Optimal. Leaf size=159 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2 a b+c} \sqrt {a x^3-b x}}{x \sqrt {2 a b+c}-a x^2+b}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}}-\frac {\tanh ^{-1}\left (\frac {\frac {a x^2}{\sqrt {2} \sqrt [4]{2 a b+c}}+\frac {x \sqrt [4]{2 a b+c}}{\sqrt {2}}-\frac {b}{\sqrt {2} \sqrt [4]{2 a b+c}}}{\sqrt {a x^3-b x}}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}} \]
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Rubi [C] time = 3.22, antiderivative size = 880, normalized size of antiderivative = 5.53, number of steps used = 21, number of rules used = 10, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2056, 1586, 6715, 6728, 406, 224, 221, 409, 1219, 1218} \begin {gather*} \frac {\sqrt [4]{b} \left (1-\frac {2 a b-c}{\sqrt {c^2-4 a^2 b^2}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a x^3-b x}}+\frac {\sqrt [4]{b} \left (\frac {2 a b-c}{\sqrt {c^2-4 a^2 b^2}}+1\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a x^3-b x}}+\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c+\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c-\sqrt {c^2-4 a^2 b^2}}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c+\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^3-b x}}+\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c+\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c-\sqrt {c^2-4 a^2 b^2}}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c+\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^3-b x}}-\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c-\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {\sqrt {c^2-4 a^2 b^2}-c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c-\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^3-b x}}-\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c-\sqrt {c^2-4 a^2 b^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {\sqrt {c^2-4 a^2 b^2}-c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {c^2-4 a^2 b^2} \left (c-\sqrt {c^2-4 a^2 b^2}\right ) \sqrt {a x^3-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 406
Rule 409
Rule 1218
Rule 1219
Rule 1586
Rule 2056
Rule 6715
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b^2+a^2 x^4}{\sqrt {-b x+a x^3} \left (b^2+c x^2+a^2 x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {-b^2+a^2 x^4}{\sqrt {x} \sqrt {-b+a x^2} \left (b^2+c x^2+a^2 x^4\right )} \, dx}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-b+a x^2}\right ) \int \frac {\sqrt {-b+a x^2} \left (b+a x^2\right )}{\sqrt {x} \left (b^2+c x^2+a^2 x^4\right )} \, dx}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b+a x^4} \left (b+a x^4\right )}{b^2+c x^4+a^2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (a+\frac {a (2 a b-c)}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {-b+a x^4}}{c-\sqrt {-4 a^2 b^2+c^2}+2 a^2 x^4}+\frac {\left (a-\frac {a (2 a b-c)}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {-b+a x^4}}{c+\sqrt {-4 a^2 b^2+c^2}+2 a^2 x^4}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (2 a \left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b+a x^4}}{c+\sqrt {-4 a^2 b^2+c^2}+2 a^2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}+\frac {\left (2 a \left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b+a x^4}}{c-\sqrt {-4 a^2 b^2+c^2}+2 a^2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=\frac {\left (\left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}+\frac {\left (\left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}-\frac {\left (\left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+a x^4} \left (c-\sqrt {-4 a^2 b^2+c^2}+2 a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}-\frac {\left (\left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+a x^4} \left (c+\sqrt {-4 a^2 b^2+c^2}+2 a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=-\frac {\left (\left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {2} a x^2}{\sqrt {-c+\sqrt {-4 a^2 b^2+c^2}}}\right ) \sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}-\frac {\left (\left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {2} a x^2}{\sqrt {-c+\sqrt {-4 a^2 b^2+c^2}}}\right ) \sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}-\frac {\left (\left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {2} a x^2}{\sqrt {-c-\sqrt {-4 a^2 b^2+c^2}}}\right ) \sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}-\frac {\left (\left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {2} a x^2}{\sqrt {-c-\sqrt {-4 a^2 b^2+c^2}}}\right ) \sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}+\frac {\left (\left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a x^4}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}+\frac {\left (\left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a x^4}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^3}}\\ &=\frac {\sqrt [4]{b} \left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}+\frac {\sqrt [4]{b} \left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}-\frac {\left (\left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {2} a x^2}{\sqrt {-c+\sqrt {-4 a^2 b^2+c^2}}}\right ) \sqrt {1-\frac {a x^4}{b}}} \, dx,x,\sqrt {x}\right )}{2 \left (c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}-\frac {\left (\left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {2} a x^2}{\sqrt {-c+\sqrt {-4 a^2 b^2+c^2}}}\right ) \sqrt {1-\frac {a x^4}{b}}} \, dx,x,\sqrt {x}\right )}{2 \left (c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}-\frac {\left (\left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {2} a x^2}{\sqrt {-c-\sqrt {-4 a^2 b^2+c^2}}}\right ) \sqrt {1-\frac {a x^4}{b}}} \, dx,x,\sqrt {x}\right )}{2 \left (c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}-\frac {\left (\left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \left (2 a b+c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {2} a x^2}{\sqrt {-c-\sqrt {-4 a^2 b^2+c^2}}}\right ) \sqrt {1-\frac {a x^4}{b}}} \, dx,x,\sqrt {x}\right )}{2 \left (c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}\\ &=\frac {\sqrt [4]{b} \left (1-\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}+\frac {\sqrt [4]{b} \left (1+\frac {2 a b-c}{\sqrt {-4 a^2 b^2+c^2}}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-b x+a x^3}}+\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c+\sqrt {-4 a^2 b^2+c^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c-\sqrt {-4 a^2 b^2+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-4 a^2 b^2+c^2} \left (c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}+\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c+\sqrt {-4 a^2 b^2+c^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c-\sqrt {-4 a^2 b^2+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-4 a^2 b^2+c^2} \left (c+\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}-\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c-\sqrt {-4 a^2 b^2+c^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c+\sqrt {-4 a^2 b^2+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-4 a^2 b^2+c^2} \left (c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}-\frac {\sqrt [4]{b} \left (4 a^2 b^2-c \left (c-\sqrt {-4 a^2 b^2+c^2}\right )\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b}}{\sqrt {-c+\sqrt {-4 a^2 b^2+c^2}}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {-4 a^2 b^2+c^2} \left (c-\sqrt {-4 a^2 b^2+c^2}\right ) \sqrt {-b x+a x^3}}\\ \end {align*}
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Mathematica [C] time = 1.49, size = 396, normalized size = 2.49 \begin {gather*} -\frac {i \sqrt {1-\frac {b}{a x^2}} \sqrt {a x^3-b x} \left (-\Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {b} \sqrt {\frac {\sqrt {c^2-4 a^2 b^2}-c}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {b} \sqrt {\frac {\sqrt {c^2-4 a^2 b^2}-c}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {\sqrt {2} \sqrt {a}}{\sqrt {b} \sqrt {-\frac {c+\sqrt {c^2-4 a^2 b^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {2} \sqrt {a}}{\sqrt {b} \sqrt {-\frac {c+\sqrt {c^2-4 a^2 b^2}}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )+2 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )\right )}{x^{3/2} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \left (a-\frac {b}{x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.99, size = 159, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2 a b+c} \sqrt {-b x+a x^3}}{b+\sqrt {2 a b+c} x-a x^2}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b}{\sqrt {2} \sqrt [4]{2 a b+c}}+\frac {\sqrt [4]{2 a b+c} x}{\sqrt {2}}+\frac {a x^2}{\sqrt {2} \sqrt [4]{2 a b+c}}}{\sqrt {-b x+a x^3}}\right )}{\sqrt {2} \sqrt [4]{2 a b+c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 401, normalized size = 2.52 \begin {gather*} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a x^{3} - b x} {\left (2 \, a b + c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}}{a x^{2} - b}\right ) + \frac {1}{4} \, \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - {\left (4 \, a b + c\right )} x^{2} + b^{2} + 2 \, \sqrt {a x^{3} - b x} {\left ({\left (2 \, a b + c\right )} x \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} - {\left (2 \, a b^{2} - {\left (2 \, a^{2} b + a c\right )} x^{2} + b c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}\right )} - 2 \, {\left ({\left (2 \, a^{2} b + a c\right )} x^{3} - {\left (2 \, a b^{2} + b c\right )} x\right )} \sqrt {-\frac {1}{2 \, a b + c}}}{a^{2} x^{4} + c x^{2} + b^{2}}\right ) - \frac {1}{4} \, \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - {\left (4 \, a b + c\right )} x^{2} + b^{2} - 2 \, \sqrt {a x^{3} - b x} {\left ({\left (2 \, a b + c\right )} x \left (-\frac {1}{2 \, a b + c}\right )^{\frac {1}{4}} - {\left (2 \, a b^{2} - {\left (2 \, a^{2} b + a c\right )} x^{2} + b c\right )} \left (-\frac {1}{2 \, a b + c}\right )^{\frac {3}{4}}\right )} - 2 \, {\left ({\left (2 \, a^{2} b + a c\right )} x^{3} - {\left (2 \, a b^{2} + b c\right )} x\right )} \sqrt {-\frac {1}{2 \, a b + c}}}{a^{2} x^{4} + c x^{2} + b^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} x^{4} - b^{2}}{{\left (a^{2} x^{4} + c x^{2} + b^{2}\right )} \sqrt {a x^{3} - b x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 374, normalized size = 2.35
method | result | size |
elliptic | \(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{4}+c \,\textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} c +2 b^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha +c \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {a b}\, a b -\sqrt {a b}\, c \right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {-\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha a b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha c +a \,b^{2}+b c}{b \left (2 a b +c \right )}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+c \right ) \left (2 a b +c \right ) \sqrt {x \left (a \,x^{2}-b \right )}}\right )}{2 a b}\) | \(374\) |
default | \(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{4}+c \,\textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} c -2 b^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha +c \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {a b}\, a b -\sqrt {a b}\, c \right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {-\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha a b -\sqrt {a b}\, \underline {\hspace {1.25 ex}}\alpha c +a \,b^{2}+b c}{b \left (2 a b +c \right )}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+c \right ) \left (2 a b +c \right ) \sqrt {x \left (a \,x^{2}-b \right )}}\right )}{2 a b}\) | \(375\) |
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*} \int \frac {a^{2} x^{4} - b^{2}}{{\left (a^{2} x^{4} + c x^{2} + b^{2}\right )} \sqrt {a x^{3} - b x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.81, size = 165, normalized size = 1.04 \begin {gather*} \frac {\ln \left (\frac {b-x\,\sqrt {-c-2\,a\,b}+2\,\sqrt {a\,x^3-b\,x}\,{\left (-c-2\,a\,b\right )}^{1/4}-a\,x^2}{b+x\,\sqrt {-c-2\,a\,b}-a\,x^2}\right )}{2\,{\left (-c-2\,a\,b\right )}^{1/4}}+\frac {\ln \left (\frac {b+x\,\sqrt {-c-2\,a\,b}-a\,x^2-\sqrt {a\,x^3-b\,x}\,{\left (-c-2\,a\,b\right )}^{1/4}\,2{}\mathrm {i}}{x\,\sqrt {-c-2\,a\,b}-b+a\,x^2}\right )\,1{}\mathrm {i}}{2\,{\left (-c-2\,a\,b\right )}^{1/4}} \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 {\left (a x^{2} - b\right ) \left (a x^{2} + b\right )}{\sqrt {x \left (a x^{2} - b\right )} \left (a^{2} x^{4} + b^{2} + c x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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