Optimal. Leaf size=137 \[ \frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{3 c^2 \sqrt {d} \sqrt {a+b x+c x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {685, 691, 689, 221} \[ \frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{3 c^2 \sqrt {d} \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 685
Rule 689
Rule 691
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx &=\frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{3 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{6 c}\\ &=\frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{3 c d}-\frac {\left (\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {-\frac {a c}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {c^2 x^2}{b^2-4 a c}}} \, dx}{6 c \sqrt {a+b x+c x^2}}\\ &=\frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{3 c d}-\frac {\left (\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{3 c^2 d \sqrt {a+b x+c x^2}}\\ &=\frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{3 c d}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{3 c^2 \sqrt {d} \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 91, normalized size = 0.66 \[ \frac {\sqrt {a+x (b+c x)} \sqrt {d (b+2 c x)} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x + a}}{\sqrt {2 \, c d x + b d}}, x\right ) \]
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 \[ \int \frac {\sqrt {c x^{2} + b x + a}}{\sqrt {2 \, c d x + b d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 364, normalized size = 2.66 \[ \frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {\left (2 c x +b \right ) d}\, \left (4 c^{3} x^{3}+6 b \,c^{2} x^{2}+4 a \,c^{2} x +2 b^{2} c x +2 a b c +4 \sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-4 a c +b^{2}}\, a c \EllipticF \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )-\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-4 a c +b^{2}}\, b^{2} \EllipticF \left (\frac {\sqrt {\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right )\right )}{6 \left (2 c^{2} x^{3}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right ) c^{2} d} \]
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 \[ \int \frac {\sqrt {c x^{2} + b x + a}}{\sqrt {2 \, c d x + b d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c\,x^2+b\,x+a}}{\sqrt {b\,d+2\,c\,d\,x}} \,d x \]
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 \[ \int \frac {\sqrt {a + b x + c x^{2}}}{\sqrt {d \left (b + 2 c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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