3.1436 \(\int x^m \left (a+b x^7\right )^2 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+8}}{m+8}+\frac{b^2 x^{m+15}}{m+15} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(8 + m))/(8 + m) + (b^2*x^(15 + m))/(15 + m)

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Rubi [A]  time = 0.0435868, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+8}}{m+8}+\frac{b^2 x^{m+15}}{m+15} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^7)^2,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(8 + m))/(8 + m) + (b^2*x^(15 + m))/(15 + m)

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Rubi in Sympy [A]  time = 7.48904, size = 36, normalized size = 0.84 \[ \frac{a^{2} x^{m + 1}}{m + 1} + \frac{2 a b x^{m + 8}}{m + 8} + \frac{b^{2} x^{m + 15}}{m + 15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**7+a)**2,x)

[Out]

a**2*x**(m + 1)/(m + 1) + 2*a*b*x**(m + 8)/(m + 8) + b**2*x**(m + 15)/(m + 15)

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Mathematica [A]  time = 0.0314434, size = 39, normalized size = 0.91 \[ x^m \left (\frac{a^2 x}{m+1}+\frac{2 a b x^8}{m+8}+\frac{b^2 x^{15}}{m+15}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^7)^2,x]

[Out]

x^m*((a^2*x)/(1 + m) + (2*a*b*x^8)/(8 + m) + (b^2*x^15)/(15 + m))

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Maple [B]  time = 0.009, size = 93, normalized size = 2.2 \[{\frac{{x}^{1+m} \left ({b}^{2}{m}^{2}{x}^{14}+9\,{b}^{2}m{x}^{14}+8\,{b}^{2}{x}^{14}+2\,ab{m}^{2}{x}^{7}+32\,abm{x}^{7}+30\,ab{x}^{7}+{a}^{2}{m}^{2}+23\,{a}^{2}m+120\,{a}^{2} \right ) }{ \left ( 1+m \right ) \left ( 8+m \right ) \left ( 15+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^7+a)^2,x)

[Out]

x^(1+m)*(b^2*m^2*x^14+9*b^2*m*x^14+8*b^2*x^14+2*a*b*m^2*x^7+32*a*b*m*x^7+30*a*b*
x^7+a^2*m^2+23*a^2*m+120*a^2)/(1+m)/(8+m)/(15+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)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238296, size = 115, normalized size = 2.67 \[ \frac{{\left ({\left (b^{2} m^{2} + 9 \, b^{2} m + 8 \, b^{2}\right )} x^{15} + 2 \,{\left (a b m^{2} + 16 \, a b m + 15 \, a b\right )} x^{8} +{\left (a^{2} m^{2} + 23 \, a^{2} m + 120 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 24 \, m^{2} + 143 \, m + 120} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^7 + a)^2*x^m,x, algorithm="fricas")

[Out]

((b^2*m^2 + 9*b^2*m + 8*b^2)*x^15 + 2*(a*b*m^2 + 16*a*b*m + 15*a*b)*x^8 + (a^2*m
^2 + 23*a^2*m + 120*a^2)*x)*x^m/(m^3 + 24*m^2 + 143*m + 120)

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Sympy [A]  time = 26.3544, size = 313, normalized size = 7.28 \[ \begin{cases} - \frac{a^{2}}{14 x^{14}} - \frac{2 a b}{7 x^{7}} + b^{2} \log{\left (x \right )} & \text{for}\: m = -15 \\- \frac{a^{2}}{7 x^{7}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{7}}{7} & \text{for}\: m = -8 \\a^{2} \log{\left (x \right )} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{14}}{14} & \text{for}\: m = -1 \\\frac{a^{2} m^{2} x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{23 a^{2} m x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{120 a^{2} x x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{2 a b m^{2} x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{32 a b m x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{30 a b x^{8} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{b^{2} m^{2} x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{9 b^{2} m x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} + \frac{8 b^{2} x^{15} x^{m}}{m^{3} + 24 m^{2} + 143 m + 120} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**7+a)**2,x)

[Out]

Piecewise((-a**2/(14*x**14) - 2*a*b/(7*x**7) + b**2*log(x), Eq(m, -15)), (-a**2/
(7*x**7) + 2*a*b*log(x) + b**2*x**7/7, Eq(m, -8)), (a**2*log(x) + 2*a*b*x**7/7 +
b**2*x**14/14, Eq(m, -1)), (a**2*m**2*x*x**m/(m**3 + 24*m**2 + 143*m + 120) + 2
3*a**2*m*x*x**m/(m**3 + 24*m**2 + 143*m + 120) + 120*a**2*x*x**m/(m**3 + 24*m**2
+ 143*m + 120) + 2*a*b*m**2*x**8*x**m/(m**3 + 24*m**2 + 143*m + 120) + 32*a*b*m
*x**8*x**m/(m**3 + 24*m**2 + 143*m + 120) + 30*a*b*x**8*x**m/(m**3 + 24*m**2 + 1
43*m + 120) + b**2*m**2*x**15*x**m/(m**3 + 24*m**2 + 143*m + 120) + 9*b**2*m*x**
15*x**m/(m**3 + 24*m**2 + 143*m + 120) + 8*b**2*x**15*x**m/(m**3 + 24*m**2 + 143
*m + 120), True))

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GIAC/XCAS [A]  time = 0.228667, size = 182, normalized size = 4.23 \[ \frac{b^{2} m^{2} x^{15} e^{\left (m{\rm ln}\left (x\right )\right )} + 9 \, b^{2} m x^{15} e^{\left (m{\rm ln}\left (x\right )\right )} + 8 \, b^{2} x^{15} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a b m^{2} x^{8} e^{\left (m{\rm ln}\left (x\right )\right )} + 32 \, a b m x^{8} e^{\left (m{\rm ln}\left (x\right )\right )} + 30 \, a b x^{8} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 23 \, a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 120 \, a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 24 \, m^{2} + 143 \, m + 120} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^7 + a)^2*x^m,x, algorithm="giac")

[Out]

(b^2*m^2*x^15*e^(m*ln(x)) + 9*b^2*m*x^15*e^(m*ln(x)) + 8*b^2*x^15*e^(m*ln(x)) +
2*a*b*m^2*x^8*e^(m*ln(x)) + 32*a*b*m*x^8*e^(m*ln(x)) + 30*a*b*x^8*e^(m*ln(x)) +
a^2*m^2*x*e^(m*ln(x)) + 23*a^2*m*x*e^(m*ln(x)) + 120*a^2*x*e^(m*ln(x)))/(m^3 + 2
4*m^2 + 143*m + 120)