3.1046 \(\int x^{15} \left (a+b x^4\right )^{5/4} \, dx\)

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

[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 = 0.107598, 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 \[ -\frac{a^3 \left (a+b x^4\right )^{9/4}}{9 b^4}+\frac{3 a^2 \left (a+b x^4\right )^{13/4}}{13 b^4}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^4}-\frac{3 a \left (a+b x^4\right )^{17/4}}{17 b^4} \]

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)

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Rubi in Sympy [A]  time = 14.2803, size = 71, normalized size = 0.89 \[ - \frac{a^{3} \left (a + b x^{4}\right )^{\frac{9}{4}}}{9 b^{4}} + \frac{3 a^{2} \left (a + b x^{4}\right )^{\frac{13}{4}}}{13 b^{4}} - \frac{3 a \left (a + b x^{4}\right )^{\frac{17}{4}}}{17 b^{4}} + \frac{\left (a + b x^{4}\right )^{\frac{21}{4}}}{21 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**15*(b*x**4+a)**(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)

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Mathematica [A]  time = 0.0498684, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^4\right )^{9/4} \left (-128 a^3+288 a^2 b x^4-468 a b^2 x^8+663 b^3 x^{12}\right )}{13923 b^4} \]

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))/(1
3923*b^4)

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Maple [A]  time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-663\,{b}^{3}{x}^{12}+468\,a{b}^{2}{x}^{8}-288\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{13923\,{b}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \]

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 = 1.44508, size = 86, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{21}{4}}}{21 \, b^{4}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a}{17 \, b^{4}} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2}}{13 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3}}{9 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^15,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 = 0.288325, size = 92, normalized size = 1.15 \[ \frac{{\left (663 \, b^{5} x^{20} + 858 \, a b^{4} x^{16} + 15 \, a^{2} b^{3} x^{12} - 20 \, a^{3} b^{2} x^{8} + 32 \, a^{4} b x^{4} - 128 \, a^{5}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{13923 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^15,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 = 121.707, size = 134, normalized size = 1.68 \[ \begin{cases} - \frac{128 a^{5} \sqrt [4]{a + b x^{4}}}{13923 b^{4}} + \frac{32 a^{4} x^{4} \sqrt [4]{a + b x^{4}}}{13923 b^{3}} - \frac{20 a^{3} x^{8} \sqrt [4]{a + b x^{4}}}{13923 b^{2}} + \frac{5 a^{2} x^{12} \sqrt [4]{a + b x^{4}}}{4641 b} + \frac{22 a x^{16} \sqrt [4]{a + b x^{4}}}{357} + \frac{b x^{20} \sqrt [4]{a + b x^{4}}}{21} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{4}} x^{16}}{16} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.215834, size = 181, normalized size = 2.26 \[ \frac{\frac{21 \,{\left (195 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} - 765 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a + 1105 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2} - 663 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3}\right )} a}{b^{3}} + \frac{3315 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} - 16380 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a + 32130 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2} - 30940 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3} + 13923 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{4}}{b^{3}}}{69615 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^15,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) - 1638
0*(b*x^4 + a)^(17/4)*a + 32130*(b*x^4 + a)^(13/4)*a^2 - 30940*(b*x^4 + a)^(9/4)*
a^3 + 13923*(b*x^4 + a)^(5/4)*a^4)/b^3)/b