\(\int \frac {x^7}{\sqrt {a+b x^4}} \, dx\) [812]

Optimal result

Integrand size = 15, antiderivative size = 38 \[ \int \frac {x^7}{\sqrt {a+b x^4}} \, dx=-\frac {a \sqrt {a+b x^4}}{2 b^2}+\frac {\left (a+b x^4\right )^{3/2}}{6 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^7}{\sqrt {a+b x^4}} \, dx=\frac {\left (a+b x^4\right )^{3/2}}{6 b^2}-\frac {a \sqrt {a+b x^4}}{2 b^2} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {x^7}{\sqrt {a+b x^4}} \, dx=\frac {\left (-2 a+b x^4\right ) \sqrt {a+b x^4}}{6 b^2} \]

[In]

Integrate[x^7/Sqrt[a + b*x^4],x]

[Out]

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

Maple [A] (verified)

Time = 4.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63

method	result	size
pseudoelliptic	\(\frac {\left (b \,x^{4}-2 a \right ) \sqrt {b \,x^{4}+a}}{6 b^{2}}\)	\(24\)
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	\(-\frac {\sqrt {b \,x^{4}+a}\, \left (-b \,x^{4}+2 a \right )}{6 b^{2}}\)	\(25\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int \frac {x^7}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b x^{4} + a} {\left (b x^{4} - 2 \, a\right )}}{6 \, b^{2}} \]

[In]

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

[Out]

1/6*sqrt(b*x^4 + a)*(b*x^4 - 2*a)/b^2

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {x^7}{\sqrt {a+b x^4}} \, dx=\begin {cases} - \frac {a \sqrt {a + b x^{4}}}{3 b^{2}} + \frac {x^{4} \sqrt {a + b x^{4}}}{6 b} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 \sqrt {a}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {x^7}{\sqrt {a+b x^4}} \, dx=\frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{6 \, b^{2}} - \frac {\sqrt {b x^{4} + a} a}{2 \, b^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {x^7}{\sqrt {a+b x^4}} \, dx=\frac {{\left (b x^{4} + a\right )}^{\frac {3}{2}}}{6 \, b^{2}} - \frac {\sqrt {b x^{4} + a} a}{2 \, b^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 5.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int \frac {x^7}{\sqrt {a+b x^4}} \, dx=-\frac {\sqrt {b\,x^4+a}\,\left (2\,a-b\,x^4\right )}{6\,b^2} \]

[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)