3.584 \(\int x^m \left (a+b x^3\right )^2 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+4}}{m+4}+\frac{b^2 x^{m+7}}{m+7} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(4 + m))/(4 + m) + (b^2*x^(7 + m))/(7 + m)

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Rubi [A]  time = 0.0411258, 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 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+4}}{m+4}+\frac{b^2 x^{m+7}}{m+7} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^3)^2,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(4 + m))/(4 + m) + (b^2*x^(7 + m))/(7 + m)

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Rubi in Sympy [A]  time = 7.23094, size = 36, normalized size = 0.84 \[ \frac{a^{2} x^{m + 1}}{m + 1} + \frac{2 a b x^{m + 4}}{m + 4} + \frac{b^{2} x^{m + 7}}{m + 7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**3+a)**2,x)

[Out]

a**2*x**(m + 1)/(m + 1) + 2*a*b*x**(m + 4)/(m + 4) + b**2*x**(m + 7)/(m + 7)

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Mathematica [A]  time = 0.0301088, size = 39, normalized size = 0.91 \[ x^m \left (\frac{a^2 x}{m+1}+\frac{2 a b x^4}{m+4}+\frac{b^2 x^7}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^3)^2,x]

[Out]

x^m*((a^2*x)/(1 + m) + (2*a*b*x^4)/(4 + m) + (b^2*x^7)/(7 + m))

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Maple [B]  time = 0.008, size = 93, normalized size = 2.2 \[{\frac{{x}^{1+m} \left ({b}^{2}{m}^{2}{x}^{6}+5\,{b}^{2}m{x}^{6}+4\,{b}^{2}{x}^{6}+2\,ab{m}^{2}{x}^{3}+16\,abm{x}^{3}+14\,ab{x}^{3}+{a}^{2}{m}^{2}+11\,{a}^{2}m+28\,{a}^{2} \right ) }{ \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^3+a)^2,x)

[Out]

x^(1+m)*(b^2*m^2*x^6+5*b^2*m*x^6+4*b^2*x^6+2*a*b*m^2*x^3+16*a*b*m*x^3+14*a*b*x^3
+a^2*m^2+11*a^2*m+28*a^2)/(7+m)/(4+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^3 + a)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247911, size = 115, normalized size = 2.67 \[ \frac{{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 4 \, b^{2}\right )} x^{7} + 2 \,{\left (a b m^{2} + 8 \, a b m + 7 \, a b\right )} x^{4} +{\left (a^{2} m^{2} + 11 \, a^{2} m + 28 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*x^m,x, algorithm="fricas")

[Out]

((b^2*m^2 + 5*b^2*m + 4*b^2)*x^7 + 2*(a*b*m^2 + 8*a*b*m + 7*a*b)*x^4 + (a^2*m^2
+ 11*a^2*m + 28*a^2)*x)*x^m/(m^3 + 12*m^2 + 39*m + 28)

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Sympy [A]  time = 4.31148, size = 313, normalized size = 7.28 \[ \begin{cases} - \frac{a^{2}}{6 x^{6}} - \frac{2 a b}{3 x^{3}} + b^{2} \log{\left (x \right )} & \text{for}\: m = -7 \\- \frac{a^{2}}{3 x^{3}} + 2 a b \log{\left (x \right )} + \frac{b^{2} x^{3}}{3} & \text{for}\: m = -4 \\a^{2} \log{\left (x \right )} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{6}}{6} & \text{for}\: m = -1 \\\frac{a^{2} m^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{11 a^{2} m x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{28 a^{2} x x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{2 a b m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{16 a b m x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{14 a b x^{4} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{b^{2} m^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{5 b^{2} m x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} + \frac{4 b^{2} x^{7} x^{m}}{m^{3} + 12 m^{2} + 39 m + 28} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**3+a)**2,x)

[Out]

Piecewise((-a**2/(6*x**6) - 2*a*b/(3*x**3) + b**2*log(x), Eq(m, -7)), (-a**2/(3*
x**3) + 2*a*b*log(x) + b**2*x**3/3, Eq(m, -4)), (a**2*log(x) + 2*a*b*x**3/3 + b*
*2*x**6/6, Eq(m, -1)), (a**2*m**2*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 11*a**2*
m*x*x**m/(m**3 + 12*m**2 + 39*m + 28) + 28*a**2*x*x**m/(m**3 + 12*m**2 + 39*m +
28) + 2*a*b*m**2*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + 16*a*b*m*x**4*x**m/(m*
*3 + 12*m**2 + 39*m + 28) + 14*a*b*x**4*x**m/(m**3 + 12*m**2 + 39*m + 28) + b**2
*m**2*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28) + 5*b**2*m*x**7*x**m/(m**3 + 12*m**
2 + 39*m + 28) + 4*b**2*x**7*x**m/(m**3 + 12*m**2 + 39*m + 28), True))

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GIAC/XCAS [A]  time = 0.220314, size = 182, normalized size = 4.23 \[ \frac{b^{2} m^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 5 \, b^{2} m x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 4 \, b^{2} x^{7} e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a b m^{2} x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 16 \, a b m x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + 14 \, a b x^{4} e^{\left (m{\rm ln}\left (x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 11 \, a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 28 \, a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 12 \, m^{2} + 39 \, m + 28} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*x^m,x, algorithm="giac")

[Out]

(b^2*m^2*x^7*e^(m*ln(x)) + 5*b^2*m*x^7*e^(m*ln(x)) + 4*b^2*x^7*e^(m*ln(x)) + 2*a
*b*m^2*x^4*e^(m*ln(x)) + 16*a*b*m*x^4*e^(m*ln(x)) + 14*a*b*x^4*e^(m*ln(x)) + a^2
*m^2*x*e^(m*ln(x)) + 11*a^2*m*x*e^(m*ln(x)) + 28*a^2*x*e^(m*ln(x)))/(m^3 + 12*m^
2 + 39*m + 28)