Optimal. Leaf size=147 \[ -\frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (2 \sqrt {a} \sqrt {b}-c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \]
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Rubi [C] time = 1.71, antiderivative size = 284, normalized size of antiderivative = 1.93, number of steps used = 13, number of rules used = 7, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2056, 6725, 329, 220, 933, 168, 537} \begin {gather*} -\frac {\sqrt {x} \left (\frac {c}{\sqrt {a}}+2 \sqrt {b}\right ) \sqrt {\frac {a x^2}{b}+1} \Pi \left (\frac {\sqrt {-a}}{\sqrt {a}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {a x^3+b x}}-\frac {\sqrt {x} \left (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {\frac {a x^2}{b}+1} \Pi \left (\frac {\sqrt {a}}{\sqrt {-a}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt [4]{b} \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 \tan ^{-1}\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*}
Warning: Unable to verify antiderivative.
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Rule 168
Rule 220
Rule 329
Rule 537
Rule 933
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {b-c x+a x^2}{\left (-b+a x^2\right ) \sqrt {b x+a x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {b-c x+a x^2}{\sqrt {x} \left (-b+a x^2\right ) \sqrt {b+a x^2}} \, 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} \left (-b+a x^2\right ) \sqrt {b+a x^2}}\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} \left (-b+a x^2\right ) \sqrt {b+a x^2}} \, dx}{\sqrt {b x+a x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {b+a x^2}\right ) \int \left (-\frac {2 b^{3/2}-\frac {b c}{\sqrt {a}}}{2 b \sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {b+a x^2}}-\frac {2 b^{3/2}+\frac {b c}{\sqrt {a}}}{2 b \sqrt {x} \left (\sqrt {b}+\sqrt {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 {\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 (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {b+a x^2}} \, dx}{2 \sqrt {b x+a x^3}}-\frac {\left (\left (2 \sqrt {b}+\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {b+a x^2}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {b+a x^2}} \, dx}{2 \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 (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}-\sqrt {a} x\right ) \sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x}{\sqrt {b}}}} \, dx}{2 \sqrt {b x+a x^3}}-\frac {\left (\left (2 \sqrt {b}+\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \int \frac {1}{\sqrt {x} \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {1-\frac {\sqrt {-a} x}{\sqrt {b}}} \sqrt {1+\frac {\sqrt {-a} x}{\sqrt {b}}}} \, dx}{2 \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 (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {b}+\sqrt {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 (\left (2 \sqrt {b}+\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {b}-\sqrt {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 {\left (2 \sqrt {b}+\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {-a}}{\sqrt {a}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {b x+a x^3}}-\frac {\left (2 \sqrt {b}-\frac {c}{\sqrt {a}}\right ) \sqrt {x} \sqrt {1+\frac {a x^2}{b}} \Pi \left (\frac {\sqrt {a}}{\sqrt {-a}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{-a} \sqrt {x}}{\sqrt [4]{b}}\right )\right |-1\right )}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {b x+a x^3}}\\ \end {align*}
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Mathematica [C] time = 0.51, size = 133, normalized size = 0.90 \begin {gather*} -\frac {2 x \sqrt {\frac {a x^2}{b}+1} \left (x \left (3 a x F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )-5 c F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )\right )+15 b F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};-\frac {a x^2}{b},\frac {a x^2}{b}\right )\right )}{15 b \sqrt {x \left (a x^2+b\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.44, size = 147, normalized size = 1.00 \begin {gather*} -\frac {\left (2 \sqrt {a} \sqrt {b}+c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (2 \sqrt {a} \sqrt {b}-c\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {a x^3+b x}}{a x^2+b}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 1553, normalized size = 10.56
result too large to display
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} - c x + b}{\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 710, normalized size = 4.83 \begin {gather*} \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 {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) c}{2 a^{2} \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}-\frac {\sqrt {-a b}\, \sqrt {\frac {x a}{\sqrt {-a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {-a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{a \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right ) b}{\sqrt {a b}\, a \sqrt {a \,x^{3}+b x}\, \left (-\frac {\sqrt {-a b}}{a}+\frac {\sqrt {a b}}{a}\right )} \end {gather*}
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} - c x + b}{\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \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 - c x}{\sqrt {x \left (a x^{2} + b\right )} \left (a x^{2} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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