Optimal. Leaf size=120 \[ \frac {1}{4} \text {RootSum}\left [2 \text {$\#$1}^8-4 \text {$\#$1}^4 a-\text {$\#$1}^4 c+2 a^2-2 a b+a c\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))+a \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-a \log (x)}{-4 \text {$\#$1}^5+4 \text {$\#$1} a+\text {$\#$1} c}\& \right ] \]
________________________________________________________________________________________
Rubi [B] time = 0.99, antiderivative size = 529, normalized size of antiderivative = 4.41, number of steps used = 10, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {6728, 377, 212, 208, 205} \begin {gather*} \frac {a^{3/4} \left (\frac {2 b-c}{\sqrt {16 a b+c^2}}+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {16 a b+c^2}+4 b-c}}{\sqrt [4]{\sqrt {16 a b+c^2}-c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {16 a b+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {16 a b+c^2}+4 b-c}}-\frac {a^{3/4} \left (1-\frac {2 b-c}{\sqrt {16 a b+c^2}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {16 a b+c^2}-4 b+c}}{\sqrt [4]{\sqrt {16 a b+c^2}+c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {16 a b+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {16 a b+c^2}-4 b+c}}+\frac {a^{3/4} \left (\frac {2 b-c}{\sqrt {16 a b+c^2}}+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {16 a b+c^2}+4 b-c}}{\sqrt [4]{\sqrt {16 a b+c^2}-c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {16 a b+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {16 a b+c^2}+4 b-c}}-\frac {a^{3/4} \left (1-\frac {2 b-c}{\sqrt {16 a b+c^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {16 a b+c^2}-4 b+c}}{\sqrt [4]{\sqrt {16 a b+c^2}+c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {16 a b+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {16 a b+c^2}-4 b+c}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 208
Rule 212
Rule 377
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-2 b-c x^4+2 a x^8\right )} \, dx &=\int \left (\frac {2 a-\frac {2 a (2 b-c)}{\sqrt {16 a b+c^2}}}{\sqrt [4]{-b+a x^4} \left (-c-\sqrt {16 a b+c^2}+4 a x^4\right )}+\frac {2 a+\frac {2 a (2 b-c)}{\sqrt {16 a b+c^2}}}{\sqrt [4]{-b+a x^4} \left (-c+\sqrt {16 a b+c^2}+4 a x^4\right )}\right ) \, dx\\ &=\left (2 a \left (1-\frac {2 b-c}{\sqrt {16 a b+c^2}}\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-c-\sqrt {16 a b+c^2}+4 a x^4\right )} \, dx+\left (2 a \left (1+\frac {2 b-c}{\sqrt {16 a b+c^2}}\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-c+\sqrt {16 a b+c^2}+4 a x^4\right )} \, dx\\ &=\left (2 a \left (1-\frac {2 b-c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c-\sqrt {16 a b+c^2}-\left (4 a b+a \left (-c-\sqrt {16 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (2 a \left (1+\frac {2 b-c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c+\sqrt {16 a b+c^2}-\left (4 a b+a \left (-c+\sqrt {16 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\left (a \left (1+\frac {2 b-c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {16 a b+c^2}}-\sqrt {a} \sqrt {4 b-c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {-c+\sqrt {16 a b+c^2}}}+\frac {\left (a \left (1+\frac {2 b-c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {16 a b+c^2}}+\sqrt {a} \sqrt {4 b-c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {-c+\sqrt {16 a b+c^2}}}-\frac {\left (a \left (1-\frac {2 b-c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {16 a b+c^2}}-\sqrt {a} \sqrt {-4 b+c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {c+\sqrt {16 a b+c^2}}}-\frac {\left (a \left (1-\frac {2 b-c}{\sqrt {16 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {16 a b+c^2}}+\sqrt {a} \sqrt {-4 b+c+\sqrt {16 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt {c+\sqrt {16 a b+c^2}}}\\ &=\frac {a^{3/4} \left (1+\frac {2 b-c}{\sqrt {16 a b+c^2}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {16 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{\left (-c+\sqrt {16 a b+c^2}\right )^{3/4} \sqrt [4]{4 b-c+\sqrt {16 a b+c^2}}}-\frac {a^{3/4} \left (1-\frac {2 b-c}{\sqrt {16 a b+c^2}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{c+\sqrt {16 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{\left (c+\sqrt {16 a b+c^2}\right )^{3/4} \sqrt [4]{-4 b+c+\sqrt {16 a b+c^2}}}+\frac {a^{3/4} \left (1+\frac {2 b-c}{\sqrt {16 a b+c^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{4 b-c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {16 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{\left (-c+\sqrt {16 a b+c^2}\right )^{3/4} \sqrt [4]{4 b-c+\sqrt {16 a b+c^2}}}-\frac {a^{3/4} \left (1-\frac {2 b-c}{\sqrt {16 a b+c^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-4 b+c+\sqrt {16 a b+c^2}} x}{\sqrt [4]{c+\sqrt {16 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{\left (c+\sqrt {16 a b+c^2}\right )^{3/4} \sqrt [4]{-4 b+c+\sqrt {16 a b+c^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.44, size = 527, normalized size = 4.39 \begin {gather*} a^{3/4} \left (\frac {\left (\sqrt {16 a b+c^2}+2 b-c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {16 a b+c^2}+4 b-c}}{\sqrt [4]{\sqrt {16 a b+c^2}-c} \sqrt [4]{a x^4-b}}\right )}{\sqrt {16 a b+c^2} \left (\sqrt {16 a b+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {16 a b+c^2}+4 b-c}}-\frac {\left (\frac {c-2 b}{\sqrt {16 a b+c^2}}+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {16 a b+c^2}-4 b+c}}{\sqrt [4]{\sqrt {16 a b+c^2}+c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {16 a b+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {16 a b+c^2}-4 b+c}}+\frac {\left (\sqrt {16 a b+c^2}+2 b-c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {16 a b+c^2}+4 b-c}}{\sqrt [4]{\sqrt {16 a b+c^2}-c} \sqrt [4]{a x^4-b}}\right )}{\sqrt {16 a b+c^2} \left (\sqrt {16 a b+c^2}-c\right )^{3/4} \sqrt [4]{\sqrt {16 a b+c^2}+4 b-c}}-\frac {\left (\frac {c-2 b}{\sqrt {16 a b+c^2}}+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {16 a b+c^2}-4 b+c}}{\sqrt [4]{\sqrt {16 a b+c^2}+c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {16 a b+c^2}+c\right )^{3/4} \sqrt [4]{\sqrt {16 a b+c^2}-4 b+c}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 121, normalized size = 1.01 \begin {gather*} \frac {1}{4} \text {RootSum}\left [2 a^2-2 a b+a c-4 a \text {$\#$1}^4-c \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-4 a \text {$\#$1}-c \text {$\#$1}+4 \text {$\#$1}^5}\&\right ] \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 {2 \, a x^{4} - b}{{\left (2 \, a x^{8} - c x^{4} - 2 \, b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{4}-b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (2 a \,x^{8}-c \,x^{4}-2 b \right )}\, dx\]
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 {2 \, a x^{4} - b}{{\left (2 \, a x^{8} - c x^{4} - 2 \, b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{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 {b-2\,a\,x^4}{{\left (a\,x^4-b\right )}^{1/4}\,\left (-2\,a\,x^8+c\,x^4+2\,b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________