3.1077 \(\int \left (a+b x^4\right )^{7/4} \, dx\)

Optimal. Leaf size=96 \[ \frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{21 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{7}{32} a x \left (a+b x^4\right )^{3/4}+\frac{1}{8} x \left (a+b x^4\right )^{7/4} \]

[Out]

(7*a*x*(a + b*x^4)^(3/4))/32 + (x*(a + b*x^4)^(7/4))/8 + (21*a^2*ArcTan[(b^(1/4)
*x)/(a + b*x^4)^(1/4)])/(64*b^(1/4)) + (21*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(
1/4)])/(64*b^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.0659831, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{21 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{21 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 \sqrt [4]{b}}+\frac{7}{32} a x \left (a+b x^4\right )^{3/4}+\frac{1}{8} x \left (a+b x^4\right )^{7/4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(7/4),x]

[Out]

(7*a*x*(a + b*x^4)^(3/4))/32 + (x*(a + b*x^4)^(7/4))/8 + (21*a^2*ArcTan[(b^(1/4)
*x)/(a + b*x^4)^(1/4)])/(64*b^(1/4)) + (21*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(
1/4)])/(64*b^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.16391, size = 90, normalized size = 0.94 \[ \frac{21 a^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 \sqrt [4]{b}} + \frac{21 a^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 \sqrt [4]{b}} + \frac{7 a x \left (a + b x^{4}\right )^{\frac{3}{4}}}{32} + \frac{x \left (a + b x^{4}\right )^{\frac{7}{4}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(7/4),x)

[Out]

21*a**2*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(64*b**(1/4)) + 21*a**2*atanh(b**(1
/4)*x/(a + b*x**4)**(1/4))/(64*b**(1/4)) + 7*a*x*(a + b*x**4)**(3/4)/32 + x*(a +
b*x**4)**(7/4)/8

_______________________________________________________________________________________

Mathematica [A]  time = 0.174423, size = 107, normalized size = 1.11 \[ \frac{21 a^2 \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{128 \sqrt [4]{b}}+\left (a+b x^4\right )^{3/4} \left (\frac{11 a x}{32}+\frac{b x^5}{8}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(7/4),x]

[Out]

(a + b*x^4)^(3/4)*((11*a*x)/32 + (b*x^5)/8) + (21*a^2*(2*ArcTan[(b^(1/4)*x)/(a +
b*x^4)^(1/4)] - Log[1 - (b^(1/4)*x)/(a + b*x^4)^(1/4)] + Log[1 + (b^(1/4)*x)/(a
+ b*x^4)^(1/4)]))/(128*b^(1/4))

_______________________________________________________________________________________

Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(7/4),x)

[Out]

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

_______________________________________________________________________________________

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)^(7/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.302404, size = 255, normalized size = 2.66 \[ \frac{1}{32} \,{\left (4 \, b x^{5} + 11 \, a x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}} + \frac{21}{32} \, \left (\frac{a^{8}}{b}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (\frac{a^{8}}{b}\right )^{\frac{3}{4}} b x}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} + x \sqrt{\frac{\sqrt{b x^{4} + a} a^{12} + \sqrt{\frac{a^{8}}{b}} a^{8} b x^{2}}{x^{2}}}}\right ) + \frac{21}{128} \, \left (\frac{a^{8}}{b}\right )^{\frac{1}{4}} \log \left (\frac{9261 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} + \left (\frac{a^{8}}{b}\right )^{\frac{3}{4}} b x\right )}}{x}\right ) - \frac{21}{128} \, \left (\frac{a^{8}}{b}\right )^{\frac{1}{4}} \log \left (\frac{9261 \,{\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} - \left (\frac{a^{8}}{b}\right )^{\frac{3}{4}} b x\right )}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(7/4),x, algorithm="fricas")

[Out]

1/32*(4*b*x^5 + 11*a*x)*(b*x^4 + a)^(3/4) + 21/32*(a^8/b)^(1/4)*arctan((a^8/b)^(
3/4)*b*x/((b*x^4 + a)^(1/4)*a^6 + x*sqrt((sqrt(b*x^4 + a)*a^12 + sqrt(a^8/b)*a^8
*b*x^2)/x^2))) + 21/128*(a^8/b)^(1/4)*log(9261*((b*x^4 + a)^(1/4)*a^6 + (a^8/b)^
(3/4)*b*x)/x) - 21/128*(a^8/b)^(1/4)*log(9261*((b*x^4 + a)^(1/4)*a^6 - (a^8/b)^(
3/4)*b*x)/x)

_______________________________________________________________________________________

Sympy [A]  time = 12.4105, size = 37, normalized size = 0.39 \[ \frac{a^{\frac{7}{4}} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(7/4),x)

[Out]

a**(7/4)*x*gamma(1/4)*hyper((-7/4, 1/4), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*ga
mma(5/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{7}{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(7/4),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(7/4), x)