3.341 \(\int x^m (a+b x^2)^3 \, dx\)

Optimal. Leaf size=61 \[ \frac{3 a^2 b x^{m+3}}{m+3}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+5}}{m+5}+\frac{b^3 x^{m+7}}{m+7} \]

[Out]

(a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(3 + m))/(3 + m) + (3*a*b^2*x^(5 + m))/(5 + m) + (b^3*x^(7 + m))/(7 + m)

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Rubi [A]  time = 0.022071, antiderivative size = 61, 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, Rules used = {270} \[ \frac{3 a^2 b x^{m+3}}{m+3}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+5}}{m+5}+\frac{b^3 x^{m+7}}{m+7} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^2)^3,x]

[Out]

(a^3*x^(1 + m))/(1 + m) + (3*a^2*b*x^(3 + m))/(3 + m) + (3*a*b^2*x^(5 + m))/(5 + m) + (b^3*x^(7 + m))/(7 + m)

Rule 270

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^2\right )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{2+m}+3 a b^2 x^{4+m}+b^3 x^{6+m}\right ) \, dx\\ &=\frac{a^3 x^{1+m}}{1+m}+\frac{3 a^2 b x^{3+m}}{3+m}+\frac{3 a b^2 x^{5+m}}{5+m}+\frac{b^3 x^{7+m}}{7+m}\\ \end{align*}

Mathematica [A]  time = 0.0296843, size = 56, normalized size = 0.92 \[ x^{m+1} \left (\frac{3 a^2 b x^2}{m+3}+\frac{a^3}{m+1}+\frac{3 a b^2 x^4}{m+5}+\frac{b^3 x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^2)^3,x]

[Out]

x^(1 + m)*(a^3/(1 + m) + (3*a^2*b*x^2)/(3 + m) + (3*a*b^2*x^4)/(5 + m) + (b^3*x^6)/(7 + m))

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Maple [B]  time = 0.004, size = 178, normalized size = 2.9 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{3}{m}^{3}{x}^{6}+9\,{b}^{3}{m}^{2}{x}^{6}+3\,a{b}^{2}{m}^{3}{x}^{4}+23\,{b}^{3}m{x}^{6}+33\,a{b}^{2}{m}^{2}{x}^{4}+15\,{b}^{3}{x}^{6}+3\,{a}^{2}b{m}^{3}{x}^{2}+93\,a{b}^{2}m{x}^{4}+39\,{a}^{2}b{m}^{2}{x}^{2}+63\,a{b}^{2}{x}^{4}+{a}^{3}{m}^{3}+141\,{a}^{2}bm{x}^{2}+15\,{a}^{3}{m}^{2}+105\,{a}^{2}b{x}^{2}+71\,{a}^{3}m+105\,{a}^{3} \right ) }{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^2+a)^3,x)

[Out]

x^(1+m)*(b^3*m^3*x^6+9*b^3*m^2*x^6+3*a*b^2*m^3*x^4+23*b^3*m*x^6+33*a*b^2*m^2*x^4+15*b^3*x^6+3*a^2*b*m^3*x^2+93
*a*b^2*m*x^4+39*a^2*b*m^2*x^2+63*a*b^2*x^4+a^3*m^3+141*a^2*b*m*x^2+15*a^3*m^2+105*a^2*b*x^2+71*a^3*m+105*a^3)/
(7+m)/(5+m)/(3+m)/(1+m)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.66446, size = 347, normalized size = 5.69 \begin{align*} \frac{{\left ({\left (b^{3} m^{3} + 9 \, b^{3} m^{2} + 23 \, b^{3} m + 15 \, b^{3}\right )} x^{7} + 3 \,{\left (a b^{2} m^{3} + 11 \, a b^{2} m^{2} + 31 \, a b^{2} m + 21 \, a b^{2}\right )} x^{5} + 3 \,{\left (a^{2} b m^{3} + 13 \, a^{2} b m^{2} + 47 \, a^{2} b m + 35 \, a^{2} b\right )} x^{3} +{\left (a^{3} m^{3} + 15 \, a^{3} m^{2} + 71 \, a^{3} m + 105 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^3*m^3 + 9*b^3*m^2 + 23*b^3*m + 15*b^3)*x^7 + 3*(a*b^2*m^3 + 11*a*b^2*m^2 + 31*a*b^2*m + 21*a*b^2)*x^5 + 3*
(a^2*b*m^3 + 13*a^2*b*m^2 + 47*a^2*b*m + 35*a^2*b)*x^3 + (a^3*m^3 + 15*a^3*m^2 + 71*a^3*m + 105*a^3)*x)*x^m/(m
^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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Sympy [A]  time = 1.53952, size = 683, normalized size = 11.2 \begin{align*} \begin{cases} - \frac{a^{3}}{6 x^{6}} - \frac{3 a^{2} b}{4 x^{4}} - \frac{3 a b^{2}}{2 x^{2}} + b^{3} \log{\left (x \right )} & \text{for}\: m = -7 \\- \frac{a^{3}}{4 x^{4}} - \frac{3 a^{2} b}{2 x^{2}} + 3 a b^{2} \log{\left (x \right )} + \frac{b^{3} x^{2}}{2} & \text{for}\: m = -5 \\- \frac{a^{3}}{2 x^{2}} + 3 a^{2} b \log{\left (x \right )} + \frac{3 a b^{2} x^{2}}{2} + \frac{b^{3} x^{4}}{4} & \text{for}\: m = -3 \\a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{6}}{6} & \text{for}\: m = -1 \\\frac{a^{3} m^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{15 a^{3} m^{2} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{71 a^{3} m x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{105 a^{3} x x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{3 a^{2} b m^{3} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{39 a^{2} b m^{2} x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{141 a^{2} b m x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{105 a^{2} b x^{3} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{3 a b^{2} m^{3} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{33 a b^{2} m^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{93 a b^{2} m x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{63 a b^{2} x^{5} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{b^{3} m^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{9 b^{3} m^{2} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{23 b^{3} m x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} + \frac{15 b^{3} x^{7} x^{m}}{m^{4} + 16 m^{3} + 86 m^{2} + 176 m + 105} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**2+a)**3,x)

[Out]

Piecewise((-a**3/(6*x**6) - 3*a**2*b/(4*x**4) - 3*a*b**2/(2*x**2) + b**3*log(x), Eq(m, -7)), (-a**3/(4*x**4) -
3*a**2*b/(2*x**2) + 3*a*b**2*log(x) + b**3*x**2/2, Eq(m, -5)), (-a**3/(2*x**2) + 3*a**2*b*log(x) + 3*a*b**2*x
**2/2 + b**3*x**4/4, Eq(m, -3)), (a**3*log(x) + 3*a**2*b*x**2/2 + 3*a*b**2*x**4/4 + b**3*x**6/6, Eq(m, -1)), (
a**3*m**3*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*a**3*m**2*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 17
6*m + 105) + 71*a**3*m*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 105*a**3*x*x**m/(m**4 + 16*m**3 + 86*
m**2 + 176*m + 105) + 3*a**2*b*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 39*a**2*b*m**2*x**3*x
**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 141*a**2*b*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105)
+ 105*a**2*b*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 3*a*b**2*m**3*x**5*x**m/(m**4 + 16*m**3 + 8
6*m**2 + 176*m + 105) + 33*a*b**2*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 93*a*b**2*m*x**5*x
**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 63*a*b**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) +
b**3*m**3*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*b**3*m**2*x**7*x**m/(m**4 + 16*m**3 + 86*m**2
+ 176*m + 105) + 23*b**3*m*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*b**3*x**7*x**m/(m**4 + 16*
m**3 + 86*m**2 + 176*m + 105), True))

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Giac [B]  time = 2.1142, size = 302, normalized size = 4.95 \begin{align*} \frac{b^{3} m^{3} x^{7} x^{m} + 9 \, b^{3} m^{2} x^{7} x^{m} + 3 \, a b^{2} m^{3} x^{5} x^{m} + 23 \, b^{3} m x^{7} x^{m} + 33 \, a b^{2} m^{2} x^{5} x^{m} + 15 \, b^{3} x^{7} x^{m} + 3 \, a^{2} b m^{3} x^{3} x^{m} + 93 \, a b^{2} m x^{5} x^{m} + 39 \, a^{2} b m^{2} x^{3} x^{m} + 63 \, a b^{2} x^{5} x^{m} + a^{3} m^{3} x x^{m} + 141 \, a^{2} b m x^{3} x^{m} + 15 \, a^{3} m^{2} x x^{m} + 105 \, a^{2} b x^{3} x^{m} + 71 \, a^{3} m x x^{m} + 105 \, a^{3} x x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*m^3*x^7*x^m + 9*b^3*m^2*x^7*x^m + 3*a*b^2*m^3*x^5*x^m + 23*b^3*m*x^7*x^m + 33*a*b^2*m^2*x^5*x^m + 15*b^3*
x^7*x^m + 3*a^2*b*m^3*x^3*x^m + 93*a*b^2*m*x^5*x^m + 39*a^2*b*m^2*x^3*x^m + 63*a*b^2*x^5*x^m + a^3*m^3*x*x^m +
141*a^2*b*m*x^3*x^m + 15*a^3*m^2*x*x^m + 105*a^2*b*x^3*x^m + 71*a^3*m*x*x^m + 105*a^3*x*x^m)/(m^4 + 16*m^3 +
86*m^2 + 176*m + 105)