\(\int \frac {(-b+a x^5)^{3/4} (4 b+a x^5)}{x^4 (-b+c x^4+a x^5)} \, dx\) [2098]

Optimal result

Integrand size = 42, antiderivative size = 152 \[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\frac {4 \left (-b+a x^5\right )^{3/4}}{3 x^3}+\sqrt {2} c^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{-\sqrt {c} x^2+\sqrt {-b+a x^5}}\right )+\sqrt {2} c^{3/4} \text {arctanh}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{-b+a x^5}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-b+a x^5\right )^{3/4} \left (4 b+a x^5\right )}{x^4 \left (-b+c x^4+a x^5\right )} \, dx=\frac {4 \left (-b+a x^5\right )^{3/4}}{3 x^3}+\sqrt {2} c^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}{-\sqrt {c} x^2+\sqrt {-b+a x^5}}\right )+\sqrt {2} c^{3/4} \text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{-b+a x^5}}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(\frac {-3 \ln \left (\frac {\sqrt {a \,x^{5}-b}-\left (a \,x^{5}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}}{\left (a \,x^{5}-b \right )^{\frac {1}{4}} x \,c^{\frac {1}{4}} \sqrt {2}+\sqrt {c}\, x^{2}+\sqrt {a \,x^{5}-b}}\right ) c^{\frac {3}{4}} \sqrt {2}\, x^{3}-6 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}+c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c^{\frac {3}{4}} \sqrt {2}\, x^{3}-6 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}-c^{\frac {1}{4}} x}{c^{\frac {1}{4}} x}\right ) c^{\frac {3}{4}} \sqrt {2}\, x^{3}+8 \left (a \,x^{5}-b \right )^{\frac {3}{4}}}{6 x^{3}}\)	\(193\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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