\(\int \frac {4 b+a x^3}{(b+a x^3) \sqrt [4]{-b-a x^3+c x^4}} \, dx\) [914]

Optimal result

Integrand size = 38, antiderivative size = 69 \[ \int \frac {4 b+a x^3}{\left (b+a x^3\right ) \sqrt [4]{-b-a x^3+c x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b-a x^3+c x^4}}\right )}{\sqrt [4]{c}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b-a x^3+c x^4}}\right )}{\sqrt [4]{c}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {4 b+a x^3}{\left (b+a x^3\right ) \sqrt [4]{-b-a x^3+c x^4}} \, dx=\frac {2 \left (\arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b-a x^3+c x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b-a x^3+c x^4}}\right )\right )}{\sqrt [4]{c}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.80 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(\frac {-2 \arctan \left (\frac {\left (c \,x^{4}-a \,x^{3}-b \right )^{\frac {1}{4}}}{c^{\frac {1}{4}} x}\right )+\ln \left (\frac {c^{\frac {1}{4}} x +\left (c \,x^{4}-a \,x^{3}-b \right )^{\frac {1}{4}}}{-c^{\frac {1}{4}} x +\left (c \,x^{4}-a \,x^{3}-b \right )^{\frac {1}{4}}}\right )}{c^{\frac {1}{4}}}\)	\(84\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.99 \[ \int \frac {4 b+a x^3}{\left (b+a x^3\right ) \sqrt [4]{-b-a x^3+c x^4}} \, dx=\frac {\log \left (\frac {c^{\frac {1}{4}} x + {\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{c^{\frac {1}{4}}} - \frac {\log \left (-\frac {c^{\frac {1}{4}} x - {\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{c^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {i \, c^{\frac {1}{4}} x + {\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{c^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {-i \, c^{\frac {1}{4}} x + {\left (c x^{4} - a x^{3} - b\right )}^{\frac {1}{4}}}{x}\right )}{c^{\frac {1}{4}}} \]

[In]

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

[Out]

log((c^(1/4)*x + (c*x^4 - a*x^3 - b)^(1/4))/x)/c^(1/4) - log(-(c^(1/4)*x - (c*x^4 - a*x^3 - b)^(1/4))/x)/c^(1/
4) - I*log((I*c^(1/4)*x + (c*x^4 - a*x^3 - b)^(1/4))/x)/c^(1/4) + I*log((-I*c^(1/4)*x + (c*x^4 - a*x^3 - b)^(1
/4))/x)/c^(1/4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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