3.312 \(\int \frac{(b x^2+c x^4)^3}{\sqrt{x}} \, dx\)

Optimal. Leaf size=51 \[ \frac{6}{17} b^2 c x^{17/2}+\frac{2}{13} b^3 x^{13/2}+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2} \]

[Out]

(2*b^3*x^(13/2))/13 + (6*b^2*c*x^(17/2))/17 + (2*b*c^2*x^(21/2))/7 + (2*c^3*x^(25/2))/25

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Rubi [A]  time = 0.0202165, antiderivative size = 51, 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} \[ \frac{6}{17} b^2 c x^{17/2}+\frac{2}{13} b^3 x^{13/2}+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/Sqrt[x],x]

[Out]

(2*b^3*x^(13/2))/13 + (6*b^2*c*x^(17/2))/17 + (2*b*c^2*x^(21/2))/7 + (2*c^3*x^(25/2))/25

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 )^3}{\sqrt{x}} \, dx &=\int x^{11/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{11/2}+3 b^2 c x^{15/2}+3 b c^2 x^{19/2}+c^3 x^{23/2}\right ) \, dx\\ &=\frac{2}{13} b^3 x^{13/2}+\frac{6}{17} b^2 c x^{17/2}+\frac{2}{7} b c^2 x^{21/2}+\frac{2}{25} c^3 x^{25/2}\\ \end{align*}

Mathematica [A]  time = 0.0112586, size = 41, normalized size = 0.8 \[ \frac{2 x^{13/2} \left (6825 b^2 c x^2+2975 b^3+5525 b c^2 x^4+1547 c^3 x^6\right )}{38675} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/Sqrt[x],x]

[Out]

(2*x^(13/2)*(2975*b^3 + 6825*b^2*c*x^2 + 5525*b*c^2*x^4 + 1547*c^3*x^6))/38675

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Maple [A]  time = 0.047, size = 38, normalized size = 0.8 \begin{align*}{\frac{3094\,{c}^{3}{x}^{6}+11050\,b{c}^{2}{x}^{4}+13650\,{b}^{2}c{x}^{2}+5950\,{b}^{3}}{38675}{x}^{{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^(1/2),x)

[Out]

2/38675*x^(13/2)*(1547*c^3*x^6+5525*b*c^2*x^4+6825*b^2*c*x^2+2975*b^3)

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Maxima [A]  time = 1.00679, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{25} \, c^{3} x^{\frac{25}{2}} + \frac{2}{7} \, b c^{2} x^{\frac{21}{2}} + \frac{6}{17} \, b^{2} c x^{\frac{17}{2}} + \frac{2}{13} \, b^{3} x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^(1/2),x, algorithm="maxima")

[Out]

2/25*c^3*x^(25/2) + 2/7*b*c^2*x^(21/2) + 6/17*b^2*c*x^(17/2) + 2/13*b^3*x^(13/2)

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Fricas [A]  time = 1.25456, size = 112, normalized size = 2.2 \begin{align*} \frac{2}{38675} \,{\left (1547 \, c^{3} x^{12} + 5525 \, b c^{2} x^{10} + 6825 \, b^{2} c x^{8} + 2975 \, b^{3} x^{6}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^(1/2),x, algorithm="fricas")

[Out]

2/38675*(1547*c^3*x^12 + 5525*b*c^2*x^10 + 6825*b^2*c*x^8 + 2975*b^3*x^6)*sqrt(x)

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Sympy [A]  time = 18.7054, size = 49, normalized size = 0.96 \begin{align*} \frac{2 b^{3} x^{\frac{13}{2}}}{13} + \frac{6 b^{2} c x^{\frac{17}{2}}}{17} + \frac{2 b c^{2} x^{\frac{21}{2}}}{7} + \frac{2 c^{3} x^{\frac{25}{2}}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**(1/2),x)

[Out]

2*b**3*x**(13/2)/13 + 6*b**2*c*x**(17/2)/17 + 2*b*c**2*x**(21/2)/7 + 2*c**3*x**(25/2)/25

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Giac [A]  time = 1.16078, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{25} \, c^{3} x^{\frac{25}{2}} + \frac{2}{7} \, b c^{2} x^{\frac{21}{2}} + \frac{6}{17} \, b^{2} c x^{\frac{17}{2}} + \frac{2}{13} \, b^{3} x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^(1/2),x, algorithm="giac")

[Out]

2/25*c^3*x^(25/2) + 2/7*b*c^2*x^(21/2) + 6/17*b^2*c*x^(17/2) + 2/13*b^3*x^(13/2)