Optimal. Leaf size=327 \[ 2 a \sqrt [4]{b x^3+a x^4}+\left (4 a^{9/4}-3 \sqrt [4]{a} b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\left (-4 a^{9/4}+3 \sqrt [4]{a} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )-\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-4 a^4 \log (x)+4 a^2 b \log (x)-b^2 \log (x)+4 a^4 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-4 a^2 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+b^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+2 a^3 \log (x) \text {$\#$1}^4-3 a b \log (x) \text {$\#$1}^4-2 a^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+3 a b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\& \right ] \]
[Out]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1092\) vs. \(2(327)=654\).
time = 1.23, antiderivative size = 1092, normalized size of antiderivative = 3.34, number of steps
used = 25, number of rules used = 12, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2081, 6860,
103, 163, 65, 338, 304, 209, 212, 95, 211, 214} \begin {gather*} \frac {\left (2 a^2-2 \sqrt {a^2+4 b} a-b\right ) \sqrt [4]{a x^4+b x^3} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right ) \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4+b x^3} \text {ArcTan}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right ) \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a^2-2 \sqrt {a^2+4 b} a-b\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right ) \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right ) \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\sqrt [4]{a x^4+b x^3} \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (b-2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{a x^4+b x^3} \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4+b x^3} \text {ArcTan}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (b-2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{a x^4+b x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 95
Rule 103
Rule 163
Rule 209
Rule 211
Rule 212
Rule 214
Rule 304
Rule 338
Rule 2081
Rule 6860
Rubi steps
\begin {align*} \int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x} (b+2 a x)}{-b+a x+x^2} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {\left (2 a-\frac {2 \left (a^2-b\right )}{\sqrt {a^2+4 b}}\right ) x^{3/4} \sqrt [4]{b+a x}}{a-\sqrt {a^2+4 b}+2 x}+\frac {\left (2 a+\frac {2 \left (a^2-b\right )}{\sqrt {a^2+4 b}}\right ) x^{3/4} \sqrt [4]{b+a x}}{a+\sqrt {a^2+4 b}+2 x}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (2 \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{b+a x}}{a-\sqrt {a^2+4 b}+2 x} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{b+a x}}{a+\sqrt {a^2+4 b}+2 x} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\frac {3}{4} b \left (a-\sqrt {a^2+4 b}\right )+\frac {1}{2} \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) x}{\sqrt [4]{x} \left (a-\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\frac {3}{4} b \left (a+\sqrt {a^2+4 b}\right )+\frac {1}{2} \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) x}{\sqrt [4]{x} \left (a+\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{4 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right ) \left (a^2-2 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a+\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{4 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (\frac {3}{2} b \left (a-\sqrt {a^2+4 b}\right )-\frac {1}{2} \left (a-\sqrt {a^2+4 b}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a-\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right ) \left (a^2-2 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2+4 b}-\left (-2 b+a \left (a+\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (\frac {3}{2} b \left (a-\sqrt {a^2+4 b}\right )-\frac {1}{2} \left (a-\sqrt {a^2+4 b}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{a-\sqrt {a^2+4 b}-\left (-2 b+a \left (a-\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right ) \sqrt {a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right ) \sqrt {a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}+\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (\frac {3}{2} b \left (a-\sqrt {a^2+4 b}\right )-\frac {1}{2} \left (a-\sqrt {a^2+4 b}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (\frac {3}{2} b \left (a-\sqrt {a^2+4 b}\right )-\frac {1}{2} \left (a-\sqrt {a^2+4 b}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}+\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2-b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2-b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (b-2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2-b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (b-2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2-b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 338, normalized size = 1.03 \begin {gather*} \frac {x^{9/4} (b+a x)^{3/4} \left (2 a x^{3/4} \sqrt [4]{b+a x}+\sqrt [4]{a} \left (4 a^2-3 b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\sqrt [4]{a} \left (-4 a^2+3 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\frac {1}{4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {4 a^4 \log (x)-4 a^2 b \log (x)+b^2 \log (x)-16 a^4 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+16 a^2 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-4 b^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-2 a^3 \log (x) \text {$\#$1}^4+3 a b \log (x) \text {$\#$1}^4+8 a^3 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4-12 a b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ]\right )}{\left (x^3 (b+a x)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (2 a x +b \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a x +x^{2}-b}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 21.39, size = 8527, normalized size = 26.08 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (2 a x + b\right )}{a x - b + x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 4.12, size = 243, normalized size = 0.74 \begin {gather*} -\frac {1}{2} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) + \frac {1}{4} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}\,\left (b+2\,a\,x\right )}{x^2+a\,x-b} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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