{4 x^{4}} - \frac{b}{2 x^{2}} + c \log{\left (x \right )}}{d^{5}} & \text{for}\: m = -5 \\\frac{- \frac{a}{2 x^{2}} + b \log{\left (x \right )} + \frac{c x^{2}}{2}}{d^{3}} & \text{for}\: m = -3 \\\frac{a \log{\left (x \right )} + \frac{b x^{2}}{2} + \frac{c x^{4}}{4}}{d} & \text{for}\: m = -1 \\\frac{a d^{m} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{8 a d^{m} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{15 a d^{m} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{b d^{m} m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{6 b d^{m} m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{5 b d^{m} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{c d^{m} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{4 c d^{m} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac{3 c d^{m} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((-a/(4*x**4) - b/(2*x**2) + c*log(x))/d**5, Eq(m, -5)), ((-a/(2*x**2)
+ b*log(x) + c*x**2/2)/d**3, Eq(m, -3)), ((a*log(x) + b*x**2/2 + c*x**4/4)/d, E
q(m, -1)), (a*d**m*m**2*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + 8*a*d**m*m*x*x**m/(
m**3 + 9*m**2 + 23*m + 15) + 15*a*d**m*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + b*d*
*m*m**2*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 6*b*d**m*m*x**3*x**m/(m**3 + 9*m
**2 + 23*m + 15) + 5*b*d**m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + c*d**m*m**2*
x**5*x**m/(m**3 + 9*m**2 + 23*m + 15) + 4*c*d**m*m*x**5*x**m/(m**3 + 9*m**2 + 23
*m + 15) + 3*c*d**m*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15), True))

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GIAC/XCAS [A]  time = 0.265979, size = 185, normalized size = 3.56 \[ \frac{c m^{2} x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 4 \, c m x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + b m^{2} x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 3 \, c x^{5} e^{\left (m{\rm ln}\left (d x\right )\right )} + 6 \, b m x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + a m^{2} x e^{\left (m{\rm ln}\left (d x\right )\right )} + 5 \, b x^{3} e^{\left (m{\rm ln}\left (d x\right )\right )} + 8 \, a m x e^{\left (m{\rm ln}\left (d x\right )\right )} + 15 \, a x e^{\left (m{\rm ln}\left (d x\right )\right )}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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