Integrand size = 37, antiderivative size = 53 \[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=-2 \arctan \left (\frac {\left (b x+a x^5\right )^{3/4}}{b+a x^4}\right )-2 \text {arctanh}\left (\frac {\left (b x+a x^5\right )^{3/4}}{b+a x^4}\right ) \]
\[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=\int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx \]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a x^4-3 b}{\left (a x^4+b-x^3\right ) \sqrt [4]{a x^5+b x}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^4+b} \int -\frac {3 b-a x^4}{\sqrt [4]{x} \sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}dx}{\sqrt [4]{a x^5+b x}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x^4+b} \int \frac {3 b-a x^4}{\sqrt [4]{x} \sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}dx}{\sqrt [4]{a x^5+b x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \int \frac {\sqrt {x} \left (3 b-a x^4\right )}{\sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^5+b x}}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \int \left (\frac {\sqrt {x} \left (4 b-x^3\right )}{\sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}-\frac {\sqrt {x}}{\sqrt [4]{a x^4+b}}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^5+b x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^4+b} \left (4 b \int \frac {\sqrt {x}}{\sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}d\sqrt [4]{x}-\int \frac {x^{7/2}}{\sqrt [4]{a x^4+b} \left (a x^4-x^3+b\right )}d\sqrt [4]{x}-\frac {x^{3/4} \sqrt [4]{\frac {a x^4}{b}+1} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},-\frac {a x^4}{b}\right )}{3 \sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a x^5+b x}}\) |
3.7.75.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 1.92 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.13
method | result | size |
pseudoelliptic | \(\ln \left (\frac {{\left (x \left (a \,x^{4}+b \right )\right )}^{\frac {1}{4}}-x}{x}\right )-\ln \left (\frac {{\left (x \left (a \,x^{4}+b \right )\right )}^{\frac {1}{4}}+x}{x}\right )+2 \arctan \left (\frac {{\left (x \left (a \,x^{4}+b \right )\right )}^{\frac {1}{4}}}{x}\right )\) | \(60\) |
Timed out. \[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=\text {Timed out} \]
\[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=\int { \frac {a x^{4} - 3 \, b}{{\left (a x^{5} + b x\right )}^{\frac {1}{4}} {\left (a x^{4} - x^{3} + b\right )}} \,d x } \]
\[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=\int { \frac {a x^{4} - 3 \, b}{{\left (a x^{5} + b x\right )}^{\frac {1}{4}} {\left (a x^{4} - x^{3} + b\right )}} \,d x } \]
Timed out. \[ \int \frac {-3 b+a x^4}{\left (b-x^3+a x^4\right ) \sqrt [4]{b x+a x^5}} \, dx=\int -\frac {3\,b-a\,x^4}{{\left (a\,x^5+b\,x\right )}^{1/4}\,\left (a\,x^4-x^3+b\right )} \,d x \]