3.857 \(\int \frac{1}{(a+b x^2)^{7/4}} \, dx\)

Optimal. Leaf size=78 \[ \frac{2 \left (\frac{b x^2}{a}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{3 \sqrt{a} \sqrt{b} \left (a+b x^2\right )^{3/4}}+\frac{2 x}{3 a \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0166227, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {199, 233, 231} \[ \frac{2 x}{3 a \left (a+b x^2\right )^{3/4}}+\frac{2 \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \sqrt{b} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(-7/4),x]

[Out]

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

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 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]

Rubi steps

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

Mathematica [C]  time = 0.0216339, size = 55, normalized size = 0.71 \[ \frac{x \left (\left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )+2\right )}{3 a \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(-7/4),x]

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2 + a)^(-7/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.1028, size = 24, normalized size = 0.31 \begin{align*} \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{7}{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*hyper((1/2, 7/4), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(7/4)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2 + a)^(-7/4), x)