3.1048 \(\int x^{15} (a+b x^4)^{5/4} \, dx\)

Optimal. Leaf size=80 \[ \frac{3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}-\frac{a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^4}-\frac{3 a \left (a+b x^4\right )^{17/4}}{17 b^4} \]

[Out]

-(a^3*(a + b*x^4)^(9/4))/(9*b^4) + (3*a^2*(a + b*x^4)^(13/4))/(13*b^4) - (3*a*(a + b*x^4)^(17/4))/(17*b^4) + (
a + b*x^4)^(21/4)/(21*b^4)

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Rubi [A]  time = 0.0435236, antiderivative size = 80, 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} \[ \frac{3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}-\frac{a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^4}-\frac{3 a \left (a+b x^4\right )^{17/4}}{17 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^15*(a + b*x^4)^(5/4),x]

[Out]

-(a^3*(a + b*x^4)^(9/4))/(9*b^4) + (3*a^2*(a + b*x^4)^(13/4))/(13*b^4) - (3*a*(a + b*x^4)^(17/4))/(17*b^4) + (
a + b*x^4)^(21/4)/(21*b^4)

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^{15} \left (a+b x^4\right )^{5/4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^3 (a+b x)^{5/4} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{5/4}}{b^3}+\frac{3 a^2 (a+b x)^{9/4}}{b^3}-\frac{3 a (a+b x)^{13/4}}{b^3}+\frac{(a+b x)^{17/4}}{b^3}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac{3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}-\frac{3 a \left (a+b x^4\right )^{17/4}}{17 b^4}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0266784, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^4\right )^{9/4} \left (288 a^2 b x^4-128 a^3-468 a b^2 x^8+663 b^3 x^{12}\right )}{13923 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15*(a + b*x^4)^(5/4),x]

[Out]

((a + b*x^4)^(9/4)*(-128*a^3 + 288*a^2*b*x^4 - 468*a*b^2*x^8 + 663*b^3*x^12))/(13923*b^4)

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Maple [A]  time = 0.006, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-663\,{b}^{3}{x}^{12}+468\,a{b}^{2}{x}^{8}-288\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{13923\,{b}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15*(b*x^4+a)^(5/4),x)

[Out]

-1/13923*(b*x^4+a)^(9/4)*(-663*b^3*x^12+468*a*b^2*x^8-288*a^2*b*x^4+128*a^3)/b^4

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Maxima [A]  time = 0.962003, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{21}{4}}}{21 \, b^{4}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a}{17 \, b^{4}} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2}}{13 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3}}{9 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(5/4),x, algorithm="maxima")

[Out]

1/21*(b*x^4 + a)^(21/4)/b^4 - 3/17*(b*x^4 + a)^(17/4)*a/b^4 + 3/13*(b*x^4 + a)^(13/4)*a^2/b^4 - 1/9*(b*x^4 + a
)^(9/4)*a^3/b^4

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Fricas [A]  time = 1.71103, size = 166, normalized size = 2.08 \begin{align*} \frac{{\left (663 \, b^{5} x^{20} + 858 \, a b^{4} x^{16} + 15 \, a^{2} b^{3} x^{12} - 20 \, a^{3} b^{2} x^{8} + 32 \, a^{4} b x^{4} - 128 \, a^{5}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{13923 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(5/4),x, algorithm="fricas")

[Out]

1/13923*(663*b^5*x^20 + 858*a*b^4*x^16 + 15*a^2*b^3*x^12 - 20*a^3*b^2*x^8 + 32*a^4*b*x^4 - 128*a^5)*(b*x^4 + a
)^(1/4)/b^4

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Sympy [A]  time = 45.6387, size = 134, normalized size = 1.68 \begin{align*} \begin{cases} - \frac{128 a^{5} \sqrt [4]{a + b x^{4}}}{13923 b^{4}} + \frac{32 a^{4} x^{4} \sqrt [4]{a + b x^{4}}}{13923 b^{3}} - \frac{20 a^{3} x^{8} \sqrt [4]{a + b x^{4}}}{13923 b^{2}} + \frac{5 a^{2} x^{12} \sqrt [4]{a + b x^{4}}}{4641 b} + \frac{22 a x^{16} \sqrt [4]{a + b x^{4}}}{357} + \frac{b x^{20} \sqrt [4]{a + b x^{4}}}{21} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{4}} x^{16}}{16} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15*(b*x**4+a)**(5/4),x)

[Out]

Piecewise((-128*a**5*(a + b*x**4)**(1/4)/(13923*b**4) + 32*a**4*x**4*(a + b*x**4)**(1/4)/(13923*b**3) - 20*a**
3*x**8*(a + b*x**4)**(1/4)/(13923*b**2) + 5*a**2*x**12*(a + b*x**4)**(1/4)/(4641*b) + 22*a*x**16*(a + b*x**4)*
*(1/4)/357 + b*x**20*(a + b*x**4)**(1/4)/21, Ne(b, 0)), (a**(5/4)*x**16/16, True))

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Giac [B]  time = 1.10225, size = 181, normalized size = 2.26 \begin{align*} \frac{\frac{21 \,{\left (195 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} - 765 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a + 1105 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2} - 663 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3}\right )} a}{b^{3}} + \frac{3315 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} - 16380 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a + 32130 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2} - 30940 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3} + 13923 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{4}}{b^{3}}}{69615 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15*(b*x^4+a)^(5/4),x, algorithm="giac")

[Out]

1/69615*(21*(195*(b*x^4 + a)^(17/4) - 765*(b*x^4 + a)^(13/4)*a + 1105*(b*x^4 + a)^(9/4)*a^2 - 663*(b*x^4 + a)^
(5/4)*a^3)*a/b^3 + (3315*(b*x^4 + a)^(21/4) - 16380*(b*x^4 + a)^(17/4)*a + 32130*(b*x^4 + a)^(13/4)*a^2 - 3094
0*(b*x^4 + a)^(9/4)*a^3 + 13923*(b*x^4 + a)^(5/4)*a^4)/b^3)/b