\(\int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx\) [2575]

Optimal result

Integrand size = 45, antiderivative size = 221 \[ \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt [4]{b}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{\sqrt [4]{b}}\right )}{\sqrt [4]{d}} \]

[Out]

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

Rubi [F]

\[ \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx=\int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 4.04 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.96 \[ \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx=-\frac {2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a (-1+x)+b x^4}{c (-1+x)+d x^4}}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {a (-1+x)+b x^4}{c (-1+x)+d x^4}}}{\sqrt [4]{b}}\right )}{\sqrt [4]{d}}-\frac {2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt [4]{\frac {a (-1+x)+b x^4}{c (-1+x)+d x^4}}}{\sqrt [4]{a}}\right )}{\sqrt [4]{c}}+\frac {2 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{d} \sqrt [4]{\frac {a (-1+x)+b x^4}{c (-1+x)+d x^4}}}{\sqrt [4]{b}}\right )}{\sqrt [4]{d}} \]

[In]

Integrate[((-4 + 3*x)*((-a + a*x + b*x^4)/(-c + c*x + d*x^4))^(1/4))/((-1 + x)*x),x]

[Out]

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

Maple [F]

\[\int \frac {\left (-4+3 x \right ) \left (\frac {b \,x^{4}+a x -a}{d \,x^{4}+c x -c}\right )^{\frac {1}{4}}}{\left (-1+x \right ) x}d x\]

[In]

int((-4+3*x)*((b*x^4+a*x-a)/(d*x^4+c*x-c))^(1/4)/(-1+x)/x,x)

[Out]

int((-4+3*x)*((b*x^4+a*x-a)/(d*x^4+c*x-c))^(1/4)/(-1+x)/x,x)

Fricas [F(-1)]

Timed out. \[ \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx=\text {Timed out} \]

[In]

integrate((-4+3*x)*((b*x^4+a*x-a)/(d*x^4+c*x-c))^(1/4)/(-1+x)/x,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx=\text {Timed out} \]

[In]

integrate((-4+3*x)*((b*x**4+a*x-a)/(d*x**4+c*x-c))**(1/4)/(-1+x)/x,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx=\int { \frac {{\left (3 \, x - 4\right )} \left (\frac {b x^{4} + a x - a}{d x^{4} + c x - c}\right )^{\frac {1}{4}}}{{\left (x - 1\right )} x} \,d x } \]

[In]

integrate((-4+3*x)*((b*x^4+a*x-a)/(d*x^4+c*x-c))^(1/4)/(-1+x)/x,x, algorithm="maxima")

[Out]

integrate((3*x - 4)*((b*x^4 + a*x - a)/(d*x^4 + c*x - c))^(1/4)/((x - 1)*x), x)

Giac [F]

\[ \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx=\int { \frac {{\left (3 \, x - 4\right )} \left (\frac {b x^{4} + a x - a}{d x^{4} + c x - c}\right )^{\frac {1}{4}}}{{\left (x - 1\right )} x} \,d x } \]

[In]

integrate((-4+3*x)*((b*x^4+a*x-a)/(d*x^4+c*x-c))^(1/4)/(-1+x)/x,x, algorithm="giac")

[Out]

integrate((3*x - 4)*((b*x^4 + a*x - a)/(d*x^4 + c*x - c))^(1/4)/((x - 1)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(-4+3 x) \sqrt [4]{\frac {-a+a x+b x^4}{-c+c x+d x^4}}}{(-1+x) x} \, dx=\int \frac {\left (3\,x-4\right )\,{\left (\frac {b\,x^4+a\,x-a}{d\,x^4+c\,x-c}\right )}^{1/4}}{x\,\left (x-1\right )} \,d x \]

[In]

int(((3*x - 4)*((a*x - a + b*x^4)/(c*x - c + d*x^4))^(1/4))/(x*(x - 1)),x)

[Out]

int(((3*x - 4)*((a*x - a + b*x^4)/(c*x - c + d*x^4))^(1/4))/(x*(x - 1)), x)