3.9.36 \(\int \frac {x^5}{\sqrt {a-b x^4}} \, dx\) [836]

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

[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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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {281, 327, 223, 209} \begin {gather*} \frac {a \text {ArcTan}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/Sqrt[a - b*x^4],x]

[Out]

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

Rule 209

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

Rule 223

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 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 327

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]

Rubi steps

\begin {align*} \int \frac {x^5}{\sqrt {a-b x^4}} \, dx &=\frac {1}{2} \text {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 \text {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 \text {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*}

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Mathematica [A]
time = 0.15, size = 55, normalized size = 1.00 \begin {gather*} -\frac {x^2 \sqrt {a-b x^4}}{4 b}-\frac {a \tan ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {b} x^2}\right )}{4 b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/Sqrt[a - b*x^4],x]

[Out]

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

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Maple [A]
time = 0.15, size = 44, normalized size = 0.80

method	result	size
default	\(\frac {a \arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {-b \,x^{4}+a}}\right )}{4 b^{\frac {3}{2}}}-\frac {x^{2} \sqrt {-b \,x^{4}+a}}{4 b}\)	\(44\)
risch	\(\frac {a \arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {-b \,x^{4}+a}}\right )}{4 b^{\frac {3}{2}}}-\frac {x^{2} \sqrt {-b \,x^{4}+a}}{4 b}\)	\(44\)
elliptic	\(\frac {a \arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {-b \,x^{4}+a}}\right )}{4 b^{\frac {3}{2}}}-\frac {x^{2} \sqrt {-b \,x^{4}+a}}{4 b}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 [A]
time = 0.51, size = 62, normalized size = 1.13 \begin {gather*} -\frac {a \arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {b} x^{2}}\right )}{4 \, b^{\frac {3}{2}}} - \frac {\sqrt {-b x^{4} + a} a}{4 \, {\left (b^{2} - \frac {{\left (b x^{4} - a\right )} b}{x^{4}}\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.39, size = 116, normalized size = 2.11 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.14, size = 128, normalized size = 2.33 \begin {gather*} \begin {cases} \frac {i \sqrt {a} x^{2}}{4 b \sqrt {-1 + \frac {b x^{4}}{a}}} - \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4 b^{\frac {3}{2}}} - \frac {i x^{6}}{4 \sqrt {a} \sqrt {-1 + \frac {b x^{4}}{a}}} & \text {for}\: \left |{\frac {b x^{4}}{a}}\right | > 1 \\- \frac {\sqrt {a} x^{2} \sqrt {1 - \frac {b x^{4}}{a}}}{4 b} + \frac {a \operatorname {asin}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{4 b^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [A]
time = 1.35, size = 53, normalized size = 0.96 \begin {gather*} -\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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^5}{\sqrt {a-b\,x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a - b*x^4)^(1/2),x)

[Out]

int(x^5/(a - b*x^4)^(1/2), x)

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