Optimal. Leaf size=133 \[ \frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a} e+\sqrt {b} c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1876, 1167, 205, 208, 1248, 635, 260} \[ \frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a} e+\sqrt {b} c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 208
Rule 260
Rule 635
Rule 1167
Rule 1248
Rule 1876
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx &=\int \left (\frac {c+e x^2}{a-b x^4}+\frac {x \left (d+f x^2\right )}{a-b x^4}\right ) \, dx\\ &=\int \frac {c+e x^2}{a-b x^4} \, dx+\int \frac {x \left (d+f x^2\right )}{a-b x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {d+f x}{a-b x^2} \, dx,x,x^2\right )+\frac {1}{2} \left (-\frac {\sqrt {b} c}{\sqrt {a}}+e\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx+\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}+e\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx\\ &=\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {1}{2} d \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )+\frac {1}{2} f \operatorname {Subst}\left (\int \frac {x}{a-b x^2} \, dx,x,x^2\right )\\ &=\frac {\left (\sqrt {b} c-\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 214, normalized size = 1.61 \[ -\frac {\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{3/4} e+\sqrt [4]{a} \sqrt {b} c+\sqrt {a} \sqrt [4]{b} d\right )}{4 a b^{3/4}}-\frac {\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-a^{3/4} e-\sqrt [4]{a} \sqrt {b} c+\sqrt {a} \sqrt [4]{b} d\right )}{4 a b^{3/4}}+\frac {\left (\sqrt [4]{a} \sqrt {b} c-a^{3/4} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a b^{3/4}}+\frac {d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.19, size = 280, normalized size = 2.11 \[ -\frac {\sqrt {2} {\left (b^{2} c - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {f \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 177, normalized size = 1.33 \[ -\frac {d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{4 \sqrt {a b}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 a}-\frac {e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}-\frac {f \ln \left (b \,x^{4}-a \right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.03, size = 174, normalized size = 1.31 \[ \frac {{\left (\sqrt {b} c - \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{4 \, \sqrt {a} b} - \frac {{\left (\sqrt {b} d + \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{4 \, \sqrt {a} b} - \frac {{\left (\sqrt {b} c + \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.66, size = 1970, normalized size = 14.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________