\(\int \frac {3 b+a x^4}{(-b-x^3+a x^4) \sqrt [4]{-b x+a x^5}} \, dx\) [763]

Optimal result

Integrand size = 40, antiderivative size = 59 \[ \int \frac {3 b+a x^4}{\left (-b-x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=-2 \arctan \left (\frac {\left (-b x+a x^5\right )^{3/4}}{-b+a x^4}\right )-2 \text {arctanh}\left (\frac {\left (-b x+a x^5\right )^{3/4}}{-b+a x^4}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(4*x*(1 - (a*x^4)/b)^(1/4)*Hypergeometric2F1[3/16, 1/4, 19/16, (a*x^4)/b])/(3*(-(b*x) + a*x^5)^(1/4)) - (16*b*
x^(1/4)*(-b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^2/((b + x^12 - a*x^16)*(-b + a*x^16)^(1/4)), x], x, x^(1/
4)])/(-(b*x) + a*x^5)^(1/4) + (4*x^(1/4)*(-b + a*x^4)^(1/4)*Defer[Subst][Defer[Int][x^14/((-b + a*x^16)^(1/4)*
(-b - x^12 + a*x^16)), x], x, x^(1/4)])/(-(b*x) + a*x^5)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \int \frac {3 b+a x^4}{\sqrt [4]{x} \sqrt [4]{-b+a x^4} \left (-b-x^3+a x^4\right )} \, dx}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (3 b+a x^{16}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \left (\frac {x^2}{\sqrt [4]{-b+a x^{16}}}+\frac {x^2 \left (4 b+x^{12}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b-x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (4 b+x^{12}\right )}{\sqrt [4]{-b+a x^{16}} \left (-b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \left (-\frac {4 b x^2}{\left (b+x^{12}-a x^{16}\right ) \sqrt [4]{-b+a x^{16}}}+\frac {x^{14}}{\sqrt [4]{-b+a x^{16}} \left (-b-x^{12}+a x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1-\frac {a x^4}{b}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1-\frac {a x^{16}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ & = \frac {4 x \sqrt [4]{1-\frac {a x^4}{b}} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},\frac {a x^4}{b}\right )}{3 \sqrt [4]{-b x+a x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{16}} \left (-b-x^{12}+a x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}}-\frac {\left (16 b \sqrt [4]{x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+x^{12}-a x^{16}\right ) \sqrt [4]{-b+a x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \frac {3 b+a x^4}{\left (-b-x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=-2 \arctan \left (\frac {\left (-b x+a x^5\right )^{3/4}}{-b+a x^4}\right )-2 \text {arctanh}\left (\frac {\left (-b x+a x^5\right )^{3/4}}{-b+a x^4}\right ) \]

[In]

Integrate[(3*b + a*x^4)/((-b - x^3 + a*x^4)*(-(b*x) + a*x^5)^(1/4)),x]

[Out]

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

Maple [A] (verified)

Time = 1.91 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\ln \left (\frac {{\left (x \left (a \,x^{4}-b \right )\right )}^{\frac {1}{4}}-x}{x}\right )-\ln \left (\frac {{\left (x \left (a \,x^{4}-b \right )\right )}^{\frac {1}{4}}+x}{x}\right )+2 \arctan \left (\frac {{\left (x \left (a \,x^{4}-b \right )\right )}^{\frac {1}{4}}}{x}\right )\)	\(66\)

[In]

int((a*x^4+3*b)/(a*x^4-x^3-b)/(a*x^5-b*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

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

[In]

integrate((a*x^4+3*b)/(a*x^4-x^3-b)/(a*x^5-b*x)^(1/4),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate((a*x**4+3*b)/(a*x**4-x**3-b)/(a*x**5-b*x)**(1/4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {3 b+a x^4}{\left (-b-x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=\int { \frac {a x^{4} + 3 \, b}{{\left (a x^{5} - b x\right )}^{\frac {1}{4}} {\left (a x^{4} - x^{3} - b\right )}} \,d x } \]

[In]

integrate((a*x^4+3*b)/(a*x^4-x^3-b)/(a*x^5-b*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^4 + 3*b)/((a*x^5 - b*x)^(1/4)*(a*x^4 - x^3 - b)), x)

Giac [F]

\[ \int \frac {3 b+a x^4}{\left (-b-x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=\int { \frac {a x^{4} + 3 \, b}{{\left (a x^{5} - b x\right )}^{\frac {1}{4}} {\left (a x^{4} - x^{3} - b\right )}} \,d x } \]

[In]

integrate((a*x^4+3*b)/(a*x^4-x^3-b)/(a*x^5-b*x)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^4 + 3*b)/((a*x^5 - b*x)^(1/4)*(a*x^4 - x^3 - b)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {3 b+a x^4}{\left (-b-x^3+a x^4\right ) \sqrt [4]{-b x+a x^5}} \, dx=\int -\frac {a\,x^4+3\,b}{{\left (a\,x^5-b\,x\right )}^{1/4}\,\left (-a\,x^4+x^3+b\right )} \,d x \]

[In]

int(-(3*b + a*x^4)/((a*x^5 - b*x)^(1/4)*(b - a*x^4 + x^3)),x)

[Out]

int(-(3*b + a*x^4)/((a*x^5 - b*x)^(1/4)*(b - a*x^4 + x^3)), x)