3.6.85 \(\int x^m (a+b x^3)^3 \, dx\) [585]

Optimal. Leaf size=61 \[ \frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{4+m}}{4+m}+\frac {3 a b^2 x^{7+m}}{7+m}+\frac {b^3 x^{10+m}}{10+m} \]

[Out]

a^3*x^(1+m)/(1+m)+3*a^2*b*x^(4+m)/(4+m)+3*a*b^2*x^(7+m)/(7+m)+b^3*x^(10+m)/(10+m)

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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \begin {gather*} \frac {a^3 x^{m+1}}{m+1}+\frac {3 a^2 b x^{m+4}}{m+4}+\frac {3 a b^2 x^{m+7}}{m+7}+\frac {b^3 x^{m+10}}{m+10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^3)^3,x]

[Out]

(a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(4 + m))/(4 + m) + (3*a*b^2*x^(7 + m))/(7 + m) + (b^3*x^(10 + m))/(10 + m
)

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^m \left (a+b x^3\right )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{3+m}+3 a b^2 x^{6+m}+b^3 x^{9+m}\right ) \, dx\\ &=\frac {a^3 x^{1+m}}{1+m}+\frac {3 a^2 b x^{4+m}}{4+m}+\frac {3 a b^2 x^{7+m}}{7+m}+\frac {b^3 x^{10+m}}{10+m}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 56, normalized size = 0.92 \begin {gather*} x^{1+m} \left (\frac {a^3}{1+m}+\frac {3 a^2 b x^3}{4+m}+\frac {3 a b^2 x^6}{7+m}+\frac {b^3 x^9}{10+m}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^3)^3,x]

[Out]

x^(1 + m)*(a^3/(1 + m) + (3*a^2*b*x^3)/(4 + m) + (3*a*b^2*x^6)/(7 + m) + (b^3*x^9)/(10 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(61)=122\).
time = 0.14, size = 177, normalized size = 2.90

method	result	size
risch	\(\frac {x \left (b^{3} m^{3} x^{9}+12 b^{3} m^{2} x^{9}+39 m \,x^{9} b^{3}+3 a \,b^{2} m^{3} x^{6}+28 b^{3} x^{9}+45 a \,b^{2} m^{2} x^{6}+162 m \,x^{6} a \,b^{2}+3 a^{2} b \,m^{3} x^{3}+120 a \,b^{2} x^{6}+54 a^{2} b \,m^{2} x^{3}+261 m \,x^{3} a^{2} b +a^{3} m^{3}+210 a^{2} b \,x^{3}+21 a^{3} m^{2}+138 m \,a^{3}+280 a^{3}\right ) x^{m}}{\left (10+m \right ) \left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\)	\(177\)
gosper	\(\frac {x^{1+m} \left (b^{3} m^{3} x^{9}+12 b^{3} m^{2} x^{9}+39 m \,x^{9} b^{3}+3 a \,b^{2} m^{3} x^{6}+28 b^{3} x^{9}+45 a \,b^{2} m^{2} x^{6}+162 m \,x^{6} a \,b^{2}+3 a^{2} b \,m^{3} x^{3}+120 a \,b^{2} x^{6}+54 a^{2} b \,m^{2} x^{3}+261 m \,x^{3} a^{2} b +a^{3} m^{3}+210 a^{2} b \,x^{3}+21 a^{3} m^{2}+138 m \,a^{3}+280 a^{3}\right )}{\left (10+m \right ) \left (7+m \right ) \left (4+m \right ) \left (1+m \right )}\)	\(178\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

x*(b^3*m^3*x^9+12*b^3*m^2*x^9+39*b^3*m*x^9+3*a*b^2*m^3*x^6+28*b^3*x^9+45*a*b^2*m^2*x^6+162*a*b^2*m*x^6+3*a^2*b
*m^3*x^3+120*a*b^2*x^6+54*a^2*b*m^2*x^3+261*a^2*b*m*x^3+a^3*m^3+210*a^2*b*x^3+21*a^3*m^2+138*a^3*m+280*a^3)*x^
m/(10+m)/(7+m)/(4+m)/(1+m)

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Maxima [A]
time = 0.30, size = 61, normalized size = 1.00 \begin {gather*} \frac {b^{3} x^{m + 10}}{m + 10} + \frac {3 \, a b^{2} x^{m + 7}}{m + 7} + \frac {3 \, a^{2} b x^{m + 4}}{m + 4} + \frac {a^{3} x^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^3+a)^3,x, algorithm="maxima")

[Out]

b^3*x^(m + 10)/(m + 10) + 3*a*b^2*x^(m + 7)/(m + 7) + 3*a^2*b*x^(m + 4)/(m + 4) + a^3*x^(m + 1)/(m + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (61) = 122\).
time = 0.37, size = 157, normalized size = 2.57 \begin {gather*} \frac {{\left ({\left (b^{3} m^{3} + 12 \, b^{3} m^{2} + 39 \, b^{3} m + 28 \, b^{3}\right )} x^{10} + 3 \, {\left (a b^{2} m^{3} + 15 \, a b^{2} m^{2} + 54 \, a b^{2} m + 40 \, a b^{2}\right )} x^{7} + 3 \, {\left (a^{2} b m^{3} + 18 \, a^{2} b m^{2} + 87 \, a^{2} b m + 70 \, a^{2} b\right )} x^{4} + {\left (a^{3} m^{3} + 21 \, a^{3} m^{2} + 138 \, a^{3} m + 280 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^3*m^3 + 12*b^3*m^2 + 39*b^3*m + 28*b^3)*x^10 + 3*(a*b^2*m^3 + 15*a*b^2*m^2 + 54*a*b^2*m + 40*a*b^2)*x^7 +
3*(a^2*b*m^3 + 18*a^2*b*m^2 + 87*a^2*b*m + 70*a^2*b)*x^4 + (a^3*m^3 + 21*a^3*m^2 + 138*a^3*m + 280*a^3)*x)*x^m
/(m^4 + 22*m^3 + 159*m^2 + 418*m + 280)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (53) = 106\).
time = 0.50, size = 666, normalized size = 10.92 \begin {gather*} \begin {cases} - \frac {a^{3}}{9 x^{9}} - \frac {a^{2} b}{2 x^{6}} - \frac {a b^{2}}{x^{3}} + b^{3} \log {\left (x \right )} & \text {for}\: m = -10 \\- \frac {a^{3}}{6 x^{6}} - \frac {a^{2} b}{x^{3}} + 3 a b^{2} \log {\left (x \right )} + \frac {b^{3} x^{3}}{3} & \text {for}\: m = -7 \\- \frac {a^{3}}{3 x^{3}} + 3 a^{2} b \log {\left (x \right )} + a b^{2} x^{3} + \frac {b^{3} x^{6}}{6} & \text {for}\: m = -4 \\a^{3} \log {\left (x \right )} + a^{2} b x^{3} + \frac {a b^{2} x^{6}}{2} + \frac {b^{3} x^{9}}{9} & \text {for}\: m = -1 \\\frac {a^{3} m^{3} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {21 a^{3} m^{2} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {138 a^{3} m x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {280 a^{3} x x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {3 a^{2} b m^{3} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {54 a^{2} b m^{2} x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {261 a^{2} b m x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {210 a^{2} b x^{4} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {3 a b^{2} m^{3} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {45 a b^{2} m^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {162 a b^{2} m x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {120 a b^{2} x^{7} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {b^{3} m^{3} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {12 b^{3} m^{2} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {39 b^{3} m x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} + \frac {28 b^{3} x^{10} x^{m}}{m^{4} + 22 m^{3} + 159 m^{2} + 418 m + 280} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**3+a)**3,x)

[Out]

Piecewise((-a**3/(9*x**9) - a**2*b/(2*x**6) - a*b**2/x**3 + b**3*log(x), Eq(m, -10)), (-a**3/(6*x**6) - a**2*b
/x**3 + 3*a*b**2*log(x) + b**3*x**3/3, Eq(m, -7)), (-a**3/(3*x**3) + 3*a**2*b*log(x) + a*b**2*x**3 + b**3*x**6
/6, Eq(m, -4)), (a**3*log(x) + a**2*b*x**3 + a*b**2*x**6/2 + b**3*x**9/9, Eq(m, -1)), (a**3*m**3*x*x**m/(m**4
+ 22*m**3 + 159*m**2 + 418*m + 280) + 21*a**3*m**2*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 138*a**3
*m*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 280*a**3*x*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280
) + 3*a**2*b*m**3*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 54*a**2*b*m**2*x**4*x**m/(m**4 + 22*m*
*3 + 159*m**2 + 418*m + 280) + 261*a**2*b*m*x**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 210*a**2*b*x
**4*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 3*a*b**2*m**3*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*
m + 280) + 45*a*b**2*m**2*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 162*a*b**2*m*x**7*x**m/(m**4 +
22*m**3 + 159*m**2 + 418*m + 280) + 120*a*b**2*x**7*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + b**3*m**
3*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 12*b**3*m**2*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 +
418*m + 280) + 39*b**3*m*x**10*x**m/(m**4 + 22*m**3 + 159*m**2 + 418*m + 280) + 28*b**3*x**10*x**m/(m**4 + 22*
m**3 + 159*m**2 + 418*m + 280), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (61) = 122\).
time = 1.23, size = 224, normalized size = 3.67 \begin {gather*} \frac {b^{3} m^{3} x^{10} x^{m} + 12 \, b^{3} m^{2} x^{10} x^{m} + 39 \, b^{3} m x^{10} x^{m} + 3 \, a b^{2} m^{3} x^{7} x^{m} + 28 \, b^{3} x^{10} x^{m} + 45 \, a b^{2} m^{2} x^{7} x^{m} + 162 \, a b^{2} m x^{7} x^{m} + 3 \, a^{2} b m^{3} x^{4} x^{m} + 120 \, a b^{2} x^{7} x^{m} + 54 \, a^{2} b m^{2} x^{4} x^{m} + 261 \, a^{2} b m x^{4} x^{m} + a^{3} m^{3} x x^{m} + 210 \, a^{2} b x^{4} x^{m} + 21 \, a^{3} m^{2} x x^{m} + 138 \, a^{3} m x x^{m} + 280 \, a^{3} x x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*m^3*x^10*x^m + 12*b^3*m^2*x^10*x^m + 39*b^3*m*x^10*x^m + 3*a*b^2*m^3*x^7*x^m + 28*b^3*x^10*x^m + 45*a*b^2
*m^2*x^7*x^m + 162*a*b^2*m*x^7*x^m + 3*a^2*b*m^3*x^4*x^m + 120*a*b^2*x^7*x^m + 54*a^2*b*m^2*x^4*x^m + 261*a^2*
b*m*x^4*x^m + a^3*m^3*x*x^m + 210*a^2*b*x^4*x^m + 21*a^3*m^2*x*x^m + 138*a^3*m*x*x^m + 280*a^3*x*x^m)/(m^4 + 2
2*m^3 + 159*m^2 + 418*m + 280)

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Mupad [B]
time = 1.35, size = 167, normalized size = 2.74 \begin {gather*} x^m\,\left (\frac {a^3\,x\,\left (m^3+21\,m^2+138\,m+280\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {b^3\,x^{10}\,\left (m^3+12\,m^2+39\,m+28\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {3\,a\,b^2\,x^7\,\left (m^3+15\,m^2+54\,m+40\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}+\frac {3\,a^2\,b\,x^4\,\left (m^3+18\,m^2+87\,m+70\right )}{m^4+22\,m^3+159\,m^2+418\,m+280}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^3)^3,x)

[Out]

x^m*((a^3*x*(138*m + 21*m^2 + m^3 + 280))/(418*m + 159*m^2 + 22*m^3 + m^4 + 280) + (b^3*x^10*(39*m + 12*m^2 +
m^3 + 28))/(418*m + 159*m^2 + 22*m^3 + m^4 + 280) + (3*a*b^2*x^7*(54*m + 15*m^2 + m^3 + 40))/(418*m + 159*m^2
+ 22*m^3 + m^4 + 280) + (3*a^2*b*x^4*(87*m + 18*m^2 + m^3 + 70))/(418*m + 159*m^2 + 22*m^3 + m^4 + 280))

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