3.10.58 \(\int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx\) [958]

Optimal. Leaf size=59 \[ -\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}}+\frac {8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}} \]

[Out]

-2/3*(-b*x^2+a)^(3/4)/a/c/(c*x)^(7/2)+8/21*(-b*x^2+a)^(7/4)/a^2/c/(c*x)^(7/2)

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Rubi [A]
time = 0.01, 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 = {279, 270} \begin {gather*} \frac {8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}}-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x]

[Out]

(-2*(a - b*x^2)^(3/4))/(3*a*c*(c*x)^(7/2)) + (8*(a - b*x^2)^(7/4))/(21*a^2*c*(c*x)^(7/2))

Rule 270

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 279

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] /; Free
Q[{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx &=-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}}-\frac {4 \int \frac {\left (a-b x^2\right )^{3/4}}{(c x)^{9/2}} \, dx}{3 a}\\ &=-\frac {2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}}+\frac {8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 37, normalized size = 0.63 \begin {gather*} -\frac {2 x \left (a-b x^2\right )^{3/4} \left (3 a+4 b x^2\right )}{21 a^2 (c x)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x]

[Out]

(-2*x*(a - b*x^2)^(3/4)*(3*a + 4*b*x^2))/(21*a^2*(c*x)^(9/2))

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Maple [A]
time = 0.05, size = 32, normalized size = 0.54

method	result	size
gosper	\(-\frac {2 x \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (4 b \,x^{2}+3 a \right )}{21 a^{2} \left (c x \right )^{\frac {9}{2}}}\)	\(32\)
risch	\(-\frac {2 \left (-b \,x^{2}+a \right )^{\frac {3}{4}} \left (4 b \,x^{2}+3 a \right )}{21 c^{4} \sqrt {c x}\, a^{2} x^{3}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-2/21*x*(-b*x^2+a)^(3/4)*(4*b*x^2+3*a)/a^2/(c*x)^(9/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(9/2)), x)

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Fricas [A]
time = 1.44, size = 36, normalized size = 0.61 \begin {gather*} -\frac {2 \, {\left (4 \, b x^{2} + 3 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{21 \, a^{2} c^{5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x, algorithm="fricas")

[Out]

-2/21*(4*b*x^2 + 3*a)*(-b*x^2 + a)^(3/4)*sqrt(c*x)/(a^2*c^5*x^4)

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Sympy [C] Result contains complex when optimal does not.
time = 25.97, size = 343, normalized size = 5.81 \begin {gather*} \begin {cases} - \frac {3 b^{\frac {3}{4}} \left (\frac {a}{b x^{2}} - 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{8 a c^{\frac {9}{2}} x^{2} \Gamma \left (\frac {1}{4}\right )} - \frac {b^{\frac {7}{4}} \left (\frac {a}{b x^{2}} - 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 a^{2} c^{\frac {9}{2}} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\- \frac {3 a^{2} b^{\frac {7}{4}} \left (- \frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 8 a^{3} b c^{\frac {9}{2}} x^{2} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {9}{2}} x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} - \frac {a b^{\frac {11}{4}} x^{2} \left (- \frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 8 a^{3} b c^{\frac {9}{2}} x^{2} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {9}{2}} x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} + \frac {4 b^{\frac {15}{4}} x^{4} \left (- \frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{- 8 a^{3} b c^{\frac {9}{2}} x^{2} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) + 8 a^{2} b^{2} c^{\frac {9}{2}} x^{4} e^{\frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(9/2)/(-b*x**2+a)**(1/4),x)

[Out]

Piecewise((-3*b**(3/4)*(a/(b*x**2) - 1)**(3/4)*gamma(-7/4)/(8*a*c**(9/2)*x**2*gamma(1/4)) - b**(7/4)*(a/(b*x**
2) - 1)**(3/4)*gamma(-7/4)/(2*a**2*c**(9/2)*gamma(1/4)), Abs(a/(b*x**2)) > 1), (-3*a**2*b**(7/4)*(-a/(b*x**2)
+ 1)**(3/4)*gamma(-7/4)/(-8*a**3*b*c**(9/2)*x**2*exp(I*pi/4)*gamma(1/4) + 8*a**2*b**2*c**(9/2)*x**4*exp(I*pi/4
)*gamma(1/4)) - a*b**(11/4)*x**2*(-a/(b*x**2) + 1)**(3/4)*gamma(-7/4)/(-8*a**3*b*c**(9/2)*x**2*exp(I*pi/4)*gam
ma(1/4) + 8*a**2*b**2*c**(9/2)*x**4*exp(I*pi/4)*gamma(1/4)) + 4*b**(15/4)*x**4*(-a/(b*x**2) + 1)**(3/4)*gamma(
-7/4)/(-8*a**3*b*c**(9/2)*x**2*exp(I*pi/4)*gamma(1/4) + 8*a**2*b**2*c**(9/2)*x**4*exp(I*pi/4)*gamma(1/4)), Tru
e))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(9/2)/(-b*x^2+a)^(1/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 + a)^(1/4)*(c*x)^(9/2)), x)

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Mupad [B]
time = 5.14, size = 41, normalized size = 0.69 \begin {gather*} -\frac {{\left (a-b\,x^2\right )}^{3/4}\,\left (\frac {2}{7\,a\,c^4}+\frac {8\,b\,x^2}{21\,a^2\,c^4}\right )}{x^3\,\sqrt {c\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(9/2)*(a - b*x^2)^(1/4)),x)

[Out]

-((a - b*x^2)^(3/4)*(2/(7*a*c^4) + (8*b*x^2)/(21*a^2*c^4)))/(x^3*(c*x)^(1/2))

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