3.10.78 \(\int \frac {1}{(c x)^{9/2} (a-b x^2)^{3/4}} \, dx\) [978]

Optimal. Leaf size=130 \[ -\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac {4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {8 b^{5/2} \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{7 a^{5/2} c^6 \left (a-b x^2\right )^{3/4}} \]

[Out]

-2/7*(-b*x^2+a)^(1/4)/a/c/(c*x)^(7/2)-4/7*b*(-b*x^2+a)^(1/4)/a^2/c^3/(c*x)^(3/2)-8/7*b^(5/2)*(1-a/b/x^2)^(3/4)
*(c*x)^(3/2)*(cos(1/2*arccsc(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccsc(x*b^(1/2)/a^(1/2)))*EllipticF(sin(1/2
*arccsc(x*b^(1/2)/a^(1/2))),2^(1/2))/a^(5/2)/c^6/(-b*x^2+a)^(3/4)

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Rubi [A]
time = 0.07, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {331, 335, 243, 342, 281, 238} \begin {gather*} -\frac {8 b^{5/2} (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{7 a^{5/2} c^6 \left (a-b x^2\right )^{3/4}}-\frac {4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(9/2)*(a - b*x^2)^(3/4)),x]

[Out]

(-2*(a - b*x^2)^(1/4))/(7*a*c*(c*x)^(7/2)) - (4*b*(a - b*x^2)^(1/4))/(7*a^2*c^3*(c*x)^(3/2)) - (8*b^(5/2)*(1 -
a/(b*x^2))^(3/4)*(c*x)^(3/2)*EllipticF[ArcCsc[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(7*a^(5/2)*c^6*(a - b*x^2)^(3/4))

Rule 238

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[-b/a, 2]))*EllipticF[(1/2)*ArcSin[Rt[-b/a,
2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 243

Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[x^3*((1 + a/(b*x^4))^(3/4)/(a + b*x^4)^(3/4)), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 342

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {1}{(c x)^{9/2} \left (a-b x^2\right )^{3/4}} \, dx &=-\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}+\frac {(6 b) \int \frac {1}{(c x)^{5/2} \left (a-b x^2\right )^{3/4}} \, dx}{7 a c^2}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac {4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac {\left (4 b^2\right ) \int \frac {1}{\sqrt {c x} \left (a-b x^2\right )^{3/4}} \, dx}{7 a^2 c^4}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac {4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt {c x}\right )}{7 a^2 c^5}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac {4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}+\frac {\left (8 b^2 \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {a c^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {c x}\right )}{7 a^2 c^5 \left (a-b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac {4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {\left (8 b^2 \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1-\frac {a c^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {c x}}\right )}{7 a^2 c^5 \left (a-b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac {4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {\left (4 b^2 \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {a c^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{c x}\right )}{7 a^2 c^5 \left (a-b x^2\right )^{3/4}}\\ &=-\frac {2 \sqrt [4]{a-b x^2}}{7 a c (c x)^{7/2}}-\frac {4 b \sqrt [4]{a-b x^2}}{7 a^2 c^3 (c x)^{3/2}}-\frac {8 b^{5/2} \left (1-\frac {a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{7 a^{5/2} c^6 \left (a-b x^2\right )^{3/4}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 57, normalized size = 0.44 \begin {gather*} -\frac {2 x \left (1-\frac {b x^2}{a}\right )^{3/4} \, _2F_1\left (-\frac {7}{4},\frac {3}{4};-\frac {3}{4};\frac {b x^2}{a}\right )}{7 (c x)^{9/2} \left (a-b x^2\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(9/2)*(a - b*x^2)^(3/4)),x]

[Out]

(-2*x*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[-7/4, 3/4, -3/4, (b*x^2)/a])/(7*(c*x)^(9/2)*(a - b*x^2)^(3/4))

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c x \right )^{\frac {9}{2}} \left (-b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(9/2)/(-b*x^2+a)^(3/4),x)

[Out]

int(1/(c*x)^(9/2)/(-b*x^2+a)^(3/4),x)

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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)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(9/2)), x)

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Fricas [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)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 52.06, size = 36, normalized size = 0.28 \begin {gather*} \frac {i e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a}{b x^{2}}} \right )}}{5 b^{\frac {3}{4}} c^{\frac {9}{2}} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(9/2)/(-b*x**2+a)**(3/4),x)

[Out]

I*exp(-I*pi/4)*hyper((3/4, 5/2), (7/2,), a/(b*x**2))/(5*b**(3/4)*c**(9/2)*x**5)

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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)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((-b*x^2 + a)^(3/4)*(c*x)^(9/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{9/2}\,{\left (a-b\,x^2\right )}^{3/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(9/2)*(a - b*x^2)^(3/4)),x)

[Out]

int(1/((c*x)^(9/2)*(a - b*x^2)^(3/4)), x)

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