Optimal. Leaf size=36 \[ -\frac {2 \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt {c}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 1.35, antiderivative size = 289, normalized size of antiderivative = 8.03, number of steps
used = 13, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 6860,
335, 226, 947, 174, 551} \begin {gather*} -\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {\frac {a x^2}{b}+1} \Pi \left (\frac {2 \sqrt {-a} \sqrt {b}}{c-\sqrt {c^2-4 a b}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt {a x^3+b x}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {\frac {a x^2}{b}+1} \Pi \left (\frac {2 \sqrt {-a} \sqrt {b}}{c+\sqrt {c^2-4 a b}};\left .\text {ArcSin}\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 {x} \left (\sqrt {a} x+\sqrt {b}\right ) \sqrt {\frac {a x^2+b}{\left (\sqrt {a} x+\sqrt {b}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 174
Rule 226
Rule 335
Rule 551
Rule 947
Rule 2081
Rule 6860
Rubi steps
\begin {align*} \int \frac {-b+a x^2}{\left (b+c x+a x^2\right ) \sqrt {b x+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {-b+a x^2}{\sqrt {x} \sqrt {b+a x^2} \left (b+c x+a x^2\right )} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {b+a x^2}}-\frac {2 b+c x}{\sqrt {x} \sqrt {b+a x^2} \left (b+c x+a x^2\right )}\right ) \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}}-\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {2 b+c x}{\sqrt {x} \sqrt {b+a x^2} \left (b+c x+a x^2\right )} \, dx}{\sqrt {b x+a x^3}}\\ &=-\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \left (\frac {c-\sqrt {-4 a b+c^2}}{\sqrt {x} \left (c-\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {b+a x^2}}+\frac {c+\sqrt {-4 a b+c^2}}{\sqrt {x} \left (c+\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {b+a x^2}}\right ) \, dx}{\sqrt {b x+a x^3}}+\frac {\left (2 \sqrt {x} \sqrt {b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (\left (c-\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (c-\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}}-\frac {\left (\left (c+\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (c+\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (\left (c-\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (c-\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x}{\sqrt {b}}}} \, dx}{\sqrt {b x+a x^3}}-\frac {\left (\left (c+\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (c+\sqrt {-4 a b+c^2}+2 a x\right ) \sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x}{\sqrt {b}}}} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}+\frac {\left (2 \left (c-\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (-c+\sqrt {-4 a b+c^2}-2 a x^2\right ) \sqrt {1-\frac {\sqrt {-a} x^2}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x^2}{\sqrt {b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}}+\frac {\left (2 \left (c+\sqrt {-4 a b+c^2}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (-c-\sqrt {-4 a b+c^2}-2 a x^2\right ) \sqrt {1-\frac {\sqrt {-a} x^2}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x^2}{\sqrt {b}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {b x+a x^3}}\\ &=\frac {\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1+\frac {a x^2}{b}} \Pi \left (\frac {2 \sqrt {-a} \sqrt {b}}{c-\sqrt {-4 a b+c^2}};\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 {2 \sqrt [4]{b} \sqrt {x} \sqrt {1+\frac {a x^2}{b}} \Pi \left (\frac {2 \sqrt {-a} \sqrt {b}}{c+\sqrt {-4 a b+c^2}};\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}}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 59, normalized size = 1.64 \begin {gather*} -\frac {2 \sqrt {x} \sqrt {b+a x^2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b+a x^2}}\right )}{\sqrt {c} \sqrt {x \left (b+a x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.90, size = 1161, normalized size = 32.25 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 159, normalized size = 4.42 \begin {gather*} \left [-\frac {\sqrt {-c} \log \left (\frac {a^{2} x^{4} - 6 \, a c x^{3} - 6 \, b c x + {\left (2 \, a b + c^{2}\right )} x^{2} + b^{2} - 4 \, \sqrt {a x^{3} + b x} {\left (a x^{2} - c x + b\right )} \sqrt {-c}}{a^{2} x^{4} + 2 \, a c x^{3} + 2 \, b c x + {\left (2 \, a b + c^{2}\right )} x^{2} + b^{2}}\right )}{2 \, c}, \frac {\arctan \left (\frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - c x + b\right )} \sqrt {c}}{2 \, {\left (a c x^{3} + b c x\right )}}\right )}{\sqrt {c}}\right ] \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 {a x^{2} - b}{\sqrt {x \left (a x^{2} + b\right )} \left (a x^{2} + b + c x\right )}\, dx \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.25, size = 51, normalized size = 1.42 \begin {gather*} \frac {\ln \left (\frac {\frac {b}{2}-\frac {c\,x}{2}+\frac {a\,x^2}{2}+\sqrt {c}\,\sqrt {a\,x^3+b\,x}\,1{}\mathrm {i}}{a\,x^2+c\,x+b}\right )\,1{}\mathrm {i}}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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