3.984 \(\int \frac {1}{(c x)^{11/2} (a-b x^2)^{3/4}} \, dx\)
Optimal. Leaf size=86 \[ -\frac {64 \left (a-b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac {16 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{9/2}} \]
[Out]
-2*(-b*x^2+a)^(1/4)/a/c/(c*x)^(9/2)+16/5*(-b*x^2+a)^(5/4)/a^2/c/(c*x)^(9/2)-64/45*(-b*x^2+a)^(9/4)/a^3/c/(c*x)
^(9/2)
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Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used =
{273, 264} \[ -\frac {64 \left (a-b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac {16 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{9/2}} \]
Antiderivative was successfully verified.
[In]
Int[1/((c*x)^(11/2)*(a - b*x^2)^(3/4)),x]
[Out]
(-2*(a - b*x^2)^(1/4))/(a*c*(c*x)^(9/2)) + (16*(a - b*x^2)^(5/4))/(5*a^2*c*(c*x)^(9/2)) - (64*(a - b*x^2)^(9/4
))/(45*a^3*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 {1}{(c x)^{11/2} \left (a-b x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{9/2}}-\frac {8 \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{11/2}} \, dx}{a}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{9/2}}+\frac {16 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}+\frac {32 \int \frac {\left (a-b x^2\right )^{5/4}}{(c x)^{11/2}} \, dx}{5 a^2}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{9/2}}+\frac {16 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac {64 \left (a-b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.62 \[ -\frac {2 \sqrt {c x} \sqrt [4]{a-b x^2} \left (5 a^2+8 a b x^2+32 b^2 x^4\right )}{45 a^3 c^6 x^5} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((c*x)^(11/2)*(a - b*x^2)^(3/4)),x]
[Out]
(-2*Sqrt[c*x]*(a - b*x^2)^(1/4)*(5*a^2 + 8*a*b*x^2 + 32*b^2*x^4))/(45*a^3*c^6*x^5)
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fricas [A] time = 1.05, size = 47, normalized size = 0.55 \[ -\frac {2 \, {\left (32 \, b^{2} x^{4} + 8 \, a b x^{2} + 5 \, a^{2}\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{45 \, a^{3} c^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c*x)^(11/2)/(-b*x^2+a)^(3/4),x, algorithm="fricas")
[Out]
-2/45*(32*b^2*x^4 + 8*a*b*x^2 + 5*a^2)*(-b*x^2 + a)^(1/4)*sqrt(c*x)/(a^3*c^6*x^5)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c*x)^(11/2)/(-b*x^2+a)^(3/4),x, algorithm="giac")
[Out]
integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(11/2)), x)
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maple [A] time = 0.01, size = 43, normalized size = 0.50 \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (32 b^{2} x^{4}+8 a b \,x^{2}+5 a^{2}\right ) x}{45 \left (c x \right )^{\frac {11}{2}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(c*x)^(11/2)/(-b*x^2+a)^(3/4),x)
[Out]
-2/45*x*(-b*x^2+a)^(1/4)*(32*b^2*x^4+8*a*b*x^2+5*a^2)/a^3/(c*x)^(11/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c*x)^(11/2)/(-b*x^2+a)^(3/4),x, algorithm="maxima")
[Out]
integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(11/2)), x)
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mupad [B] time = 5.15, size = 55, normalized size = 0.64 \[ -\frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2}{9\,a\,c^5}+\frac {16\,b\,x^2}{45\,a^2\,c^5}+\frac {64\,b^2\,x^4}{45\,a^3\,c^5}\right )}{x^4\,\sqrt {c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((c*x)^(11/2)*(a - b*x^2)^(3/4)),x)
[Out]
-((a - b*x^2)^(1/4)*(2/(9*a*c^5) + (16*b*x^2)/(45*a^2*c^5) + (64*b^2*x^4)/(45*a^3*c^5)))/(x^4*(c*x)^(1/2))
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sympy [C] time = 171.23, size = 1263, normalized size = 14.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(c*x)**(11/2)/(-b*x**2+a)**(3/4),x)
[Out]
Piecewise((-5*a**4*b**(17/4)*(a/(b*x**2) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp
(3*I*pi/4)*gamma(3/4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3*I*pi/4)*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*exp
(3*I*pi/4)*gamma(3/4)) + 2*a**3*b**(21/4)*x**2*(a/(b*x**2) - 1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(32*a**5*b**4*
c**(11/2)*x**4*exp(3*I*pi/4)*gamma(3/4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3*I*pi/4)*gamma(3/4) + 32*a**3*b**6*
c**(11/2)*x**8*exp(3*I*pi/4)*gamma(3/4)) - 21*a**2*b**(25/4)*x**4*(a/(b*x**2) - 1)**(1/4)*exp(-I*pi/4)*gamma(-
9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp(3*I*pi/4)*gamma(3/4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3*I*pi/4)*gamma(3
/4) + 32*a**3*b**6*c**(11/2)*x**8*exp(3*I*pi/4)*gamma(3/4)) + 56*a*b**(29/4)*x**6*(a/(b*x**2) - 1)**(1/4)*exp(
-I*pi/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp(3*I*pi/4)*gamma(3/4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3
*I*pi/4)*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*exp(3*I*pi/4)*gamma(3/4)) - 32*b**(33/4)*x**8*(a/(b*x**2) -
1)**(1/4)*exp(-I*pi/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp(3*I*pi/4)*gamma(3/4) - 64*a**4*b**5*c**(11
/2)*x**6*exp(3*I*pi/4)*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*exp(3*I*pi/4)*gamma(3/4)), Abs(a/(b*x**2)) > 1
), (-5*a**4*b**(17/4)*(-a/(b*x**2) + 1)**(1/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp(3*I*pi/4)*gamma(3/
4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3*I*pi/4)*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*exp(3*I*pi/4)*gamma(3/
4)) + 2*a**3*b**(21/4)*x**2*(-a/(b*x**2) + 1)**(1/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp(3*I*pi/4)*ga
mma(3/4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3*I*pi/4)*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*exp(3*I*pi/4)*ga
mma(3/4)) - 21*a**2*b**(25/4)*x**4*(-a/(b*x**2) + 1)**(1/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp(3*I*p
i/4)*gamma(3/4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3*I*pi/4)*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*exp(3*I*p
i/4)*gamma(3/4)) + 56*a*b**(29/4)*x**6*(-a/(b*x**2) + 1)**(1/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp(3
*I*pi/4)*gamma(3/4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3*I*pi/4)*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*exp(3
*I*pi/4)*gamma(3/4)) - 32*b**(33/4)*x**8*(-a/(b*x**2) + 1)**(1/4)*gamma(-9/4)/(32*a**5*b**4*c**(11/2)*x**4*exp
(3*I*pi/4)*gamma(3/4) - 64*a**4*b**5*c**(11/2)*x**6*exp(3*I*pi/4)*gamma(3/4) + 32*a**3*b**6*c**(11/2)*x**8*exp
(3*I*pi/4)*gamma(3/4)), True))
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