\(\int \frac {(-4 b+a x^5) (b+a x^5)^{3/4}}{x^4 (2 b+c x^4+2 a x^5)} \, dx\) [2123]

Optimal result

Integrand size = 41, antiderivative size = 154 \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\frac {2 \left (b+a x^5\right )^{3/4}}{3 x^3}+\frac {c^{3/4} \arctan \left (\frac {2^{3/4} \sqrt [4]{c} x \sqrt [4]{b+a x^5}}{-\sqrt {c} x^2+\sqrt {2} \sqrt {b+a x^5}}\right )}{2 \sqrt [4]{2}}+\frac {c^{3/4} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{2^{3/4}}+\frac {\sqrt {b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c}}}{x \sqrt [4]{b+a x^5}}\right )}{2 \sqrt [4]{2}} \]

[Out]

2/3*(a*x^5+b)^(3/4)/x^3+1/4*c^(3/4)*arctan(2^(3/4)*c^(1/4)*x*(a*x^5+b)^(1/4)/(-c^(1/2)*x^2+2^(1/2)*(a*x^5+b)^(
1/2)))*2^(3/4)+1/4*c^(3/4)*arctanh((1/2*c^(1/4)*x^2*2^(1/4)+1/2*(a*x^5+b)^(1/2)*2^(3/4)/c^(1/4))/x/(a*x^5+b)^(
1/4))*2^(3/4)

Rubi [F]

\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx \]

[In]

Int[((-4*b + a*x^5)*(b + a*x^5)^(3/4))/(x^4*(2*b + c*x^4 + 2*a*x^5)),x]

[Out]

(2*(b + a*x^5)^(3/4)*Hypergeometric2F1[-3/4, -3/5, 2/5, -((a*x^5)/b)])/(3*x^3*(1 + (a*x^5)/b)^(3/4)) + 2*c*Def
er[Int][(b + a*x^5)^(3/4)/(2*b + c*x^4 + 2*a*x^5), x] + 5*a*Defer[Int][(x*(b + a*x^5)^(3/4))/(2*b + c*x^4 + 2*
a*x^5), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \left (b+a x^5\right )^{3/4}}{x^4}+\frac {(2 c+5 a x) \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}\right ) \, dx \\ & = -\left (2 \int \frac {\left (b+a x^5\right )^{3/4}}{x^4} \, dx\right )+\int \frac {(2 c+5 a x) \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5} \, dx \\ & = -\frac {\left (2 \left (b+a x^5\right )^{3/4}\right ) \int \frac {\left (1+\frac {a x^5}{b}\right )^{3/4}}{x^4} \, dx}{\left (1+\frac {a x^5}{b}\right )^{3/4}}+\int \left (\frac {2 c \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}+\frac {5 a x \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5}\right ) \, dx \\ & = \frac {2 \left (b+a x^5\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {3}{5},\frac {2}{5},-\frac {a x^5}{b}\right )}{3 x^3 \left (1+\frac {a x^5}{b}\right )^{3/4}}+(5 a) \int \frac {x \left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5} \, dx+(2 c) \int \frac {\left (b+a x^5\right )^{3/4}}{2 b+c x^4+2 a x^5} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\frac {1}{12} \left (\frac {8 \left (b+a x^5\right )^{3/4}}{x^3}+3\ 2^{3/4} c^{3/4} \arctan \left (\frac {\sqrt [4]{c} x}{2^{3/4} \sqrt [4]{b+a x^5}}-\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c} x}\right )+3\ 2^{3/4} c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{c} x}{2^{3/4} \sqrt [4]{b+a x^5}}+\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{2} \sqrt [4]{c} x}\right )\right ) \]

[In]

Integrate[((-4*b + a*x^5)*(b + a*x^5)^(3/4))/(x^4*(2*b + c*x^4 + 2*a*x^5)),x]

[Out]

((8*(b + a*x^5)^(3/4))/x^3 + 3*2^(3/4)*c^(3/4)*ArcTan[(c^(1/4)*x)/(2^(3/4)*(b + a*x^5)^(1/4)) - (b + a*x^5)^(1
/4)/(2^(1/4)*c^(1/4)*x)] + 3*2^(3/4)*c^(3/4)*ArcTanh[(c^(1/4)*x)/(2^(3/4)*(b + a*x^5)^(1/4)) + (b + a*x^5)^(1/
4)/(2^(1/4)*c^(1/4)*x)])/12

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.21

method	result	size
pseudoelliptic	\(\frac {\left (8 \left (a \,x^{5}+b \right )^{\frac {3}{4}} 2^{\frac {1}{4}} c^{\frac {1}{4}}-3 \ln \left (\frac {\sqrt {2}\, \sqrt {c}\, x^{2}-2 \,2^{\frac {1}{4}} c^{\frac {1}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} x +2 \sqrt {a \,x^{5}+b}}{\sqrt {2}\, \sqrt {c}\, x^{2}+2 \,2^{\frac {1}{4}} c^{\frac {1}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} x +2 \sqrt {a \,x^{5}+b}}\right ) c \,x^{3}-6 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c \,x^{3}-6 \arctan \left (\frac {2^{\frac {3}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c \,x^{3}\right ) 2^{\frac {3}{4}}}{24 x^{3} c^{\frac {1}{4}}}\)	\(187\)

[In]

int((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x,method=_RETURNVERBOSE)

[Out]

1/24*(8*(a*x^5+b)^(3/4)*2^(1/4)*c^(1/4)-3*ln((2^(1/2)*c^(1/2)*x^2-2*2^(1/4)*c^(1/4)*(a*x^5+b)^(1/4)*x+2*(a*x^5
+b)^(1/2))/(2^(1/2)*c^(1/2)*x^2+2*2^(1/4)*c^(1/4)*(a*x^5+b)^(1/4)*x+2*(a*x^5+b)^(1/2)))*c*x^3-6*arctan((2^(3/4
)*(a*x^5+b)^(1/4)+c^(1/4)*x)/c^(1/4)/x)*c*x^3-6*arctan((2^(3/4)*(a*x^5+b)^(1/4)-c^(1/4)*x)/c^(1/4)/x)*c*x^3)*2
^(3/4)/x^3/c^(1/4)

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int \frac {\left (a x^{5} - 4 b\right ) \left (a x^{5} + b\right )^{\frac {3}{4}}}{x^{4} \cdot \left (2 a x^{5} + 2 b + c x^{4}\right )}\, dx \]

[In]

integrate((a*x**5-4*b)*(a*x**5+b)**(3/4)/x**4/(2*a*x**5+c*x**4+2*b),x)

[Out]

Integral((a*x**5 - 4*b)*(a*x**5 + b)**(3/4)/(x**4*(2*a*x**5 + 2*b + c*x**4)), x)

Maxima [F]

\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int { \frac {{\left (a x^{5} + b\right )}^{\frac {3}{4}} {\left (a x^{5} - 4 \, b\right )}}{{\left (2 \, a x^{5} + c x^{4} + 2 \, b\right )} x^{4}} \,d x } \]

[In]

integrate((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x, algorithm="maxima")

[Out]

integrate((a*x^5 + b)^(3/4)*(a*x^5 - 4*b)/((2*a*x^5 + c*x^4 + 2*b)*x^4), x)

Giac [F]

\[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int { \frac {{\left (a x^{5} + b\right )}^{\frac {3}{4}} {\left (a x^{5} - 4 \, b\right )}}{{\left (2 \, a x^{5} + c x^{4} + 2 \, b\right )} x^{4}} \,d x } \]

[In]

integrate((a*x^5-4*b)*(a*x^5+b)^(3/4)/x^4/(2*a*x^5+c*x^4+2*b),x, algorithm="giac")

[Out]

integrate((a*x^5 + b)^(3/4)*(a*x^5 - 4*b)/((2*a*x^5 + c*x^4 + 2*b)*x^4), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-4 b+a x^5\right ) \left (b+a x^5\right )^{3/4}}{x^4 \left (2 b+c x^4+2 a x^5\right )} \, dx=\int -\frac {{\left (a\,x^5+b\right )}^{3/4}\,\left (4\,b-a\,x^5\right )}{x^4\,\left (2\,a\,x^5+c\,x^4+2\,b\right )} \,d x \]

[In]

int(-((b + a*x^5)^(3/4)*(4*b - a*x^5))/(x^4*(2*b + 2*a*x^5 + c*x^4)),x)

[Out]

int(-((b + a*x^5)^(3/4)*(4*b - a*x^5))/(x^4*(2*b + 2*a*x^5 + c*x^4)), x)