3.9.32 \(\int \frac {x^7}{\sqrt {a-b x^4}} \, dx\) [832]

Optimal. Leaf size=40 \[ -\frac {a \sqrt {a-b x^4}}{2 b^2}+\frac {\left (a-b x^4\right )^{3/2}}{6 b^2} \]

[Out]

1/6*(-b*x^4+a)^(3/2)/b^2-1/2*a*(-b*x^4+a)^(1/2)/b^2

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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {272, 45} \begin {gather*} \frac {\left (a-b x^4\right )^{3/2}}{6 b^2}-\frac {a \sqrt {a-b x^4}}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/Sqrt[a - b*x^4],x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rubi steps

\begin {align*} \int \frac {x^7}{\sqrt {a-b x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{\sqrt {a-b x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a}{b \sqrt {a-b x}}-\frac {\sqrt {a-b x}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac {a \sqrt {a-b x^4}}{2 b^2}+\frac {\left (a-b x^4\right )^{3/2}}{6 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 29, normalized size = 0.72 \begin {gather*} \frac {\left (-2 a-b x^4\right ) \sqrt {a-b x^4}}{6 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/Sqrt[a - b*x^4],x]

[Out]

((-2*a - b*x^4)*Sqrt[a - b*x^4])/(6*b^2)

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Maple [A]
time = 0.14, size = 25, normalized size = 0.62

method	result	size
gosper	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (b \,x^{4}+2 a \right )}{6 b^{2}}\)	\(25\)
default	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (b \,x^{4}+2 a \right )}{6 b^{2}}\)	\(25\)
trager	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (b \,x^{4}+2 a \right )}{6 b^{2}}\)	\(25\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (b \,x^{4}+2 a \right )}{6 b^{2}}\)	\(25\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}\, \left (b \,x^{4}+2 a \right )}{6 b^{2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-b*x^4+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 32, normalized size = 0.80 \begin {gather*} \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{6 \, b^{2}} - \frac {\sqrt {-b x^{4} + a} a}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-b*x^4 + a)^(3/2)/b^2 - 1/2*sqrt(-b*x^4 + a)*a/b^2

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Fricas [A]
time = 0.36, size = 24, normalized size = 0.60 \begin {gather*} -\frac {{\left (b x^{4} + 2 \, a\right )} \sqrt {-b x^{4} + a}}{6 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(b*x^4 + 2*a)*sqrt(-b*x^4 + a)/b^2

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Sympy [A]
time = 0.30, size = 44, normalized size = 1.10 \begin {gather*} \begin {cases} - \frac {a \sqrt {a - b x^{4}}}{3 b^{2}} - \frac {x^{4} \sqrt {a - b x^{4}}}{6 b} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 \sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*sqrt(a - b*x**4)/(3*b**2) - x**4*sqrt(a - b*x**4)/(6*b), Ne(b, 0)), (x**8/(8*sqrt(a)), True))

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Giac [A]
time = 1.80, size = 32, normalized size = 0.80 \begin {gather*} \frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{6 \, b^{2}} - \frac {\sqrt {-b x^{4} + a} a}{2 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(-b*x^4 + a)^(3/2)/b^2 - 1/2*sqrt(-b*x^4 + a)*a/b^2

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Mupad [B]
time = 1.17, size = 24, normalized size = 0.60 \begin {gather*} -\frac {\sqrt {a-b\,x^4}\,\left (b\,x^4+2\,a\right )}{6\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a - b*x^4)^(1/2),x)

[Out]

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

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