3.834 \(\int \frac {1}{x^4 (a+b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=102 \[ \frac {5 b^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {5 b \sqrt [4]{a+b x^2}}{6 a^2 x}-\frac {\sqrt [4]{a+b x^2}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {325, 233, 231} \[ \frac {5 b^{3/2} \left (\frac {b x^2}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}+\frac {5 b \sqrt [4]{a+b x^2}}{6 a^2 x}-\frac {\sqrt [4]{a+b x^2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(a + b*x^2)^(3/4)),x]

[Out]

-(a + b*x^2)^(1/4)/(3*a*x^3) + (5*b*(a + b*x^2)^(1/4))/(6*a^2*x) + (5*b^(3/2)*(1 + (b*x^2)/a)^(3/4)*EllipticF[
ArcTan[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(6*a^(3/2)*(a + b*x^2)^(3/4))

Rule 231

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

Rule 233

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(1 + (b*x^2)/a)^(3/4)/(a + b*x^2)^(3/4), Int[1/(1 + (b*x^2
)/a)^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 325

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]

Rubi steps

\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a+b x^2}}{3 a x^3}-\frac {(5 b) \int \frac {1}{x^2 \left (a+b x^2\right )^{3/4}} \, dx}{6 a}\\ &=-\frac {\sqrt [4]{a+b x^2}}{3 a x^3}+\frac {5 b \sqrt [4]{a+b x^2}}{6 a^2 x}+\frac {\left (5 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{12 a^2}\\ &=-\frac {\sqrt [4]{a+b x^2}}{3 a x^3}+\frac {5 b \sqrt [4]{a+b x^2}}{6 a^2 x}+\frac {\left (5 b^2 \left (1+\frac {b x^2}{a}\right )^{3/4}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx}{12 a^2 \left (a+b x^2\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^2}}{3 a x^3}+\frac {5 b \sqrt [4]{a+b x^2}}{6 a^2 x}+\frac {5 b^{3/2} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 51, normalized size = 0.50 \[ -\frac {\left (\frac {b x^2}{a}+1\right )^{3/4} \, _2F_1\left (-\frac {3}{2},\frac {3}{4};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b x^{6} + a x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2 + a)^(1/4)/(b*x^6 + a*x^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b*x^2)^(3/4)),x)

[Out]

int(1/(x^4*(a + b*x^2)^(3/4)), x)

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sympy [C]  time = 1.09, size = 32, normalized size = 0.31 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{4}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-hyper((-3/2, 3/4), (-1/2,), b*x**2*exp_polar(I*pi)/a)/(3*a**(3/4)*x**3)

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