3.273 \(\int x^{5/2} (a+b x^2)^2 \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{11} a b x^{11/2}+\frac{2}{15} b^2 x^{15/2} \]

[Out]

(2*a^2*x^(7/2))/7 + (4*a*b*x^(11/2))/11 + (2*b^2*x^(15/2))/15

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Rubi [A]  time = 0.0084177, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{11} a b x^{11/2}+\frac{2}{15} b^2 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + b*x^2)^2,x]

[Out]

(2*a^2*x^(7/2))/7 + (4*a*b*x^(11/2))/11 + (2*b^2*x^(15/2))/15

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^{5/2} \left (a+b x^2\right )^2 \, dx &=\int \left (a^2 x^{5/2}+2 a b x^{9/2}+b^2 x^{13/2}\right ) \, dx\\ &=\frac{2}{7} a^2 x^{7/2}+\frac{4}{11} a b x^{11/2}+\frac{2}{15} b^2 x^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.007443, size = 30, normalized size = 0.83 \[ \frac{2 x^{7/2} \left (165 a^2+210 a b x^2+77 b^2 x^4\right )}{1155} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + b*x^2)^2,x]

[Out]

(2*x^(7/2)*(165*a^2 + 210*a*b*x^2 + 77*b^2*x^4))/1155

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*}{\frac{154\,{b}^{2}{x}^{4}+420\,ab{x}^{2}+330\,{a}^{2}}{1155}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(b*x^2+a)^2,x)

[Out]

2/1155*x^(7/2)*(77*b^2*x^4+210*a*b*x^2+165*a^2)

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Maxima [A]  time = 1.50366, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{15} \, b^{2} x^{\frac{15}{2}} + \frac{4}{11} \, a b x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^2,x, algorithm="maxima")

[Out]

2/15*b^2*x^(15/2) + 4/11*a*b*x^(11/2) + 2/7*a^2*x^(7/2)

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Fricas [A]  time = 1.21277, size = 77, normalized size = 2.14 \begin{align*} \frac{2}{1155} \,{\left (77 \, b^{2} x^{7} + 210 \, a b x^{5} + 165 \, a^{2} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^2,x, algorithm="fricas")

[Out]

2/1155*(77*b^2*x^7 + 210*a*b*x^5 + 165*a^2*x^3)*sqrt(x)

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Sympy [A]  time = 5.69772, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b x^{\frac{11}{2}}}{11} + \frac{2 b^{2} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(b*x**2+a)**2,x)

[Out]

2*a**2*x**(7/2)/7 + 4*a*b*x**(11/2)/11 + 2*b**2*x**(15/2)/15

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Giac [A]  time = 2.75891, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{15} \, b^{2} x^{\frac{15}{2}} + \frac{4}{11} \, a b x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(b*x^2+a)^2,x, algorithm="giac")

[Out]

2/15*b^2*x^(15/2) + 4/11*a*b*x^(11/2) + 2/7*a^2*x^(7/2)