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

Optimal. Leaf size=43 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+5}}{m+5}+\frac{b^2 x^{m+9}}{m+9} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(5 + m))/(5 + m) + (b^2*x^(9 + m))/(9 + m)

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Rubi [A]  time = 0.0438201, 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 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+5}}{m+5}+\frac{b^2 x^{m+9}}{m+9} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^4)^2,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(5 + m))/(5 + m) + (b^2*x^(9 + m))/(9 + m)

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Rubi in Sympy [A]  time = 7.24812, size = 36, normalized size = 0.84 \[ \frac{a^{2} x^{m + 1}}{m + 1} + \frac{2 a b x^{m + 5}}{m + 5} + \frac{b^{2} x^{m + 9}}{m + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**(m + 1)/(m + 1) + 2*a*b*x**(m + 5)/(m + 5) + b**2*x**(m + 9)/(m + 9)

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Mathematica [A]  time = 0.0283918, size = 39, normalized size = 0.91 \[ x^m \left (\frac{a^2 x}{m+1}+\frac{2 a b x^5}{m+5}+\frac{b^2 x^9}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^4)^2,x]

[Out]

x^m*((a^2*x)/(1 + m) + (2*a*b*x^5)/(5 + m) + (b^2*x^9)/(9 + m))

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Maple [B]  time = 0.008, size = 93, normalized size = 2.2 \[{\frac{{x}^{1+m} \left ({b}^{2}{m}^{2}{x}^{8}+6\,{b}^{2}m{x}^{8}+5\,{b}^{2}{x}^{8}+2\,ab{m}^{2}{x}^{4}+20\,abm{x}^{4}+18\,ab{x}^{4}+{a}^{2}{m}^{2}+14\,{a}^{2}m+45\,{a}^{2} \right ) }{ \left ( 9+m \right ) \left ( 5+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^4+a)^2,x)

[Out]

x^(1+m)*(b^2*m^2*x^8+6*b^2*m*x^8+5*b^2*x^8+2*a*b*m^2*x^4+20*a*b*m*x^4+18*a*b*x^4
+a^2*m^2+14*a^2*m+45*a^2)/(9+m)/(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)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246231, size = 115, normalized size = 2.67 \[ \frac{{\left ({\left (b^{2} m^{2} + 6 \, b^{2} m + 5 \, b^{2}\right )} x^{9} + 2 \,{\left (a b m^{2} + 10 \, a b m + 9 \, a b\right )} x^{5} +{\left (a^{2} m^{2} + 14 \, a^{2} m + 45 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 15 \, m^{2} + 59 \, m + 45} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b^2*m^2 + 6*b^2*m + 5*b^2)*x^9 + 2*(a*b*m^2 + 10*a*b*m + 9*a*b)*x^5 + (a^2*m^2
+ 14*a^2*m + 45*a^2)*x)*x^m/(m^3 + 15*m^2 + 59*m + 45)

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Sympy [A]  time = 7.12518, size = 309, normalized size = 7.19 \[ \begin{cases} - \frac{a^{2}}{8 x^{8}} - \frac{a b}{2 x^{4}} + b^{2} \log{\left (x \right )} & \text{for}\: m = -9 \\- \frac{a^{2}}{4 x^{4}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{4}}{4} & \text{for}\: m = -5 \\a^{2} \log{\left (x \right )} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{8}}{8} & \text{for}\: m = -1 \\\frac{a^{2} m^{2} x x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac{14 a^{2} m x x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac{45 a^{2} x x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac{2 a b m^{2} x^{5} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac{20 a b m x^{5} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac{18 a b x^{5} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac{b^{2} m^{2} x^{9} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac{6 b^{2} m x^{9} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} + \frac{5 b^{2} x^{9} x^{m}}{m^{3} + 15 m^{2} + 59 m + 45} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**4+a)**2,x)

[Out]

Piecewise((-a**2/(8*x**8) - a*b/(2*x**4) + b**2*log(x), Eq(m, -9)), (-a**2/(4*x*
*4) + 2*a*b*log(x) + b**2*x**4/4, Eq(m, -5)), (a**2*log(x) + a*b*x**4/2 + b**2*x
**8/8, Eq(m, -1)), (a**2*m**2*x*x**m/(m**3 + 15*m**2 + 59*m + 45) + 14*a**2*m*x*
x**m/(m**3 + 15*m**2 + 59*m + 45) + 45*a**2*x*x**m/(m**3 + 15*m**2 + 59*m + 45)
+ 2*a*b*m**2*x**5*x**m/(m**3 + 15*m**2 + 59*m + 45) + 20*a*b*m*x**5*x**m/(m**3 +
15*m**2 + 59*m + 45) + 18*a*b*x**5*x**m/(m**3 + 15*m**2 + 59*m + 45) + b**2*m**
2*x**9*x**m/(m**3 + 15*m**2 + 59*m + 45) + 6*b**2*m*x**9*x**m/(m**3 + 15*m**2 +
59*m + 45) + 5*b**2*x**9*x**m/(m**3 + 15*m**2 + 59*m + 45), True))

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GIAC/XCAS [A]  time = 0.228233, size = 182, normalized size = 4.23 \[ \frac{b^{2} m^{2} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 6 \, b^{2} m x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, b^{2} x^{9} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a b m^{2} x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 20 \, a b m x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + 18 \, a b x^{5} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 14 \, a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 45 \, a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 15 \, m^{2} + 59 \, m + 45} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*m^2*x^9*e^(m*ln(x)) + 6*b^2*m*x^9*e^(m*ln(x)) + 5*b^2*x^9*e^(m*ln(x)) + 2*a
*b*m^2*x^5*e^(m*ln(x)) + 20*a*b*m*x^5*e^(m*ln(x)) + 18*a*b*x^5*e^(m*ln(x)) + a^2
*m^2*x*e^(m*ln(x)) + 14*a^2*m*x*e^(m*ln(x)) + 45*a^2*x*e^(m*ln(x)))/(m^3 + 15*m^
2 + 59*m + 45)