Integrand size = 28, antiderivative size = 236 \[ \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx=\frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}-\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{5 c^2 \sqrt {a+b x+c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {699, 705, 704, 313, 227, 1213, 435} \[ \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {d} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{5 c^2 \sqrt {a+b x+c x^2}}-\frac {\sqrt {d} \left (b^2-4 a c\right )^{7/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{5 c^2 \sqrt {a+b x+c x^2}}+\frac {\sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{5 c d} \]
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Rule 227
Rule 313
Rule 435
Rule 699
Rule 704
Rule 705
Rule 1213
Rubi steps \begin{align*} \text {integral}& = \frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{10 c} \\ & = \frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 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 {\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}{10 c \sqrt {a+b x+c x^2}} \\ & = \frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 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 ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 c^2 d \sqrt {a+b x+c x^2}} \\ & = \frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}+\frac {\left (\left (b^2-4 a c\right )^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {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 )}{5 c^2 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}{\sqrt {1-\frac {x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 c^2 \sqrt {a+b x+c x^2}} \\ & = \frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}+\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {d} \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 )}{5 c^2 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x^2}{\sqrt {b^2-4 a c} d}}}{\sqrt {1-\frac {x^2}{\sqrt {b^2-4 a c} d}}} \, dx,x,\sqrt {b d+2 c d x}\right )}{5 c^2 \sqrt {a+b x+c x^2}} \\ & = \frac {(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{5 c d}-\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 )}{5 c^2 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{7/4} \sqrt {d} \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 )}{5 c^2 \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.39 \[ \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx=\frac {(d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{6 c d \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(491\) vs. \(2(198)=396\).
Time = 2.71 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.08
method | result | size |
default | \(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, \left (16 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a^{2} c^{2}-8 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,b^{2} c +\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, E\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{4}+8 c^{4} x^{4}+16 b \,c^{3} x^{3}+8 x^{2} c^{3} a +10 b^{2} c^{2} x^{2}+8 a b \,c^{2} x +2 b^{3} c x +2 a \,b^{2} c \right )}{10 \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right ) c^{2}}\) | \(492\) |
elliptic | \(\text {Expression too large to display}\) | \(1070\) |
risch | \(\text {Expression too large to display}\) | \(1358\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.42 \[ \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {2} \sqrt {c^{2} d} {\left (b^{2} - 4 \, a c\right )} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (2 \, c^{2} x + b c\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{5 \, c^{2}} \]
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\[ \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx=\int \sqrt {d \left (b + 2 c x\right )} \sqrt {a + b x + c x^{2}}\, dx \]
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\[ \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a} \,d x } \]
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\[ \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a} \,d x } \]
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Timed out. \[ \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx=\int \sqrt {b\,d+2\,c\,d\,x}\,\sqrt {c\,x^2+b\,x+a} \,d x \]
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