3.943 \(\int \frac{\sqrt [4]{a-b x^2}}{(c x)^{19/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{256 \left (a-b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac{64 \left (a-b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac{8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}} \]

[Out]

(-2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(17/2)) + (8*(a - b*x^2)^(9/4))/(15*a^2*c*(c*x)^(17/2)) - (64*(a - b*x^2)^
(13/4))/(195*a^3*c*(c*x)^(17/2)) + (256*(a - b*x^2)^(17/4))/(3315*a^4*c*(c*x)^(17/2))

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Rubi [A]  time = 0.0402291, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {273, 264} \[ \frac{256 \left (a-b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}-\frac{64 \left (a-b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac{8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(1/4)/(c*x)^(19/2),x]

[Out]

(-2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(17/2)) + (8*(a - b*x^2)^(9/4))/(15*a^2*c*(c*x)^(17/2)) - (64*(a - b*x^2)^
(13/4))/(195*a^3*c*(c*x)^(17/2)) + (256*(a - b*x^2)^(17/4))/(3315*a^4*c*(c*x)^(17/2))

Rule 273

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt [4]{a-b x^2}}{(c x)^{19/2}} \, dx &=-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}}-\frac{12 \int \frac{\left (a-b x^2\right )^{5/4}}{(c x)^{19/2}} \, dx}{5 a}\\ &=-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}}+\frac{8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}+\frac{32 \int \frac{\left (a-b x^2\right )^{9/4}}{(c x)^{19/2}} \, dx}{15 a^2}\\ &=-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}}+\frac{8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac{64 \left (a-b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}-\frac{128 \int \frac{\left (a-b x^2\right )^{13/4}}{(c x)^{19/2}} \, dx}{195 a^3}\\ &=-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{17/2}}+\frac{8 \left (a-b x^2\right )^{9/4}}{15 a^2 c (c x)^{17/2}}-\frac{64 \left (a-b x^2\right )^{13/4}}{195 a^3 c (c x)^{17/2}}+\frac{256 \left (a-b x^2\right )^{17/4}}{3315 a^4 c (c x)^{17/2}}\\ \end{align*}

Mathematica [A]  time = 0.0165824, size = 64, normalized size = 0.55 \[ -\frac{2 \left (a-b x^2\right )^{5/4} \left (180 a^2 b x^2+195 a^3+160 a b^2 x^4+128 b^3 x^6\right )}{3315 a^4 c^9 x^8 \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(1/4)/(c*x)^(19/2),x]

[Out]

(-2*(a - b*x^2)^(5/4)*(195*a^3 + 180*a^2*b*x^2 + 160*a*b^2*x^4 + 128*b^3*x^6))/(3315*a^4*c^9*x^8*Sqrt[c*x])

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Maple [A]  time = 0.005, size = 54, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( 128\,{b}^{3}{x}^{6}+160\,a{b}^{2}{x}^{4}+180\,{a}^{2}b{x}^{2}+195\,{a}^{3} \right ) }{3315\,{a}^{4}} \left ( -b{x}^{2}+a \right ) ^{{\frac{5}{4}}} \left ( cx \right ) ^{-{\frac{19}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^(1/4)/(c*x)^(19/2),x)

[Out]

-2/3315*x*(-b*x^2+a)^(5/4)*(128*b^3*x^6+160*a*b^2*x^4+180*a^2*b*x^2+195*a^3)/a^4/(c*x)^(19/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{19}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(19/2),x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(19/2), x)

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Fricas [A]  time = 1.569, size = 166, normalized size = 1.42 \begin{align*} \frac{2 \,{\left (128 \, b^{4} x^{8} + 32 \, a b^{3} x^{6} + 20 \, a^{2} b^{2} x^{4} + 15 \, a^{3} b x^{2} - 195 \, a^{4}\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{3315 \, a^{4} c^{10} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(19/2),x, algorithm="fricas")

[Out]

2/3315*(128*b^4*x^8 + 32*a*b^3*x^6 + 20*a^2*b^2*x^4 + 15*a^3*b*x^2 - 195*a^4)*(-b*x^2 + a)^(1/4)*sqrt(c*x)/(a^
4*c^10*x^9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**2+a)**(1/4)/(c*x)**(19/2),x)

[Out]

Timed out

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Giac [B]  time = 2.12245, size = 367, normalized size = 3.14 \begin{align*} \frac{2 \,{\left (\frac{663 \,{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}{\left (b c^{2} - \frac{a c^{2}}{x^{2}}\right )} b^{3} c^{6}}{\sqrt{c x}} - \frac{1105 \,{\left (b^{2} c^{8} x^{4} - 2 \, a b c^{8} x^{2} + a^{2} c^{8}\right )}{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}} b^{2}}{\sqrt{c x} x^{4}} + \frac{765 \,{\left (b^{3} c^{12} x^{6} - 3 \, a b^{2} c^{12} x^{4} + 3 \, a^{2} b c^{12} x^{2} - a^{3} c^{12}\right )}{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}} b}{\sqrt{c x} c^{4} x^{6}} - \frac{195 \,{\left (b^{4} c^{16} x^{8} - 4 \, a b^{3} c^{16} x^{6} + 6 \, a^{2} b^{2} c^{16} x^{4} - 4 \, a^{3} b c^{16} x^{2} + a^{4} c^{16}\right )}{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}}{\sqrt{c x} c^{8} x^{8}}\right )}}{3315 \, a^{4} c^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(19/2),x, algorithm="giac")

[Out]

2/3315*(663*(-b*c^4*x^2 + a*c^4)^(1/4)*(b*c^2 - a*c^2/x^2)*b^3*c^6/sqrt(c*x) - 1105*(b^2*c^8*x^4 - 2*a*b*c^8*x
^2 + a^2*c^8)*(-b*c^4*x^2 + a*c^4)^(1/4)*b^2/(sqrt(c*x)*x^4) + 765*(b^3*c^12*x^6 - 3*a*b^2*c^12*x^4 + 3*a^2*b*
c^12*x^2 - a^3*c^12)*(-b*c^4*x^2 + a*c^4)^(1/4)*b/(sqrt(c*x)*c^4*x^6) - 195*(b^4*c^16*x^8 - 4*a*b^3*c^16*x^6 +
6*a^2*b^2*c^16*x^4 - 4*a^3*b*c^16*x^2 + a^4*c^16)*(-b*c^4*x^2 + a*c^4)^(1/4)/(sqrt(c*x)*c^8*x^8))/(a^4*c^18)