Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {20 c \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {40 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 \left (b^2-4 a c\right )^{7/4} \sqrt {d} \sqrt {a+b x+c x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {701, 705, 703, 227} \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {40 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 \sqrt {d} \left (b^2-4 a c\right )^{7/4} \sqrt {a+b x+c x^2}}+\frac {20 c \sqrt {b d+2 c d x}}{3 d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {b d+2 c d x}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rule 227
Rule 701
Rule 703
Rule 705
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}-\frac {(10 c) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )} \\ & = -\frac {2 \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {20 c \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {\left (20 c^2\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2} \\ & = -\frac {2 \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {20 c \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {\left (20 c^2 \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}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {20 c \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {\left (40 c \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 )}{3 \left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {20 c \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {40 c \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 \left (b^2-4 a c\right )^{7/4} \sqrt {d} \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 = 10.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d (b+2 c x)} \left (b^2-10 b c x-2 c \left (7 a+5 c x^2\right )-20 c (a+x (b+c x)) \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 \left (b^2-4 a c\right )^2 d (a+x (b+c x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(490\) vs. \(2(159)=318\).
Time = 6.37 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.63
method | result | size |
default | \(\frac {2 \sqrt {d \left (2 c x +b \right )}\, \left (10 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, c^{2} x^{2}+10 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, b c x +10 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a c +20 c^{3} x^{3}+30 b \,c^{2} x^{2}+28 a \,c^{2} x +8 b^{2} c x +14 a b c -b^{3}\right )}{3 d \left (2 c x +b \right ) \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(491\) |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{3 \left (4 a c -b^{2}\right ) d \,c^{2} \left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right )^{2}}+\frac {\frac {40}{3} c^{2} d x +\frac {20}{3} b c d}{\left (4 a c -b^{2}\right )^{2} d \sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (2 c^{2} d x +b c d \right )}}+\frac {40 c^{2} \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(556\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (10 \, \sqrt {2} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (10 \, c^{2} x^{2} + 10 \, b c x - b^{2} + 14 \, a c\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x + {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}} \]
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\[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {d \left (b + 2 c x\right )} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{\sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{\sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {b\,d+2\,c\,d\,x}\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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