3.304 \(\int \frac{(b x^2+c x^4)^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{9} b^2 x^{9/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2} \]

[Out]

(2*b^2*x^(9/2))/9 + (4*b*c*x^(13/2))/13 + (2*c^2*x^(17/2))/17

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Rubi [A]  time = 0.0143971, 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} \[ \frac{2}{9} b^2 x^{9/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/Sqrt[x],x]

[Out]

(2*b^2*x^(9/2))/9 + (4*b*c*x^(13/2))/13 + (2*c^2*x^(17/2))/17

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}{\sqrt{x}} \, dx &=\int x^{7/2} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 x^{7/2}+2 b c x^{11/2}+c^2 x^{15/2}\right ) \, dx\\ &=\frac{2}{9} b^2 x^{9/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0081585, size = 30, normalized size = 0.83 \[ \frac{2 x^{9/2} \left (221 b^2+306 b c x^2+117 c^2 x^4\right )}{1989} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(221*b^2 + 306*b*c*x^2 + 117*c^2*x^4))/1989

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Maple [A]  time = 0.046, size = 27, normalized size = 0.8 \begin{align*}{\frac{234\,{c}^{2}{x}^{4}+612\,bc{x}^{2}+442\,{b}^{2}}{1989}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^(1/2),x)

[Out]

2/1989*x^(9/2)*(117*c^2*x^4+306*b*c*x^2+221*b^2)

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Maxima [A]  time = 1.00119, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b c x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^(1/2),x, algorithm="maxima")

[Out]

2/17*c^2*x^(17/2) + 4/13*b*c*x^(13/2) + 2/9*b^2*x^(9/2)

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Fricas [A]  time = 1.21497, size = 78, normalized size = 2.17 \begin{align*} \frac{2}{1989} \,{\left (117 \, c^{2} x^{8} + 306 \, b c x^{6} + 221 \, b^{2} x^{4}\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^(1/2),x, algorithm="fricas")

[Out]

2/1989*(117*c^2*x^8 + 306*b*c*x^6 + 221*b^2*x^4)*sqrt(x)

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Sympy [A]  time = 5.49748, size = 34, normalized size = 0.94 \begin{align*} \frac{2 b^{2} x^{\frac{9}{2}}}{9} + \frac{4 b c x^{\frac{13}{2}}}{13} + \frac{2 c^{2} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**(1/2),x)

[Out]

2*b**2*x**(9/2)/9 + 4*b*c*x**(13/2)/13 + 2*c**2*x**(17/2)/17

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Giac [A]  time = 1.1709, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b c x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^(1/2),x, algorithm="giac")

[Out]

2/17*c^2*x^(17/2) + 4/13*b*c*x^(13/2) + 2/9*b^2*x^(9/2)