3.836 \(\int \frac{x^5}{\sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=55 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{4 b^{3/2}}-\frac{x^2 \sqrt{a-b x^4}}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.029561, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {275, 321, 217, 203} \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{4 b^{3/2}}-\frac{x^2 \sqrt{a-b x^4}}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[a - b*x^4],x]

[Out]

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

Rule 275

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{x^5}{\sqrt{a-b x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a-b x^2}} \, dx,x,x^2\right )\\ &=-\frac{x^2 \sqrt{a-b x^4}}{4 b}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b x^2}} \, dx,x,x^2\right )}{4 b}\\ &=-\frac{x^2 \sqrt{a-b x^4}}{4 b}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x^2}{\sqrt{a-b x^4}}\right )}{4 b}\\ &=-\frac{x^2 \sqrt{a-b x^4}}{4 b}+\frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{4 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0228632, size = 55, normalized size = 1. \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{4 b^{3/2}}-\frac{x^2 \sqrt{a-b x^4}}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[a - b*x^4],x]

[Out]

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

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Maple [A]  time = 0.012, size = 44, normalized size = 0.8 \begin{align*}{\frac{a}{4}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){b}^{-{\frac{3}{2}}}}-{\frac{{x}^{2}}{4\,b}\sqrt{-b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(-b*x^4+a)^(1/2),x)

[Out]

1/4*a*arctan(x^2*b^(1/2)/(-b*x^4+a)^(1/2))/b^(3/2)-1/4*x^2*(-b*x^4+a)^(1/2)/b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55756, size = 271, normalized size = 4.93 \begin{align*} \left [-\frac{2 \, \sqrt{-b x^{4} + a} b x^{2} + a \sqrt{-b} \log \left (2 \, b x^{4} - 2 \, \sqrt{-b x^{4} + a} \sqrt{-b} x^{2} - a\right )}{8 \, b^{2}}, -\frac{\sqrt{-b x^{4} + a} b x^{2} + a \sqrt{b} \arctan \left (\frac{\sqrt{-b x^{4} + a} \sqrt{b} x^{2}}{b x^{4} - a}\right )}{4 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/8*(2*sqrt(-b*x^4 + a)*b*x^2 + a*sqrt(-b)*log(2*b*x^4 - 2*sqrt(-b*x^4 + a)*sqrt(-b)*x^2 - a))/b^2, -1/4*(sq
rt(-b*x^4 + a)*b*x^2 + a*sqrt(b)*arctan(sqrt(-b*x^4 + a)*sqrt(b)*x^2/(b*x^4 - a)))/b^2]

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Sympy [A]  time = 3.03096, size = 129, normalized size = 2.35 \begin{align*} \begin{cases} - \frac{i \sqrt{a} x^{2} \sqrt{-1 + \frac{b x^{4}}{a}}}{4 b} - \frac{i a \operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} & \text{for}\: \frac{\left |{b x^{4}}\right |}{\left |{a}\right |} > 1 \\- \frac{\sqrt{a} x^{2}}{4 b \sqrt{1 - \frac{b x^{4}}{a}}} + \frac{a \operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} + \frac{x^{6}}{4 \sqrt{a} \sqrt{1 - \frac{b x^{4}}{a}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-I*sqrt(a)*x**2*sqrt(-1 + b*x**4/a)/(4*b) - I*a*acosh(sqrt(b)*x**2/sqrt(a))/(4*b**(3/2)), Abs(b*x**
4)/Abs(a) > 1), (-sqrt(a)*x**2/(4*b*sqrt(1 - b*x**4/a)) + a*asin(sqrt(b)*x**2/sqrt(a))/(4*b**(3/2)) + x**6/(4*
sqrt(a)*sqrt(1 - b*x**4/a)), True))

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Giac [A]  time = 1.13279, size = 72, normalized size = 1.31 \begin{align*} -\frac{\sqrt{-b x^{4} + a} x^{2}}{4 \, b} - \frac{a \log \left ({\left | -\sqrt{-b} x^{2} + \sqrt{-b x^{4} + a} \right |}\right )}{4 \, \sqrt{-b} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(-b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

-1/4*sqrt(-b*x^4 + a)*x^2/b - 1/4*a*log(abs(-sqrt(-b)*x^2 + sqrt(-b*x^4 + a)))/(sqrt(-b)*b)