3.1060 \(\int x^{10} \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=148 \[ -\frac{35 a^4 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{11/4}}+\frac{35 a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{11/4}}-\frac{35 a^3 x^3 \sqrt [4]{a+b x^4}}{6144 b^2}+\frac{5 a^2 x^7 \sqrt [4]{a+b x^4}}{1536 b}+\frac{1}{16} x^{11} \left (a+b x^4\right )^{5/4}+\frac{5}{192} a x^{11} \sqrt [4]{a+b x^4} \]

[Out]

(-35*a^3*x^3*(a + b*x^4)^(1/4))/(6144*b^2) + (5*a^2*x^7*(a + b*x^4)^(1/4))/(1536
*b) + (5*a*x^11*(a + b*x^4)^(1/4))/192 + (x^11*(a + b*x^4)^(5/4))/16 - (35*a^4*A
rcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(4096*b^(11/4)) + (35*a^4*ArcTanh[(b^(1/4)
*x)/(a + b*x^4)^(1/4)])/(4096*b^(11/4))

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Rubi [A]  time = 0.179925, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{35 a^4 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{11/4}}+\frac{35 a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4096 b^{11/4}}-\frac{35 a^3 x^3 \sqrt [4]{a+b x^4}}{6144 b^2}+\frac{5 a^2 x^7 \sqrt [4]{a+b x^4}}{1536 b}+\frac{1}{16} x^{11} \left (a+b x^4\right )^{5/4}+\frac{5}{192} a x^{11} \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^10*(a + b*x^4)^(5/4),x]

[Out]

(-35*a^3*x^3*(a + b*x^4)^(1/4))/(6144*b^2) + (5*a^2*x^7*(a + b*x^4)^(1/4))/(1536
*b) + (5*a*x^11*(a + b*x^4)^(1/4))/192 + (x^11*(a + b*x^4)^(5/4))/16 - (35*a^4*A
rcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(4096*b^(11/4)) + (35*a^4*ArcTanh[(b^(1/4)
*x)/(a + b*x^4)^(1/4)])/(4096*b^(11/4))

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Rubi in Sympy [A]  time = 22.2403, size = 139, normalized size = 0.94 \[ - \frac{35 a^{4} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{4096 b^{\frac{11}{4}}} + \frac{35 a^{4} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{4096 b^{\frac{11}{4}}} - \frac{35 a^{3} x^{3} \sqrt [4]{a + b x^{4}}}{6144 b^{2}} + \frac{5 a^{2} x^{7} \sqrt [4]{a + b x^{4}}}{1536 b} + \frac{5 a x^{11} \sqrt [4]{a + b x^{4}}}{192} + \frac{x^{11} \left (a + b x^{4}\right )^{\frac{5}{4}}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(b*x**4+a)**(5/4),x)

[Out]

-35*a**4*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(4096*b**(11/4)) + 35*a**4*atanh(b
**(1/4)*x/(a + b*x**4)**(1/4))/(4096*b**(11/4)) - 35*a**3*x**3*(a + b*x**4)**(1/
4)/(6144*b**2) + 5*a**2*x**7*(a + b*x**4)**(1/4)/(1536*b) + 5*a*x**11*(a + b*x**
4)**(1/4)/192 + x**11*(a + b*x**4)**(5/4)/16

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Mathematica [C]  time = 0.0714935, size = 102, normalized size = 0.69 \[ \frac{x^3 \left (35 a^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-35 a^4-15 a^3 b x^4+564 a^2 b^2 x^8+928 a b^3 x^{12}+384 b^4 x^{16}\right )}{6144 b^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10*(a + b*x^4)^(5/4),x]

[Out]

(x^3*(-35*a^4 - 15*a^3*b*x^4 + 564*a^2*b^2*x^8 + 928*a*b^3*x^12 + 384*b^4*x^16 +
35*a^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -((b*x^4)/a)]))/(
6144*b^2*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{x}^{10} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(b*x^4+a)^(5/4),x)

[Out]

int(x^10*(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277714, size = 313, normalized size = 2.11 \[ -\frac{420 \, \left (\frac{a^{16}}{b^{11}}\right )^{\frac{1}{4}} b^{2} \arctan \left (\frac{\left (\frac{a^{16}}{b^{11}}\right )^{\frac{1}{4}} b^{3} x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4} + x \sqrt{\frac{\sqrt{b x^{4} + a} a^{8} + \sqrt{\frac{a^{16}}{b^{11}}} b^{6} x^{2}}{x^{2}}}}\right ) - 105 \, \left (\frac{a^{16}}{b^{11}}\right )^{\frac{1}{4}} b^{2} \log \left (\frac{35 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4} + \left (\frac{a^{16}}{b^{11}}\right )^{\frac{1}{4}} b^{3} x\right )}}{x}\right ) + 105 \, \left (\frac{a^{16}}{b^{11}}\right )^{\frac{1}{4}} b^{2} \log \left (\frac{35 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4} - \left (\frac{a^{16}}{b^{11}}\right )^{\frac{1}{4}} b^{3} x\right )}}{x}\right ) - 4 \,{\left (384 \, b^{3} x^{15} + 544 \, a b^{2} x^{11} + 20 \, a^{2} b x^{7} - 35 \, a^{3} x^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{24576 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^10,x, algorithm="fricas")

[Out]

-1/24576*(420*(a^16/b^11)^(1/4)*b^2*arctan((a^16/b^11)^(1/4)*b^3*x/((b*x^4 + a)^
(1/4)*a^4 + x*sqrt((sqrt(b*x^4 + a)*a^8 + sqrt(a^16/b^11)*b^6*x^2)/x^2))) - 105*
(a^16/b^11)^(1/4)*b^2*log(35*((b*x^4 + a)^(1/4)*a^4 + (a^16/b^11)^(1/4)*b^3*x)/x
) + 105*(a^16/b^11)^(1/4)*b^2*log(35*((b*x^4 + a)^(1/4)*a^4 - (a^16/b^11)^(1/4)*
b^3*x)/x) - 4*(384*b^3*x^15 + 544*a*b^2*x^11 + 20*a^2*b*x^7 - 35*a^3*x^3)*(b*x^4
+ a)^(1/4))/b^2

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Sympy [A]  time = 28.7002, size = 39, normalized size = 0.26 \[ \frac{a^{\frac{5}{4}} x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(b*x**4+a)**(5/4),x)

[Out]

a**(5/4)*x**11*gamma(11/4)*hyper((-5/4, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a
)/(4*gamma(15/4))

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GIAC/XCAS [A]  time = 0.24454, size = 459, normalized size = 3.1 \[ \frac{1}{49152} \,{\left (\frac{8 \,{\left (\frac{399 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )} b^{2}}{x} - \frac{105 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}}{x} + \frac{125 \,{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x^{9}} - \frac{35 \,{\left (b^{3} x^{12} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{4} + a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x^{13}}\right )} x^{16}}{a^{4} b^{2}} + \frac{210 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b^{3}} + \frac{210 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right )}{b^{3}} + \frac{105 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b^{3}} - \frac{105 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right )}{b^{3}}\right )} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^10,x, algorithm="giac")

[Out]

1/49152*(8*(399*(b*x^4 + a)^(1/4)*(b + a/x^4)*b^2/x - 105*(b*x^4 + a)^(1/4)*b^3/
x + 125*(b^2*x^8 + 2*a*b*x^4 + a^2)*(b*x^4 + a)^(1/4)*b/x^9 - 35*(b^3*x^12 + 3*a
*b^2*x^8 + 3*a^2*b*x^4 + a^3)*(b*x^4 + a)^(1/4)/x^13)*x^16/(a^4*b^2) + 210*sqrt(
2)*(-b)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) + 2*(b*x^4 + a)^(1/4)/x)/(-
b)^(1/4))/b^3 + 210*sqrt(2)*(-b)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) -
2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4))/b^3 + 105*sqrt(2)*(-b)^(1/4)*ln(sqrt(-b) + s
qrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)/x^2)/b^3 - 105*sqrt(2)*(
-b)^(1/4)*ln(sqrt(-b) - sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)
/x^2)/b^3)*a^4