3.11.83 \(\int \frac {x^7}{\sqrt [4]{a+b x^4}} \, dx\) [1083]

Optimal. Leaf size=38 \[ -\frac {a \left (a+b x^4\right )^{3/4}}{3 b^2}+\frac {\left (a+b x^4\right )^{7/4}}{7 b^2} \]

[Out]

-1/3*a*(b*x^4+a)^(3/4)/b^2+1/7*(b*x^4+a)^(7/4)/b^2

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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \begin {gather*} \frac {\left (a+b x^4\right )^{7/4}}{7 b^2}-\frac {a \left (a+b x^4\right )^{3/4}}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a + b*x^4)^(1/4),x]

[Out]

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

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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.74 \begin {gather*} \frac {\left (a+b x^4\right )^{3/4} \left (-4 a+3 b x^4\right )}{21 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(3/4)*(-4*a + 3*b*x^4))/(21*b^2)

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Maple [A]
time = 0.15, size = 25, normalized size = 0.66

method	result	size
gosper	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-3 b \,x^{4}+4 a \right )}{21 b^{2}}\)	\(25\)
trager	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-3 b \,x^{4}+4 a \right )}{21 b^{2}}\)	\(25\)
risch	\(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-3 b \,x^{4}+4 a \right )}{21 b^{2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(b*x^4+a)^(3/4)*(-3*b*x^4+4*a)/b^2

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Maxima [A]
time = 0.29, size = 30, normalized size = 0.79 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}}}{7 \, b^{2}} - \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} a}{3 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*(b*x^4 + a)^(7/4)/b^2 - 1/3*(b*x^4 + a)^(3/4)*a/b^2

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Fricas [A]
time = 0.37, size = 24, normalized size = 0.63 \begin {gather*} \frac {{\left (3 \, b x^{4} - 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{21 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(3*b*x^4 - 4*a)*(b*x^4 + a)^(3/4)/b^2

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Sympy [A]
time = 0.31, size = 44, normalized size = 1.16 \begin {gather*} \begin {cases} - \frac {4 a \left (a + b x^{4}\right )^{\frac {3}{4}}}{21 b^{2}} + \frac {x^{4} \left (a + b x^{4}\right )^{\frac {3}{4}}}{7 b} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 \sqrt [4]{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)**(1/4),x)

[Out]

Piecewise((-4*a*(a + b*x**4)**(3/4)/(21*b**2) + x**4*(a + b*x**4)**(3/4)/(7*b), Ne(b, 0)), (x**8/(8*a**(1/4)),
True))

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Giac [A]
time = 1.05, size = 29, normalized size = 0.76 \begin {gather*} \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}} - 7 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} a}{21 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(3*(b*x^4 + a)^(7/4) - 7*(b*x^4 + a)^(3/4)*a)/b^2

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Mupad [B]
time = 1.11, size = 26, normalized size = 0.68 \begin {gather*} -{\left (b\,x^4+a\right )}^{3/4}\,\left (\frac {4\,a}{21\,b^2}-\frac {x^4}{7\,b}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x^4)^(1/4),x)

[Out]

-(a + b*x^4)^(3/4)*((4*a)/(21*b^2) - x^4/(7*b))

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