3.983 \(\int \frac {1}{(c x)^{7/2} (a-b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=57 \[ \frac {8 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 57, 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} \[ \frac {8 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(7/2)*(a - b*x^2)^(3/4)),x]

[Out]

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

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)^{7/2} \left (a-b x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{5/2}}-\frac {4 \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{7/2}} \, dx}{a}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{a c (c x)^{5/2}}+\frac {8 \left (a-b x^2\right )^{5/4}}{5 a^2 c (c x)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 35, normalized size = 0.61 \[ -\frac {2 x \sqrt [4]{a-b x^2} \left (a+4 b x^2\right )}{5 a^2 (c x)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(7/2)*(a - b*x^2)^(3/4)),x]

[Out]

(-2*x*(a - b*x^2)^(1/4)*(a + 4*b*x^2))/(5*a^2*(c*x)^(7/2))

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fricas [A]  time = 0.89, size = 34, normalized size = 0.60 \[ -\frac {2 \, {\left (4 \, b x^{2} + a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{5 \, a^{2} c^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(7/2)/(-b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

-2/5*(4*b*x^2 + a)*(-b*x^2 + a)^(1/4)*sqrt(c*x)/(a^2*c^4*x^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(7/2)/(-b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 30, normalized size = 0.53 \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (4 b \,x^{2}+a \right ) x}{5 \left (c x \right )^{\frac {7}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*x*(-b*x^2+a)^(1/4)*(4*b*x^2+a)/a^2/(c*x)^(7/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(7/2)/(-b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.10, size = 41, normalized size = 0.72 \[ -\frac {{\left (a-b\,x^2\right )}^{1/4}\,\left (\frac {2}{5\,a\,c^3}+\frac {8\,b\,x^2}{5\,a^2\,c^3}\right )}{x^2\,\sqrt {c\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(7/2)*(a - b*x^2)^(3/4)),x)

[Out]

-((a - b*x^2)^(1/4)*(2/(5*a*c^3) + (8*b*x^2)/(5*a^2*c^3)))/(x^2*(c*x)^(1/2))

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sympy [A]  time = 25.45, size = 352, normalized size = 6.18 \[ \begin {cases} - \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {5}{4}\right )}{8 a c^{\frac {7}{2}} x^{2} \Gamma \left (\frac {3}{4}\right )} - \frac {b^{\frac {5}{4}} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {5}{4}\right )}{2 a^{2} c^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\- \frac {a^{2} b^{\frac {5}{4}} \sqrt [4]{- \frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{- 8 a^{3} b c^{\frac {7}{2}} x^{2} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 8 a^{2} b^{2} c^{\frac {7}{2}} x^{4} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} - \frac {3 a b^{\frac {9}{4}} x^{2} \sqrt [4]{- \frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{- 8 a^{3} b c^{\frac {7}{2}} x^{2} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 8 a^{2} b^{2} c^{\frac {7}{2}} x^{4} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} + \frac {4 b^{\frac {13}{4}} x^{4} \sqrt [4]{- \frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{- 8 a^{3} b c^{\frac {7}{2}} x^{2} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right ) + 8 a^{2} b^{2} c^{\frac {7}{2}} x^{4} e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(7/2)/(-b*x**2+a)**(3/4),x)

[Out]

Piecewise((-b**(1/4)*(a/(b*x**2) - 1)**(1/4)*gamma(-5/4)/(8*a*c**(7/2)*x**2*gamma(3/4)) - b**(5/4)*(a/(b*x**2)
- 1)**(1/4)*gamma(-5/4)/(2*a**2*c**(7/2)*gamma(3/4)), Abs(a/(b*x**2)) > 1), (-a**2*b**(5/4)*(-a/(b*x**2) + 1)
**(1/4)*gamma(-5/4)/(-8*a**3*b*c**(7/2)*x**2*exp(3*I*pi/4)*gamma(3/4) + 8*a**2*b**2*c**(7/2)*x**4*exp(3*I*pi/4
)*gamma(3/4)) - 3*a*b**(9/4)*x**2*(-a/(b*x**2) + 1)**(1/4)*gamma(-5/4)/(-8*a**3*b*c**(7/2)*x**2*exp(3*I*pi/4)*
gamma(3/4) + 8*a**2*b**2*c**(7/2)*x**4*exp(3*I*pi/4)*gamma(3/4)) + 4*b**(13/4)*x**4*(-a/(b*x**2) + 1)**(1/4)*g
amma(-5/4)/(-8*a**3*b*c**(7/2)*x**2*exp(3*I*pi/4)*gamma(3/4) + 8*a**2*b**2*c**(7/2)*x**4*exp(3*I*pi/4)*gamma(3
/4)), True))

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