3.314 \(\int \frac{(b x^2+c x^4)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac{6}{13} b^2 c x^{13/2}+\frac{2}{9} b^3 x^{9/2}+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

[Out]

(2*b^3*x^(9/2))/9 + (6*b^2*c*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (2*c^3*x^(21/2))/21

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Rubi [A]  time = 0.0189882, 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} \[ \frac{6}{13} b^2 c x^{13/2}+\frac{2}{9} b^3 x^{9/2}+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(2*b^3*x^(9/2))/9 + (6*b^2*c*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (2*c^3*x^(21/2))/21

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}{x^{5/2}} \, dx &=\int x^{7/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{7/2}+3 b^2 c x^{11/2}+3 b c^2 x^{15/2}+c^3 x^{19/2}\right ) \, dx\\ &=\frac{2}{9} b^3 x^{9/2}+\frac{6}{13} b^2 c x^{13/2}+\frac{6}{17} b c^2 x^{17/2}+\frac{2}{21} c^3 x^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0106717, size = 41, normalized size = 0.8 \[ \frac{2 x^{9/2} \left (3213 b^2 c x^2+1547 b^3+2457 b c^2 x^4+663 c^3 x^6\right )}{13923} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(2*x^(9/2)*(1547*b^3 + 3213*b^2*c*x^2 + 2457*b*c^2*x^4 + 663*c^3*x^6))/13923

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Maple [A]  time = 0.048, size = 38, normalized size = 0.8 \begin{align*}{\frac{1326\,{c}^{3}{x}^{6}+4914\,b{c}^{2}{x}^{4}+6426\,{b}^{2}c{x}^{2}+3094\,{b}^{3}}{13923}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^(5/2),x)

[Out]

2/13923*x^(9/2)*(663*c^3*x^6+2457*b*c^2*x^4+3213*b^2*c*x^2+1547*b^3)

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Maxima [A]  time = 1.01459, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b c^{2} x^{\frac{17}{2}} + \frac{6}{13} \, b^{2} c x^{\frac{13}{2}} + \frac{2}{9} \, b^{3} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 2/9*b^3*x^(9/2)

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Fricas [A]  time = 1.27837, size = 109, normalized size = 2.14 \begin{align*} \frac{2}{13923} \,{\left (663 \, c^{3} x^{10} + 2457 \, b c^{2} x^{8} + 3213 \, b^{2} c x^{6} + 1547 \, b^{3} 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)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/13923*(663*c^3*x^10 + 2457*b*c^2*x^8 + 3213*b^2*c*x^6 + 1547*b^3*x^4)*sqrt(x)

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Sympy [A]  time = 22.7064, size = 49, normalized size = 0.96 \begin{align*} \frac{2 b^{3} x^{\frac{9}{2}}}{9} + \frac{6 b^{2} c x^{\frac{13}{2}}}{13} + \frac{6 b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 c^{3} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**(5/2),x)

[Out]

2*b**3*x**(9/2)/9 + 6*b**2*c*x**(13/2)/13 + 6*b*c**2*x**(17/2)/17 + 2*c**3*x**(21/2)/21

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Giac [A]  time = 1.13714, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b c^{2} x^{\frac{17}{2}} + \frac{6}{13} \, b^{2} c x^{\frac{13}{2}} + \frac{2}{9} \, b^{3} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^(5/2),x, algorithm="giac")

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 2/9*b^3*x^(9/2)