Optimal. Leaf size=106 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b x^2}}{\sqrt {a} x}\right )}{\sqrt {2} \sqrt {a} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b x^2}}{\sqrt {a} x}\right )}{\sqrt {2} \sqrt {a} b^{3/4}} \]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 170, normalized size of antiderivative = 1.60, number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2056, 107, 106, 490, 1211, 220, 1699, 203, 206} \begin {gather*} \frac {\sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x-b}}{\sqrt [4]{b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {2} a \sqrt [4]{b} \sqrt [4]{a x^3-b x^2}}-\frac {\sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x-b}}{\sqrt [4]{b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {2} a \sqrt [4]{b} \sqrt [4]{a x^3-b x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 106
Rule 107
Rule 203
Rule 206
Rule 220
Rule 490
Rule 1211
Rule 1699
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {x} (-2 b+a x) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \int \frac {1}{\sqrt {\frac {a x}{b}} (-2 b+a x) \sqrt [4]{-b+a x}} \, dx}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (4 \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a b-a x^4\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{\sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (2 \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {b}-x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{a \sqrt [4]{-b x^2+a x^3}}+\frac {\left (2 \sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {b}+x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{a \sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\left (\sqrt {b}+x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{a \sqrt {b} \sqrt [4]{-b x^2+a x^3}}-\frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\left (\sqrt {b}-x^2\right ) \sqrt {1+\frac {x^4}{b}}} \, dx,x,\sqrt [4]{-b+a x}\right )}{a \sqrt {b} \sqrt [4]{-b x^2+a x^3}}\\ &=-\frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-2 x^2} \, dx,x,\frac {\sqrt [4]{-b+a x}}{\sqrt {\frac {a x}{b}}}\right )}{a \sqrt [4]{-b x^2+a x^3}}+\frac {\left (\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+2 x^2} \, dx,x,\frac {\sqrt [4]{-b+a x}}{\sqrt {\frac {a x}{b}}}\right )}{a \sqrt [4]{-b x^2+a x^3}}\\ &=\frac {\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-b+a x}}{\sqrt [4]{b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {2} a \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}-\frac {\sqrt {\frac {a x}{b}} \sqrt [4]{-b+a x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-b+a x}}{\sqrt [4]{b} \sqrt {\frac {a x}{b}}}\right )}{\sqrt {2} a \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.05, size = 61, normalized size = 0.58 \begin {gather*} -\frac {x \sqrt [4]{\frac {b-a x}{b}} F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {a x}{b},\frac {a x}{2 b}\right )}{b \sqrt [4]{x^2 (a x-b)}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.34, size = 106, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}{\sqrt {a} x}\right )}{\sqrt {2} \sqrt {a} b^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}{\sqrt {a} x}\right )}{\sqrt {2} \sqrt {a} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x - 2 \, b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a x -2 b \right ) \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x - 2 \, b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{\left (2\,b-a\,x\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (a x - b\right )} \left (a x - 2 b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________