3.306 \(\int \frac{(b x^2+c x^4)^2}{x^{5/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{5} b^2 x^{5/2}+\frac{4}{9} b c x^{9/2}+\frac{2}{13} c^2 x^{13/2} \]

[Out]

(2*b^2*x^(5/2))/5 + (4*b*c*x^(9/2))/9 + (2*c^2*x^(13/2))/13

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Rubi [A]  time = 0.0150283, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1584, 270} \[ \frac{2}{5} b^2 x^{5/2}+\frac{4}{9} b c x^{9/2}+\frac{2}{13} c^2 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^(5/2),x]

[Out]

(2*b^2*x^(5/2))/5 + (4*b*c*x^(9/2))/9 + (2*c^2*x^(13/2))/13

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 \frac{\left (b x^2+c x^4\right )^2}{x^{5/2}} \, dx &=\int x^{3/2} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^{3/2}+2 b c x^{7/2}+c^2 x^{11/2}\right ) \, dx\\ &=\frac{2}{5} b^2 x^{5/2}+\frac{4}{9} b c x^{9/2}+\frac{2}{13} c^2 x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0079996, size = 30, normalized size = 0.83 \[ \frac{2}{585} x^{5/2} \left (117 b^2+130 b c x^2+45 c^2 x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^(5/2),x]

[Out]

(2*x^(5/2)*(117*b^2 + 130*b*c*x^2 + 45*c^2*x^4))/585

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Maple [A]  time = 0.046, size = 27, normalized size = 0.8 \begin{align*}{\frac{90\,{c}^{2}{x}^{4}+260\,bc{x}^{2}+234\,{b}^{2}}{585}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^(5/2),x)

[Out]

2/585*x^(5/2)*(45*c^2*x^4+130*b*c*x^2+117*b^2)

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Maxima [A]  time = 0.989313, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{13} \, c^{2} x^{\frac{13}{2}} + \frac{4}{9} \, b c x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^(5/2),x, algorithm="maxima")

[Out]

2/13*c^2*x^(13/2) + 4/9*b*c*x^(9/2) + 2/5*b^2*x^(5/2)

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Fricas [A]  time = 1.26497, size = 76, normalized size = 2.11 \begin{align*} \frac{2}{585} \,{\left (45 \, c^{2} x^{6} + 130 \, b c x^{4} + 117 \, b^{2} x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^(5/2),x, algorithm="fricas")

[Out]

2/585*(45*c^2*x^6 + 130*b*c*x^4 + 117*b^2*x^2)*sqrt(x)

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Sympy [A]  time = 7.10723, size = 34, normalized size = 0.94 \begin{align*} \frac{2 b^{2} x^{\frac{5}{2}}}{5} + \frac{4 b c x^{\frac{9}{2}}}{9} + \frac{2 c^{2} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**(5/2),x)

[Out]

2*b**2*x**(5/2)/5 + 4*b*c*x**(9/2)/9 + 2*c**2*x**(13/2)/13

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Giac [A]  time = 1.16087, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{13} \, c^{2} x^{\frac{13}{2}} + \frac{4}{9} \, b c x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^(5/2),x, algorithm="giac")

[Out]

2/13*c^2*x^(13/2) + 4/9*b*c*x^(9/2) + 2/5*b^2*x^(5/2)