3.1142 \(\int \frac{x^7}{(a+b x^4)^{5/4}} \, dx\)

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

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Rubi [A]  time = 0.0213331, antiderivative size = 35, normalized size of antiderivative = 1., 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 = {266, 43} \[ \frac{a}{b^2 \sqrt [4]{a+b x^4}}+\frac{\left (a+b x^4\right )^{3/4}}{3 b^2} \]

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 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{x^7}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{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*}

Mathematica [A]  time = 0.010949, size = 27, normalized size = 0.77 \[ \frac{4 a+b x^4}{3 b^2 \sqrt [4]{a+b x^4}} \]

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 = 0.003, size = 24, normalized size = 0.7 \begin{align*}{\frac{b{x}^{4}+4\,a}{3\,{b}^{2}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.979252, size = 39, normalized size = 1.11 \begin{align*} \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{align*}

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 = 1.47996, size = 74, normalized size = 2.11 \begin{align*} \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{align*}

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 = 1.13166, size = 44, normalized size = 1.26 \begin{align*} \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{align*}

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 = 1.07925, size = 36, normalized size = 1.03 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} + \frac{3 \, a}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}}{3 \, b^{2}} \end{align*}

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) + 3*a/(b*x^4 + a)^(1/4))/b^2