Integrand size = 28, antiderivative size = 174 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx=-\frac {\sqrt {a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {\sqrt [4]{b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{14 c^3 d^{9/2} \sqrt {a+b x+c x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 174, 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 = {698, 705, 703, 227} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx=\frac {\sqrt [4]{b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{14 c^3 d^{9/2} \sqrt {a+b x+c x^2}}-\frac {\sqrt {a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}} \]
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Rule 227
Rule 698
Rule 703
Rule 705
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {3 \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{5/2}} \, dx}{14 c d^2} \\ & = -\frac {\sqrt {a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {\int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{28 c^2 d^4} \\ & = -\frac {\sqrt {a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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}{28 c^2 d^4 \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {\sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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 )}{14 c^3 d^5 \sqrt {a+b x+c x^2}} \\ & = -\frac {\sqrt {a+b x+c x^2}}{14 c^2 d^3 (b d+2 c d x)^{3/2}}-\frac {\left (a+b x+c x^2\right )^{3/2}}{7 c d (b d+2 c d x)^{7/2}}+\frac {\sqrt [4]{b^2-4 a 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 )}{14 c^3 d^{9/2} \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 = 10.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx=\frac {\left (b^2-4 a c\right ) \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {3}{2},-\frac {3}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{56 c^2 d^5 (b+2 c x)^4 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(532\) vs. \(2(146)=292\).
Time = 4.24 (sec) , antiderivative size = 533, normalized size of antiderivative = 3.06
method | result | size |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {\left (4 a c -b^{2}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{448 c^{6} d^{5} \left (x +\frac {b}{2 c}\right )^{4}}-\frac {3 \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{112 d^{5} c^{4} \left (x +\frac {b}{2 c}\right )^{2}}+\frac {\left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{14 d^{4} c^{2} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(533\) |
default | \(\frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {d \left (2 c x +b \right )}\, \left (8 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) c^{3} x^{3}+12 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b \,c^{2} x^{2}+6 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{2} c x +\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{3}-12 c^{4} x^{4}-24 b \,c^{3} x^{3}-16 x^{2} c^{3} a -14 b^{2} c^{2} x^{2}-16 a b \,c^{2} x -2 b^{3} c x -4 a^{2} c^{2}-2 a \,b^{2} c \right )}{28 d^{5} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right ) \left (2 c x +b \right )^{3} c^{3}}\) | \(678\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx=\frac {\sqrt {2} {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (6 \, c^{4} x^{2} + 6 \, b c^{3} x + b^{2} c^{2} + 2 \, a c^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{28 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac {9}{2}}}\, dx \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{9/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{9/2}} \,d x \]
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