\(\int \frac {4 b+a x^5}{(-b+a x^5) \sqrt [4]{-b+c x^4+a x^5}} \, dx\) [885]

Optimal result

Integrand size = 39, antiderivative size = 67 \[ \int \frac {4 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{\sqrt [4]{c}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^5}}\right )}{\sqrt [4]{c}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.64 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87 \[ \int \frac {4 b+a x^5}{\left (-b+a x^5\right ) \sqrt [4]{-b+c x^4+a x^5}} \, dx=-\frac {2 \left (\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+x^4 (c+a x)}}\right )\right )}{\sqrt [4]{c}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.96 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.24

method	result	size
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}\right )-\ln \left (\frac {c^{\frac {1}{4}} x +\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}{-c^{\frac {1}{4}} x +\left (a \,x^{5}+c \,x^{4}-b \right )^{\frac {1}{4}}}\right )}{c^{\frac {1}{4}}}\)	\(83\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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