\(\int \frac {x^4 (4 b+a x^5)}{(-b+a x^5)^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx\) [1483]

Optimal result

Integrand size = 42, antiderivative size = 104 \[ \int \frac {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx=\frac {x \left (-b+c x^4+a x^5\right )^{3/4}}{c \left (b-a x^5\right )}+\frac {\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{2 c^{5/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{2 c^{5/4}} \]

[Out]

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

Rubi [F]

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

[In]

Int[(x^4*(4*b + a*x^5))/((-b + a*x^5)^2*(-b + c*x^4 + a*x^5)^(1/4)),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 1.86 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.91 \[ \int \frac {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx=\frac {\frac {2 \sqrt [4]{c} x \left (-b+x^4 (c+a x)\right )^{3/4}}{b-a x^5}+\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )}{2 c^{5/4}} \]

[In]

Integrate[(x^4*(4*b + a*x^5))/((-b + a*x^5)^2*(-b + c*x^4 + a*x^5)^(1/4)),x]

[Out]

((2*c^(1/4)*x*(-b + x^4*(c + a*x))^(3/4))/(b - a*x^5) + ArcTan[(c^(1/4)*x)/(-b + x^4*(c + a*x))^(1/4)] + ArcTa
nh[(c^(1/4)*x)/(-b + x^4*(c + a*x))^(1/4)])/(2*c^(5/4))

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.26

method	result	size
pseudoelliptic	\(-\frac {2 \left (2 \left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {3}{4}} x \,c^{\frac {1}{4}}+\left (\arctan \left (\frac {\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}\right )-\frac {\ln \left (\frac {-c^{\frac {1}{4}} x -\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x -\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}\right )}{2}\right ) \left (a \,x^{5}-b \right )\right )}{c^{\frac {5}{4}} \left (4 a \,x^{5}-4 b \right )}\)	\(131\)

[In]

int(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

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

[In]

integrate(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate(x**4*(a*x**5+4*b)/(a*x**5-b)**2/(a*x**5+c*x**4-b)**(1/4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx=\int { \frac {{\left (a x^{5} + 4 \, b\right )} x^{4}}{{\left (a x^{5} + c x^{4} - b\right )}^{\frac {1}{4}} {\left (a x^{5} - b\right )}^{2}} \,d x } \]

[In]

integrate(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^5 + 4*b)*x^4/((a*x^5 + c*x^4 - b)^(1/4)*(a*x^5 - b)^2), x)

Giac [F]

\[ \int \frac {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx=\int { \frac {{\left (a x^{5} + 4 \, b\right )} x^{4}}{{\left (a x^{5} + c x^{4} - b\right )}^{\frac {1}{4}} {\left (a x^{5} - b\right )}^{2}} \,d x } \]

[In]

integrate(x^4*(a*x^5+4*b)/(a*x^5-b)^2/(a*x^5+c*x^4-b)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^5 + 4*b)*x^4/((a*x^5 + c*x^4 - b)^(1/4)*(a*x^5 - b)^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (4 b+a x^5\right )}{\left (-b+a x^5\right )^2 \sqrt [4]{-b+c x^4+a x^5}} \, dx=\int \frac {x^4\,\left (a\,x^5+4\,b\right )}{{\left (b-a\,x^5\right )}^2\,{\left (a\,x^5+c\,x^4-b\right )}^{1/4}} \,d x \]

[In]

int((x^4*(4*b + a*x^5))/((b - a*x^5)^2*(a*x^5 - b + c*x^4)^(1/4)),x)

[Out]

int((x^4*(4*b + a*x^5))/((b - a*x^5)^2*(a*x^5 - b + c*x^4)^(1/4)), x)