3.941 \(\int \frac {\sqrt [4]{a-b x^2}}{(c x)^{11/2}} \, dx\)

Optimal. Leaf size=59 \[ \frac {8 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \]

[Out]

-2/5*(-b*x^2+a)^(5/4)/a/c/(c*x)^(9/2)+8/45*(-b*x^2+a)^(9/4)/a^2/c/(c*x)^(9/2)

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Rubi [A]  time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {273, 264} \[ \frac {8 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(1/4)/(c*x)^(11/2),x]

[Out]

(-2*(a - b*x^2)^(5/4))/(5*a*c*(c*x)^(9/2)) + (8*(a - b*x^2)^(9/4))/(45*a^2*c*(c*x)^(9/2))

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]

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]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{11/2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}-\frac {4 \int \frac {\left (a-b x^2\right )^{5/4}}{(c x)^{11/2}} \, dx}{5 a}\\ &=-\frac {2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{9/2}}+\frac {8 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 42, normalized size = 0.71 \[ -\frac {2 \sqrt {c x} \left (a-b x^2\right )^{5/4} \left (5 a+4 b x^2\right )}{45 a^2 c^6 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(1/4)/(c*x)^(11/2),x]

[Out]

(-2*Sqrt[c*x]*(a - b*x^2)^(5/4)*(5*a + 4*b*x^2))/(45*a^2*c^6*x^5)

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fricas [A]  time = 0.86, size = 46, normalized size = 0.78 \[ \frac {2 \, {\left (4 \, b^{2} x^{4} + a b x^{2} - 5 \, a^{2}\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{45 \, a^{2} c^{6} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(11/2),x, algorithm="fricas")

[Out]

2/45*(4*b^2*x^4 + a*b*x^2 - 5*a^2)*(-b*x^2 + a)^(1/4)*sqrt(c*x)/(a^2*c^6*x^5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(11/2),x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(11/2), x)

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maple [A]  time = 0.00, size = 32, normalized size = 0.54 \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {5}{4}} \left (4 b \,x^{2}+5 a \right ) x}{45 \left (c x \right )^{\frac {11}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^(1/4)/(c*x)^(11/2),x)

[Out]

-2/45*x*(-b*x^2+a)^(5/4)*(4*b*x^2+5*a)/a^2/(c*x)^(11/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(1/4)/(c*x)^(11/2),x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(1/4)/(c*x)^(11/2), x)

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mupad [B]  time = 4.89, size = 51, normalized size = 0.86 \[ \frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2\,b\,x^2}{45\,a\,c^5}-\frac {2}{9\,c^5}+\frac {8\,b^2\,x^4}{45\,a^2\,c^5}\right )}{x^4\,\sqrt {c\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x^2)^(1/4)/(c*x)^(11/2),x)

[Out]

((a - b*x^2)^(1/4)*((2*b*x^2)/(45*a*c^5) - 2/(9*c^5) + (8*b^2*x^4)/(45*a^2*c^5)))/(x^4*(c*x)^(1/2))

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sympy [B]  time = 68.25, size = 462, normalized size = 7.83 \[ \begin {cases} - \frac {5 \sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {9}{4}\right )}{8 c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )} + \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {9}{4}\right )}{8 a c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right )} + \frac {b^{\frac {9}{4}} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {9}{4}\right )}{2 a^{2} c^{\frac {11}{2}} \Gamma \left (- \frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\\frac {5 a^{3} b^{\frac {5}{4}} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{x^{2} \left (- 8 a^{3} b c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )\right )} - \frac {6 a^{2} b^{\frac {9}{4}} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 8 a^{3} b c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )} - \frac {3 a b^{\frac {13}{4}} x^{2} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 8 a^{3} b c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )} + \frac {4 b^{\frac {17}{4}} x^{4} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {9}{4}\right )}{- 8 a^{3} b c^{\frac {11}{2}} x^{2} \Gamma \left (- \frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {11}{2}} x^{4} \Gamma \left (- \frac {1}{4}\right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**2+a)**(1/4)/(c*x)**(11/2),x)

[Out]

Piecewise((-5*b**(1/4)*(a/(b*x**2) - 1)**(1/4)*gamma(-9/4)/(8*c**(11/2)*x**4*gamma(-1/4)) + b**(5/4)*(a/(b*x**
2) - 1)**(1/4)*gamma(-9/4)/(8*a*c**(11/2)*x**2*gamma(-1/4)) + b**(9/4)*(a/(b*x**2) - 1)**(1/4)*gamma(-9/4)/(2*
a**2*c**(11/2)*gamma(-1/4)), Abs(a/(b*x**2)) > 1), (5*a**3*b**(5/4)*(-a/(b*x**2) + 1)**(1/4)*exp(I*pi/4)*gamma
(-9/4)/(x**2*(-8*a**3*b*c**(11/2)*x**2*gamma(-1/4) + 8*a**2*b**2*c**(11/2)*x**4*gamma(-1/4))) - 6*a**2*b**(9/4
)*(-a/(b*x**2) + 1)**(1/4)*exp(I*pi/4)*gamma(-9/4)/(-8*a**3*b*c**(11/2)*x**2*gamma(-1/4) + 8*a**2*b**2*c**(11/
2)*x**4*gamma(-1/4)) - 3*a*b**(13/4)*x**2*(-a/(b*x**2) + 1)**(1/4)*exp(I*pi/4)*gamma(-9/4)/(-8*a**3*b*c**(11/2
)*x**2*gamma(-1/4) + 8*a**2*b**2*c**(11/2)*x**4*gamma(-1/4)) + 4*b**(17/4)*x**4*(-a/(b*x**2) + 1)**(1/4)*exp(I
*pi/4)*gamma(-9/4)/(-8*a**3*b*c**(11/2)*x**2*gamma(-1/4) + 8*a**2*b**2*c**(11/2)*x**4*gamma(-1/4)), True))

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