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

Optimal. Leaf size=43 \[ \frac {a^2 x^{m+1}}{m+1}+\frac {2 a b x^{m+3}}{m+3}+\frac {b^2 x^{m+5}}{m+5} \]

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Rubi [A]  time = 0.02, antiderivative size = 43, 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 = {270} \begin {gather*} \frac {a^2 x^{m+1}}{m+1}+\frac {2 a b x^{m+3}}{m+3}+\frac {b^2 x^{m+5}}{m+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^2)^2,x]

[Out]

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

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Mathematica [A]  time = 0.02, size = 40, normalized size = 0.93 \begin {gather*} x^{m+1} \left (\frac {a^2}{m+1}+\frac {2 a b x^2}{m+3}+\frac {b^2 x^4}{m+5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^2)^2,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (a+b x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^m*(a + b*x^2)^2,x]

[Out]

Defer[IntegrateAlgebraic][x^m*(a + b*x^2)^2, x]

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fricas [A]  time = 0.93, size = 85, normalized size = 1.98 \begin {gather*} \frac {{\left ({\left (b^{2} m^{2} + 4 \, b^{2} m + 3 \, b^{2}\right )} x^{5} + 2 \, {\left (a b m^{2} + 6 \, a b m + 5 \, a b\right )} x^{3} + {\left (a^{2} m^{2} + 8 \, a^{2} m + 15 \, a^{2}\right )} x\right )} x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*m^2 + 4*b^2*m + 3*b^2)*x^5 + 2*(a*b*m^2 + 6*a*b*m + 5*a*b)*x^3 + (a^2*m^2 + 8*a^2*m + 15*a^2)*x)*x^m/(m^
3 + 9*m^2 + 23*m + 15)

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giac [B]  time = 0.65, size = 117, normalized size = 2.72 \begin {gather*} \frac {b^{2} m^{2} x^{5} x^{m} + 4 \, b^{2} m x^{5} x^{m} + 2 \, a b m^{2} x^{3} x^{m} + 3 \, b^{2} x^{5} x^{m} + 12 \, a b m x^{3} x^{m} + a^{2} m^{2} x x^{m} + 10 \, a b x^{3} x^{m} + 8 \, a^{2} m x x^{m} + 15 \, a^{2} x x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*m^2*x^5*x^m + 4*b^2*m*x^5*x^m + 2*a*b*m^2*x^3*x^m + 3*b^2*x^5*x^m + 12*a*b*m*x^3*x^m + a^2*m^2*x*x^m + 10
*a*b*x^3*x^m + 8*a^2*m*x*x^m + 15*a^2*x*x^m)/(m^3 + 9*m^2 + 23*m + 15)

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maple [B]  time = 0.01, size = 93, normalized size = 2.16 \begin {gather*} \frac {\left (b^{2} m^{2} x^{4}+4 b^{2} m \,x^{4}+2 a b \,m^{2} x^{2}+3 b^{2} x^{4}+12 a b m \,x^{2}+a^{2} m^{2}+10 a b \,x^{2}+8 a^{2} m +15 a^{2}\right ) x^{m +1}}{\left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(m+1)*(b^2*m^2*x^4+4*b^2*m*x^4+2*a*b*m^2*x^2+3*b^2*x^4+12*a*b*m*x^2+a^2*m^2+10*a*b*x^2+8*a^2*m+15*a^2)/(m+5)
/(m+3)/(m+1)

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maxima [A]  time = 1.31, size = 43, normalized size = 1.00 \begin {gather*} \frac {b^{2} x^{m + 5}}{m + 5} + \frac {2 \, a b x^{m + 3}}{m + 3} + \frac {a^{2} x^{m + 1}}{m + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.75, size = 93, normalized size = 2.16 \begin {gather*} x^m\,\left (\frac {a^2\,x\,\left (m^2+8\,m+15\right )}{m^3+9\,m^2+23\,m+15}+\frac {b^2\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {2\,a\,b\,x^3\,\left (m^2+6\,m+5\right )}{m^3+9\,m^2+23\,m+15}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a + b*x^2)^2,x)

[Out]

x^m*((a^2*x*(8*m + m^2 + 15))/(23*m + 9*m^2 + m^3 + 15) + (b^2*x^5*(4*m + m^2 + 3))/(23*m + 9*m^2 + m^3 + 15)
+ (2*a*b*x^3*(6*m + m^2 + 5))/(23*m + 9*m^2 + m^3 + 15))

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sympy [A]  time = 0.94, size = 306, normalized size = 7.12 \begin {gather*} \begin {cases} - \frac {a^{2}}{4 x^{4}} - \frac {a b}{x^{2}} + b^{2} \log {\relax (x )} & \text {for}\: m = -5 \\- \frac {a^{2}}{2 x^{2}} + 2 a b \log {\relax (x )} + \frac {b^{2} x^{2}}{2} & \text {for}\: m = -3 \\a^{2} \log {\relax (x )} + a b x^{2} + \frac {b^{2} x^{4}}{4} & \text {for}\: m = -1 \\\frac {a^{2} m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {8 a^{2} m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {15 a^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {2 a b m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {12 a b m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {10 a b x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {b^{2} m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {4 b^{2} m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {3 b^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2/(4*x**4) - a*b/x**2 + b**2*log(x), Eq(m, -5)), (-a**2/(2*x**2) + 2*a*b*log(x) + b**2*x**2/2,
Eq(m, -3)), (a**2*log(x) + a*b*x**2 + b**2*x**4/4, Eq(m, -1)), (a**2*m**2*x*x**m/(m**3 + 9*m**2 + 23*m + 15) +
8*a**2*m*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + 15*a**2*x*x**m/(m**3 + 9*m**2 + 23*m + 15) + 2*a*b*m**2*x**3*x*
*m/(m**3 + 9*m**2 + 23*m + 15) + 12*a*b*m*x**3*x**m/(m**3 + 9*m**2 + 23*m + 15) + 10*a*b*x**3*x**m/(m**3 + 9*m
**2 + 23*m + 15) + b**2*m**2*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15) + 4*b**2*m*x**5*x**m/(m**3 + 9*m**2 + 23*m
+ 15) + 3*b**2*x**5*x**m/(m**3 + 9*m**2 + 23*m + 15), True))

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