Integrand size = 28, antiderivative size = 328 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (b^2-4 a c\right )^{15/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{156 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{15/4} \sqrt {d} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{156 c^4 \sqrt {a+b x+c x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {699, 705, 704, 313, 227, 1213, 435} \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {d} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{156 c^4 \sqrt {a+b x+c x^2}}-\frac {\sqrt {d} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{156 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}}{234 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^{3/2}}{13 c d} \]
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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} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx}{26 c} \\ & = -\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}+\frac {\left (5 \left (b^2-4 a c\right )^2\right ) \int \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2} \, dx}{156 c^2} \\ & = \frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (b^2-4 a c\right )^3 \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{312 c^3} \\ & = \frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (\left (b^2-4 a c\right )^3 \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}{312 c^3 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (\left (b^2-4 a c\right )^3 \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 )}{156 c^4 d \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}+\frac {\left (\left (b^2-4 a c\right )^{7/2} \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 )}{156 c^4 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{7/2} \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 )}{156 c^4 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}+\frac {\left (b^2-4 a c\right )^{15/4} \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 )}{156 c^4 \sqrt {a+b x+c x^2}}-\frac {\left (\left (b^2-4 a c\right )^{7/2} \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 )}{156 c^4 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{156 c^3 d}-\frac {5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d}+\frac {(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 c d}-\frac {\left (b^2-4 a c\right )^{15/4} \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 )}{156 c^4 \sqrt {a+b x+c x^2}}+\frac {\left (b^2-4 a c\right )^{15/4} \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 )}{156 c^4 \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 = 101, normalized size of antiderivative = 0.31 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\left (b^2-4 a c\right )^2 (d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3}{4},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{96 c^3 d \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(923\) vs. \(2(278)=556\).
Time = 3.09 (sec) , antiderivative size = 924, normalized size of antiderivative = 2.82
method | result | size |
default | \(\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, \left (1184 a \,c^{7} x^{6}+1888 a^{2} c^{6} x^{4}+992 a^{3} c^{5} x^{2}+768 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 ) \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}}}}\, a^{4} c^{4}+3 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 ) \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}}}}\, b^{8}-68 a^{2} b^{4} c^{2}+1072 x^{3} a \,b^{3} c^{4}-768 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 ) \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}}}}\, a^{3} b^{2} c^{3}+288 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 ) \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}}}}\, a^{2} b^{4} c^{2}-48 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 ) \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}}}}\, a \,b^{6} c +200 a^{2} c^{3} b^{3} x -120 x^{2} a \,b^{4} c^{3}+248 a^{3} b^{2} c^{3}+2088 x^{2} a^{2} b^{2} c^{4}+268 b^{4} c^{4} x^{4}+1720 b^{2} c^{6} x^{6}+10 b^{6} c^{2} x^{2}+1128 b^{3} c^{5} x^{5}+1152 b \,c^{7} x^{7}+288 c^{8} x^{8}+992 a^{3} b \,c^{4} x -64 a \,b^{5} c^{2} x +6 a \,b^{6} c +6 b^{7} c x +3552 a b \,c^{6} x^{5}+3496 a \,b^{2} c^{5} x^{4}+3776 a^{2} b \,c^{5} x^{3}\right )}{936 c^{4} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(924\) |
risch | \(\text {Expression too large to display}\) | \(2230\) |
elliptic | \(\text {Expression too large to display}\) | \(3160\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.67 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {3 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (72 \, c^{6} x^{5} + 180 \, b c^{5} x^{4} + 3 \, b^{5} c - 34 \, a b^{3} c^{2} + 124 \, a^{2} b c^{3} + 4 \, {\left (31 \, b^{2} c^{4} + 56 \, a c^{5}\right )} x^{3} + 6 \, {\left (b^{3} c^{3} + 56 \, a b c^{4}\right )} x^{2} - 4 \, {\left (b^{4} c^{2} - 11 \, a b^{2} c^{3} - 62 \, a^{2} c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{468 \, c^{4}} \]
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\[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\int \sqrt {d \left (b + 2 c x\right )} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \]
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\[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\int { \sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \,d x } \]
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\[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\int { \sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\int \sqrt {b\,d+2\,c\,d\,x}\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]
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