3.13.37 \(\int \frac {x^7}{(a-b x^4)^{3/4}} \, dx\) [1237]

Optimal. Leaf size=38 \[ -\frac {a \sqrt [4]{a-b x^4}}{b^2}+\frac {\left (a-b x^4\right )^{5/4}}{5 b^2} \]

[Out]

-a*(-b*x^4+a)^(1/4)/b^2+1/5*(-b*x^4+a)^(5/4)/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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {272, 45} \begin {gather*} \frac {\left (a-b x^4\right )^{5/4}}{5 b^2}-\frac {a \sqrt [4]{a-b x^4}}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/(a - b*x^4)^(3/4),x]

[Out]

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

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Mathematica [A]
time = 0.03, size = 29, normalized size = 0.76 \begin {gather*} \frac {\left (-4 a-b x^4\right ) \sqrt [4]{a-b x^4}}{5 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a - b*x^4)^(3/4),x]

[Out]

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

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Maple [A]
time = 0.17, size = 25, normalized size = 0.66

method	result	size
gosper	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (b \,x^{4}+4 a \right )}{5 b^{2}}\)	\(25\)
trager	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (b \,x^{4}+4 a \right )}{5 b^{2}}\)	\(25\)
risch	\(-\frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}} \left (\left (-b \,x^{4}+a \right )^{3}\right )^{\frac {1}{4}} \left (b \,x^{4}+4 a \right )}{5 b^{2} \left (-\left (b \,x^{4}-a \right )^{3}\right )^{\frac {1}{4}}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-b*x^4+a)^(3/4),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 32, normalized size = 0.84 \begin {gather*} \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, b^{2}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 24, normalized size = 0.63 \begin {gather*} -\frac {{\left (b x^{4} + 4 \, a\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{5 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.32, size = 46, normalized size = 1.21 \begin {gather*} \begin {cases} - \frac {4 a \sqrt [4]{a - b x^{4}}}{5 b^{2}} - \frac {x^{4} \sqrt [4]{a - b x^{4}}}{5 b} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {3}{4}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(-b*x**4+a)**(3/4),x)

[Out]

Piecewise((-4*a*(a - b*x**4)**(1/4)/(5*b**2) - x**4*(a - b*x**4)**(1/4)/(5*b), Ne(b, 0)), (x**8/(8*a**(3/4)),
True))

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Giac [A]
time = 1.28, size = 32, normalized size = 0.84 \begin {gather*} \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, b^{2}} - \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-b*x^4+a)^(3/4),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.15, size = 24, normalized size = 0.63 \begin {gather*} -\frac {{\left (a-b\,x^4\right )}^{1/4}\,\left (b\,x^4+4\,a\right )}{5\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a - b*x^4)^(3/4),x)

[Out]

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

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