3.1050 \(\int x^7 (a+b x^4)^{5/4} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{13/4}}{13 b^2}-\frac{a \left (a+b x^4\right )^{9/4}}{9 b^2} \]

[Out]

-(a*(a + b*x^4)^(9/4))/(9*b^2) + (a + b*x^4)^(13/4)/(13*b^2)

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Rubi [A]  time = 0.021901, antiderivative size = 38, 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} \[ \frac{\left (a+b x^4\right )^{13/4}}{13 b^2}-\frac{a \left (a+b x^4\right )^{9/4}}{9 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*x^4)^(5/4),x]

[Out]

-(a*(a + b*x^4)^(9/4))/(9*b^2) + (a + b*x^4)^(13/4)/(13*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 x^7 \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x (a+b x)^{5/4} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{5/4}}{b}+\frac{(a+b x)^{9/4}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \left (a+b x^4\right )^{9/4}}{9 b^2}+\frac{\left (a+b x^4\right )^{13/4}}{13 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0151007, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^4\right )^{9/4} \left (9 b x^4-4 a\right )}{117 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*x^4)^(5/4),x]

[Out]

((a + b*x^4)^(9/4)*(-4*a + 9*b*x^4))/(117*b^2)

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Maple [A]  time = 0.005, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-9\,b{x}^{4}+4\,a}{117\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/117*(b*x^4+a)^(9/4)*(-9*b*x^4+4*a)/b^2

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Maxima [A]  time = 0.950788, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{13}{4}}}{13 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}} a}{9 \, 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]

1/13*(b*x^4 + a)^(13/4)/b^2 - 1/9*(b*x^4 + a)^(9/4)*a/b^2

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Fricas [A]  time = 1.65533, size = 104, normalized size = 2.74 \begin{align*} \frac{{\left (9 \, b^{3} x^{12} + 14 \, a b^{2} x^{8} + a^{2} b x^{4} - 4 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{117 \, 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="fricas")

[Out]

1/117*(9*b^3*x^12 + 14*a*b^2*x^8 + a^2*b*x^4 - 4*a^3)*(b*x^4 + a)^(1/4)/b^2

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Sympy [A]  time = 11.9775, size = 85, normalized size = 2.24 \begin{align*} \begin{cases} - \frac{4 a^{3} \sqrt [4]{a + b x^{4}}}{117 b^{2}} + \frac{a^{2} x^{4} \sqrt [4]{a + b x^{4}}}{117 b} + \frac{14 a x^{8} \sqrt [4]{a + b x^{4}}}{117} + \frac{b x^{12} \sqrt [4]{a + b x^{4}}}{13} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{4}} x^{8}}{8} & \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*(a + b*x**4)**(1/4)/(117*b**2) + a**2*x**4*(a + b*x**4)**(1/4)/(117*b) + 14*a*x**8*(a + b*x
**4)**(1/4)/117 + b*x**12*(a + b*x**4)**(1/4)/13, Ne(b, 0)), (a**(5/4)*x**8/8, True))

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Giac [B]  time = 1.10738, size = 105, normalized size = 2.76 \begin{align*} \frac{\frac{13 \,{\left (5 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} - 9 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a\right )} a}{b} + \frac{45 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} - 130 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a + 117 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{2}}{b}}{585 \, b} \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]

1/585*(13*(5*(b*x^4 + a)^(9/4) - 9*(b*x^4 + a)^(5/4)*a)*a/b + (45*(b*x^4 + a)^(13/4) - 130*(b*x^4 + a)^(9/4)*a
+ 117*(b*x^4 + a)^(5/4)*a^2)/b)/b