3.1433 \(\int x^m \left (a+b x^7\right ) \, dx\)

Optimal. Leaf size=25 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+8}}{m+8} \]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(8 + m))/(8 + m)

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Rubi [A]  time = 0.0211544, 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 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+8}}{m+8} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^7),x]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(8 + m))/(8 + m)

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Rubi in Sympy [A]  time = 3.96495, size = 19, normalized size = 0.76 \[ \frac{a x^{m + 1}}{m + 1} + \frac{b x^{m + 8}}{m + 8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**7+a),x)

[Out]

a*x**(m + 1)/(m + 1) + b*x**(m + 8)/(m + 8)

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Mathematica [A]  time = 0.0274158, size = 23, normalized size = 0.92 \[ x^m \left (\frac{a x}{m+1}+\frac{b x^8}{m+8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^7),x]

[Out]

x^m*((a*x)/(1 + m) + (b*x^8)/(8 + m))

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Maple [A]  time = 0.004, size = 35, normalized size = 1.4 \[{\frac{{x}^{1+m} \left ( bm{x}^{7}+b{x}^{7}+am+8\,a \right ) }{ \left ( 8+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^7+a),x)

[Out]

x^(1+m)*(b*m*x^7+b*x^7+a*m+8*a)/(8+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^7 + a)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233144, size = 45, normalized size = 1.8 \[ \frac{{\left ({\left (b m + b\right )} x^{8} +{\left (a m + 8 \, a\right )} x\right )} x^{m}}{m^{2} + 9 \, m + 8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^7 + a)*x^m,x, algorithm="fricas")

[Out]

((b*m + b)*x^8 + (a*m + 8*a)*x)*x^m/(m^2 + 9*m + 8)

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Sympy [A]  time = 5.047, size = 94, normalized size = 3.76 \[ \begin{cases} - \frac{a}{7 x^{7}} + b \log{\left (x \right )} & \text{for}\: m = -8 \\a \log{\left (x \right )} + \frac{b x^{7}}{7} & \text{for}\: m = -1 \\\frac{a m x x^{m}}{m^{2} + 9 m + 8} + \frac{8 a x x^{m}}{m^{2} + 9 m + 8} + \frac{b m x^{8} x^{m}}{m^{2} + 9 m + 8} + \frac{b x^{8} x^{m}}{m^{2} + 9 m + 8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**7+a),x)

[Out]

Piecewise((-a/(7*x**7) + b*log(x), Eq(m, -8)), (a*log(x) + b*x**7/7, Eq(m, -1)),
(a*m*x*x**m/(m**2 + 9*m + 8) + 8*a*x*x**m/(m**2 + 9*m + 8) + b*m*x**8*x**m/(m**
2 + 9*m + 8) + b*x**8*x**m/(m**2 + 9*m + 8), True))

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GIAC/XCAS [A]  time = 0.2244, size = 69, normalized size = 2.76 \[ \frac{b m x^{8} e^{\left (m{\rm ln}\left (x\right )\right )} + b x^{8} e^{\left (m{\rm ln}\left (x\right )\right )} + a m x e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, a x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{2} + 9 \, m + 8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^7 + a)*x^m,x, algorithm="giac")

[Out]

(b*m*x^8*e^(m*ln(x)) + b*x^8*e^(m*ln(x)) + a*m*x*e^(m*ln(x)) + 8*a*x*e^(m*ln(x))
)/(m^2 + 9*m + 8)