\(\int \frac {(4 b+a x^5) (-b+c x^4+a x^5)}{x^2 (-b+a x^5)^{3/4} (-b-c x^4+a x^5)} \, dx\) [1320]

Optimal result

Integrand size = 57, antiderivative size = 95 \[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\frac {4 \sqrt [4]{-b+a x^5}}{x}-4 \sqrt [4]{c} \arctan \left (\frac {\sqrt [4]{c} x \left (-b+a x^5\right )^{3/4}}{b-a x^5}\right )+4 \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} x \left (-b+a x^5\right )^{3/4}}{b-a x^5}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-4*b*(1 - (a*x^5)/b)^(3/4)*Hypergeometric2F1[-1/5, 3/4, 4/5, (a*x^5)/b])/(x*(-b + a*x^5)^(3/4)) + (2*c^3*x*(1
- (a*x^5)/b)^(3/4)*Hypergeometric2F1[1/5, 3/4, 6/5, (a*x^5)/b])/(a^2*(-b + a*x^5)^(3/4)) + (c^2*x^2*(1 - (a*x
^5)/b)^(3/4)*Hypergeometric2F1[2/5, 3/4, 7/5, (a*x^5)/b])/(a*(-b + a*x^5)^(3/4)) + (2*c*x^3*(1 - (a*x^5)/b)^(3
/4)*Hypergeometric2F1[3/5, 3/4, 8/5, (a*x^5)/b])/(3*(-b + a*x^5)^(3/4)) + (a*x^4*(1 - (a*x^5)/b)^(3/4)*Hyperge
ometric2F1[3/4, 4/5, 9/5, (a*x^5)/b])/(4*(-b + a*x^5)^(3/4)) - (2*b*c^3*Defer[Int][1/((b + c*x^4 - a*x^5)*(-b
+ a*x^5)^(3/4)), x])/a^2 - (2*c^4*Defer[Int][x^4/((b + c*x^4 - a*x^5)*(-b + a*x^5)^(3/4)), x])/a^2 + (2*b*c^2*
Defer[Int][x/((-b + a*x^5)^(3/4)*(-b - c*x^4 + a*x^5)), x])/a + 10*b*c*Defer[Int][x^2/((-b + a*x^5)^(3/4)*(-b
- c*x^4 + a*x^5)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 c^3}{a^2 \left (-b+a x^5\right )^{3/4}}+\frac {4 b}{x^2 \left (-b+a x^5\right )^{3/4}}+\frac {2 c^2 x}{a \left (-b+a x^5\right )^{3/4}}+\frac {2 c x^2}{\left (-b+a x^5\right )^{3/4}}+\frac {a x^3}{\left (-b+a x^5\right )^{3/4}}+\frac {2 \left (b c^3+a b c^2 x+5 a^2 b c x^2+c^4 x^4\right )}{a^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )}\right ) \, dx \\ & = \frac {2 \int \frac {b c^3+a b c^2 x+5 a^2 b c x^2+c^4 x^4}{\left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx}{a^2}+a \int \frac {x^3}{\left (-b+a x^5\right )^{3/4}} \, dx+(4 b) \int \frac {1}{x^2 \left (-b+a x^5\right )^{3/4}} \, dx+(2 c) \int \frac {x^2}{\left (-b+a x^5\right )^{3/4}} \, dx+\frac {\left (2 c^2\right ) \int \frac {x}{\left (-b+a x^5\right )^{3/4}} \, dx}{a}+\frac {\left (2 c^3\right ) \int \frac {1}{\left (-b+a x^5\right )^{3/4}} \, dx}{a^2} \\ & = \frac {2 \int \left (-\frac {b c^3}{\left (b+c x^4-a x^5\right ) \left (-b+a x^5\right )^{3/4}}-\frac {c^4 x^4}{\left (b+c x^4-a x^5\right ) \left (-b+a x^5\right )^{3/4}}+\frac {a b c^2 x}{\left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )}+\frac {5 a^2 b c x^2}{\left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )}\right ) \, dx}{a^2}+\frac {\left (a \left (1-\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {x^3}{\left (1-\frac {a x^5}{b}\right )^{3/4}} \, dx}{\left (-b+a x^5\right )^{3/4}}+\frac {\left (4 b \left (1-\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1-\frac {a x^5}{b}\right )^{3/4}} \, dx}{\left (-b+a x^5\right )^{3/4}}+\frac {\left (2 c \left (1-\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {x^2}{\left (1-\frac {a x^5}{b}\right )^{3/4}} \, dx}{\left (-b+a x^5\right )^{3/4}}+\frac {\left (2 c^2 \left (1-\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {x}{\left (1-\frac {a x^5}{b}\right )^{3/4}} \, dx}{a \left (-b+a x^5\right )^{3/4}}+\frac {\left (2 c^3 \left (1-\frac {a x^5}{b}\right )^{3/4}\right ) \int \frac {1}{\left (1-\frac {a x^5}{b}\right )^{3/4}} \, dx}{a^2 \left (-b+a x^5\right )^{3/4}} \\ & = -\frac {4 b \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{5},\frac {3}{4},\frac {4}{5},\frac {a x^5}{b}\right )}{x \left (-b+a x^5\right )^{3/4}}+\frac {2 c^3 x \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {3}{4},\frac {6}{5},\frac {a x^5}{b}\right )}{a^2 \left (-b+a x^5\right )^{3/4}}+\frac {c^2 x^2 \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {2}{5},\frac {3}{4},\frac {7}{5},\frac {a x^5}{b}\right )}{a \left (-b+a x^5\right )^{3/4}}+\frac {2 c x^3 \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{5},\frac {3}{4},\frac {8}{5},\frac {a x^5}{b}\right )}{3 \left (-b+a x^5\right )^{3/4}}+\frac {a x^4 \left (1-\frac {a x^5}{b}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {4}{5},\frac {9}{5},\frac {a x^5}{b}\right )}{4 \left (-b+a x^5\right )^{3/4}}+(10 b c) \int \frac {x^2}{\left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx+\frac {\left (2 b c^2\right ) \int \frac {x}{\left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx}{a}-\frac {\left (2 b c^3\right ) \int \frac {1}{\left (b+c x^4-a x^5\right ) \left (-b+a x^5\right )^{3/4}} \, dx}{a^2}-\frac {\left (2 c^4\right ) \int \frac {x^4}{\left (b+c x^4-a x^5\right ) \left (-b+a x^5\right )^{3/4}} \, dx}{a^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.86 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.79 \[ \int \frac {\left (4 b+a x^5\right ) \left (-b+c x^4+a x^5\right )}{x^2 \left (-b+a x^5\right )^{3/4} \left (-b-c x^4+a x^5\right )} \, dx=\frac {4 \sqrt [4]{-b+a x^5}}{x}+4 \sqrt [4]{c} \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^5}}\right )-4 \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+a x^5}}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.92 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91

method	result	size
pseudoelliptic	\(\frac {-2 x \left (\ln \left (\frac {c^{\frac {1}{4}} x +\left (a \,x^{5}-b \right )^{\frac {1}{4}}}{-c^{\frac {1}{4}} x +\left (a \,x^{5}-b \right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (a \,x^{5}-b \right )^{\frac {1}{4}}}{x \,c^{\frac {1}{4}}}\right )\right ) c^{\frac {1}{4}}+4 \left (a \,x^{5}-b \right )^{\frac {1}{4}}}{x}\)	\(86\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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