3.601 \(\int x^m \left (a+b x^4\right ) \, dx\)

Optimal. Leaf size=25 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+5}}{m+5} \]

[Out]

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

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Rubi [A]  time = 0.0217521, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+5}}{m+5} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.83116, size = 19, normalized size = 0.76 \[ \frac{a x^{m + 1}}{m + 1} + \frac{b x^{m + 5}}{m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**(m + 1)/(m + 1) + b*x**(m + 5)/(m + 5)

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Mathematica [A]  time = 0.0264495, size = 23, normalized size = 0.92 \[ x^m \left (\frac{a x}{m+1}+\frac{b x^5}{m+5}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^4),x]

[Out]

x^m*((a*x)/(1 + m) + (b*x^5)/(5 + m))

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Maple [A]  time = 0.005, size = 35, normalized size = 1.4 \[{\frac{{x}^{1+m} \left ( bm{x}^{4}+b{x}^{4}+am+5\,a \right ) }{ \left ( 5+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^4+a),x)

[Out]

x^(1+m)*(b*m*x^4+b*x^4+a*m+5*a)/(5+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^4 + a)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248959, size = 45, normalized size = 1.8 \[ \frac{{\left ({\left (b m + b\right )} x^{5} +{\left (a m + 5 \, a\right )} x\right )} x^{m}}{m^{2} + 6 \, m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b*m + b)*x^5 + (a*m + 5*a)*x)*x^m/(m^2 + 6*m + 5)

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Sympy [A]  time = 1.98056, size = 94, normalized size = 3.76 \[ \begin{cases} - \frac{a}{4 x^{4}} + b \log{\left (x \right )} & \text{for}\: m = -5 \\a \log{\left (x \right )} + \frac{b x^{4}}{4} & \text{for}\: m = -1 \\\frac{a m x x^{m}}{m^{2} + 6 m + 5} + \frac{5 a x x^{m}}{m^{2} + 6 m + 5} + \frac{b m x^{5} x^{m}}{m^{2} + 6 m + 5} + \frac{b x^{5} x^{m}}{m^{2} + 6 m + 5} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**4+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214973, size = 69, normalized size = 2.76 \[ \frac{b m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + b x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + a m x e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, a x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{2} + 6 \, m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*m*x^5*e^(m*ln(x)) + b*x^5*e^(m*ln(x)) + a*m*x*e^(m*ln(x)) + 5*a*x*e^(m*ln(x))
)/(m^2 + 6*m + 5)