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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 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]

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]

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 = 0.01, 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

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 29, normalized size = 0.81 \[ \frac {2}{585} \, {\left (45 \, c^{2} x^{6} + 130 \, b c x^{4} + 117 \, b^{2} x^{2}\right )} \sqrt {x} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 24, normalized size = 0.67 \[ \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} x^{\frac {5}{2}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 27, normalized size = 0.75 \[ \frac {2 \left (45 c^{2} x^{4}+130 b c \,x^{2}+117 b^{2}\right ) x^{\frac {5}{2}}}{585} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.35, size = 24, normalized size = 0.67 \[ \frac {2}{13} \, c^{2} x^{\frac {13}{2}} + \frac {4}{9} \, b c x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} x^{\frac {5}{2}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 26, normalized size = 0.72 \[ \frac {2\,x^{5/2}\,\left (117\,b^2+130\,b\,c\,x^2+45\,c^2\,x^4\right )}{585} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^2/x^(5/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 5.99, size = 34, normalized size = 0.94 \[ \frac {2 b^{2} x^{\frac {5}{2}}}{5} + \frac {4 b c x^{\frac {9}{2}}}{9} + \frac {2 c^{2} x^{\frac {13}{2}}}{13} \]

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

________________________________________________________________________________________