3.12.42 \(\int \frac {x^7}{(a+b x^4)^{5/4}} \, dx\) [1142]

Optimal. Leaf size=35 \[ \frac {a}{b^2 \sqrt [4]{a+b x^4}}+\frac {\left (a+b x^4\right )^{3/4}}{3 b^2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 35, 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 {a}{b^2 \sqrt [4]{a+b x^4}}+\frac {\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)^(5/4),x]

[Out]

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

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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.77 \begin {gather*} \frac {4 a+b x^4}{3 b^2 \sqrt [4]{a+b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(a + b*x^4)^(5/4),x]

[Out]

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

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Maple [A]
time = 0.17, size = 24, normalized size = 0.69

method	result	size
gosper	\(\frac {b \,x^{4}+4 a}{3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2}}\)	\(24\)
trager	\(\frac {b \,x^{4}+4 a}{3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2}}\)	\(24\)
risch	\(\frac {a}{b^{2} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{3 b^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(b*x^4+a)^(5/4),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 29, normalized size = 0.83 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{3 \, b^{2}} + \frac {a}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 35, normalized size = 1.00 \begin {gather*} \frac {{\left (b x^{4} + 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{3 \, {\left (b^{3} x^{4} + a b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.35, size = 44, normalized size = 1.26 \begin {gather*} \begin {cases} \frac {4 a}{3 b^{2} \sqrt [4]{a + b x^{4}}} + \frac {x^{4}}{3 b \sqrt [4]{a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {5}{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)**(5/4),x)

[Out]

Piecewise((4*a/(3*b**2*(a + b*x**4)**(1/4)) + x**4/(3*b*(a + b*x**4)**(1/4)), Ne(b, 0)), (x**8/(8*a**(5/4)), T
rue))

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Giac [A]
time = 1.04, size = 34, normalized size = 0.97 \begin {gather*} \frac {\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b} + \frac {3 \, a}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b}}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.15, size = 23, normalized size = 0.66 \begin {gather*} \frac {b\,x^4+4\,a}{3\,b^2\,{\left (b\,x^4+a\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(a + b*x^4)^(5/4),x)

[Out]

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

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