Optimal. Leaf size=153 \[ -\frac {a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}}{\sqrt {a x^3-b}-\sqrt {b}}\right )}{6 \sqrt {2} b^{5/4}}-\frac {a \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^3-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt [4]{a x^3-b}}\right )}{6 \sqrt {2} b^{5/4}}+\frac {\left (a x^3-b\right )^{3/4}}{3 b x^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 228, normalized size of antiderivative = 1.49, number of steps used = 12, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {266, 51, 63, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {a \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {a x^3-b}+\sqrt {b}\right )}{12 \sqrt {2} b^{5/4}}-\frac {a \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^3-b}+\sqrt {a x^3-b}+\sqrt {b}\right )}{12 \sqrt {2} b^{5/4}}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{5/4}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^3-b}}{\sqrt [4]{b}}+1\right )}{6 \sqrt {2} b^{5/4}}+\frac {\left (a x^3-b\right )^{3/4}}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 204
Rule 266
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x}} \, dx,x,x^3\right )\\ &=\frac {\left (-b+a x^3\right )^{3/4}}{3 b x^3}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^3\right )}{12 b}\\ &=\frac {\left (-b+a x^3\right )^{3/4}}{3 b x^3}+\frac {\operatorname {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 b}\\ &=\frac {\left (-b+a x^3\right )^{3/4}}{3 b x^3}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{6 b}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{6 b}\\ &=\frac {\left (-b+a x^3\right )^{3/4}}{3 b x^3}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 \sqrt {2} b^{5/4}}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 \sqrt {2} b^{5/4}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 b}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{12 b}\\ &=\frac {\left (-b+a x^3\right )^{3/4}}{3 b x^3}+\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{5/4}}-\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{5/4}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{5/4}}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{5/4}}\\ &=\frac {\left (-b+a x^3\right )^{3/4}}{3 b x^3}-\frac {a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{5/4}}+\frac {a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{6 \sqrt {2} b^{5/4}}+\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{5/4}}-\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{12 \sqrt {2} b^{5/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 40, normalized size = 0.26 \begin {gather*} \frac {4 a \left (a x^3-b\right )^{3/4} \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};1-\frac {a x^3}{b}\right )}{9 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.33, size = 152, normalized size = 0.99 \begin {gather*} \frac {\left (-b+a x^3\right )^{3/4}}{3 b x^3}+\frac {a \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^3}}\right )}{6 \sqrt {2} b^{5/4}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}}{\sqrt {b}+\sqrt {-b+a x^3}}\right )}{6 \sqrt {2} b^{5/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 214, normalized size = 1.40 \begin {gather*} -\frac {4 \, b x^{3} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{3} b \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} - \sqrt {-a^{4} b^{3} \sqrt {-\frac {a^{4}}{b^{5}}} + \sqrt {a x^{3} - b} a^{6}} b \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}}}{a^{4}}\right ) - b x^{3} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (b^{4} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{3}\right ) + b x^{3} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {1}{4}} \log \left (-b^{4} \left (-\frac {a^{4}}{b^{5}}\right )^{\frac {3}{4}} + {\left (a x^{3} - b\right )}^{\frac {1}{4}} a^{3}\right ) - 4 \, {\left (a x^{3} - b\right )}^{\frac {3}{4}}}{12 \, b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 198, normalized size = 1.29 \begin {gather*} \frac {\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {5}{4}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {5}{4}}} - \frac {\sqrt {2} a^{2} \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {5}{4}}} + \frac {\sqrt {2} a^{2} \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {5}{4}}} + \frac {8 \, {\left (a x^{3} - b\right )}^{\frac {3}{4}} a}{b x^{3}}}{24 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (a \,x^{3}-b \right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 199, normalized size = 1.30 \begin {gather*} \frac {{\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{3} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{3} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{3} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} a}{24 \, b} + \frac {{\left (a x^{3} - b\right )}^{\frac {3}{4}} a}{3 \, {\left ({\left (a x^{3} - b\right )} b + b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.25, size = 72, normalized size = 0.47 \begin {gather*} \frac {{\left (a\,x^3-b\right )}^{3/4}}{3\,b\,x^3}-\frac {a\,\mathrm {atan}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{5/4}}+\frac {a\,\mathrm {atanh}\left (\frac {{\left (a\,x^3-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{6\,{\left (-b\right )}^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.14, size = 42, normalized size = 0.27 \begin {gather*} - \frac {\Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [4]{a} x^{\frac {15}{4}} \Gamma \left (\frac {9}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________