Optimal. Leaf size=39 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x^3-b x}}{a x^2-b}\right )}{\sqrt {c}} \]
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Rubi [C] time = 1.99, antiderivative size = 254, normalized size of antiderivative = 6.51, number of steps used = 14, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2056, 6728, 329, 224, 221, 933, 168, 537} \begin {gather*} -\frac {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {2 \sqrt {a} \sqrt {b}}{c-\sqrt {c^2+4 a b}};\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 {2 \sqrt [4]{b} \sqrt {x} \sqrt {1-\frac {a x^2}{b}} \Pi \left (-\frac {2 \sqrt {a} \sqrt {b}}{c+\sqrt {c^2+4 a b}};\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 {2 \sqrt [4]{b} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 168
Rule 221
Rule 224
Rule 329
Rule 537
Rule 933
Rule 2056
Rule 6728
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 ) \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 (-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 {\left (2 \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 {2 \sqrt [4]{b} \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 (-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 {2 \sqrt [4]{b} \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 (2 \left (-c-\sqrt {4 a b+c^2}\right ) \sqrt {x} \sqrt {1-\frac {a x^2}{b}}\right ) \operatorname {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 ) \operatorname {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 {2 \sqrt [4]{b} \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 {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 [C] time = 1.65, size = 204, normalized size = 5.23 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {1-\frac {b}{a x^2}} \left (-\Pi \left (\frac {2 \sqrt {a} \sqrt {b}}{c-\sqrt {c^2+4 a b}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {2 \sqrt {a} \sqrt {b}}{c+\sqrt {c^2+4 a b}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )+F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \sqrt {a x^3-b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 39, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x^3-b x}}{a x^2-b}\right )}{\sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 172, normalized size = 4.41 \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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} + b}{\sqrt {a x^{3} - b x} {\left (a x^{2} + c x - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 1106, normalized size = 28.36
result too large to display
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 x^{2} + b}{\sqrt {a x^{3} - b x} {\left (a x^{2} + c x - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.19, size = 51, normalized size = 1.31 \begin {gather*} \frac {\ln \left (\frac {b+c\,x-a\,x^2-\sqrt {c}\,\sqrt {a\,x^3-b\,x}\,2{}\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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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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