3.1051 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^5} \, dx\)

Optimal. Leaf size=91 \[ -\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}+\frac{5}{4} b \sqrt [4]{a+b x^4}-\frac{5}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{5}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \]

[Out]

(5*b*(a + b*x^4)^(1/4))/4 - (a + b*x^4)^(5/4)/(4*x^4) - (5*a^(1/4)*b*ArcTan[(a +
b*x^4)^(1/4)/a^(1/4)])/8 - (5*a^(1/4)*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/8

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Rubi [A]  time = 0.137889, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{\left (a+b x^4\right )^{5/4}}{4 x^4}+\frac{5}{4} b \sqrt [4]{a+b x^4}-\frac{5}{8} \sqrt [4]{a} b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{5}{8} \sqrt [4]{a} b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^5,x]

[Out]

(5*b*(a + b*x^4)^(1/4))/4 - (a + b*x^4)^(5/4)/(4*x^4) - (5*a^(1/4)*b*ArcTan[(a +
b*x^4)^(1/4)/a^(1/4)])/8 - (5*a^(1/4)*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/8

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Rubi in Sympy [A]  time = 13.2157, size = 83, normalized size = 0.91 \[ - \frac{5 \sqrt [4]{a} b \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8} - \frac{5 \sqrt [4]{a} b \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8} + \frac{5 b \sqrt [4]{a + b x^{4}}}{4} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**(1/4)*b*atan((a + b*x**4)**(1/4)/a**(1/4))/8 - 5*a**(1/4)*b*atanh((a + b*x
**4)**(1/4)/a**(1/4))/8 + 5*b*(a + b*x**4)**(1/4)/4 - (a + b*x**4)**(5/4)/(4*x**
4)

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Mathematica [C]  time = 0.0980166, size = 73, normalized size = 0.8 \[ \left (b-\frac{a}{4 x^4}\right ) \sqrt [4]{a+b x^4}-\frac{5 a b \left (\frac{a}{b x^4}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{a}{b x^4}\right )}{12 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(b - a/(4*x^4))*(a + b*x^4)^(1/4) - (5*a*b*(1 + a/(b*x^4))^(3/4)*Hypergeometric2
F1[3/4, 3/4, 7/4, -(a/(b*x^4))])/(12*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^5,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298826, size = 209, normalized size = 2.3 \[ \frac{20 \, \left (a b^{4}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{\left (a b^{4}\right )^{\frac{1}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b + \sqrt{\sqrt{b x^{4} + a} b^{2} + \sqrt{a b^{4}}}}\right ) - 5 \, \left (a b^{4}\right )^{\frac{1}{4}} x^{4} \log \left (5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b + 5 \, \left (a b^{4}\right )^{\frac{1}{4}}\right ) + 5 \, \left (a b^{4}\right )^{\frac{1}{4}} x^{4} \log \left (5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b - 5 \, \left (a b^{4}\right )^{\frac{1}{4}}\right ) + 4 \,{\left (4 \, b x^{4} - a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{16 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(20*(a*b^4)^(1/4)*x^4*arctan((a*b^4)^(1/4)/((b*x^4 + a)^(1/4)*b + sqrt(sqrt
(b*x^4 + a)*b^2 + sqrt(a*b^4)))) - 5*(a*b^4)^(1/4)*x^4*log(5*(b*x^4 + a)^(1/4)*b
+ 5*(a*b^4)^(1/4)) + 5*(a*b^4)^(1/4)*x^4*log(5*(b*x^4 + a)^(1/4)*b - 5*(a*b^4)^
(1/4)) + 4*(4*b*x^4 - a)*(b*x^4 + a)^(1/4))/x^4

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Sympy [A]  time = 8.26234, size = 42, normalized size = 0.46 \[ - \frac{b^{\frac{5}{4}} x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(5/4)*x*gamma(-1/4)*hyper((-5/4, -1/4), (3/4,), a*exp_polar(I*pi)/(b*x**4))/
(4*gamma(3/4))

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GIAC/XCAS [A]  time = 0.22545, size = 278, normalized size = 3.05 \[ -\frac{1}{32} \,{\left (10 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) + 10 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) + 5 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) - 5 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) - 32 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} + \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{b x^{4}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(10*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^4 +
a)^(1/4))/(-a)^(1/4)) + 10*sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)
^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4)) + 5*sqrt(2)*(-a)^(1/4)*ln(sqrt(2)*(b*x
^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a)) - 5*sqrt(2)*(-a)^(1/4)*ln
(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a)) - 32*(b*x^4
+ a)^(1/4) + 8*(b*x^4 + a)^(1/4)*a/(b*x^4))*b