Integrand size = 28, antiderivative size = 174 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx=\frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{6 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{6 c^3 d^{5/2} \sqrt {a+b x+c x^2}} \]
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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 = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {698, 699, 705, 703, 227} \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx=-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{6 c^3 d^{5/2} \sqrt {a+b x+c x^2}}+\frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{6 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}} \]
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Rule 227
Rule 698
Rule 699
Rule 703
Rule 705
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx}{2 c d^2} \\ & = \frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{6 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\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}{12 c^2 d^2} \\ & = \frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{6 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\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}{12 c^2 d^2 \sqrt {a+b x+c x^2}} \\ & = \frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{6 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\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 {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 )}{6 c^3 d^3 \sqrt {a+b x+c x^2}} \\ & = \frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{6 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\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 )}{6 c^3 d^{5/2} \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 = 99, normalized size of antiderivative = 0.57 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx=\frac {\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},\frac {1}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{24 c^2 d (d (b+2 c x))^{3/2} \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(572\) vs. \(2(146)=292\).
Time = 3.92 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.29
method | result | size |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {\left (4 a c -b^{2}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{48 d^{3} c^{4} \left (x +\frac {b}{2 c}\right )^{2}}+\frac {\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{12 d^{3} c^{2}}+\frac {2 \left (\frac {8 a c -b^{2}}{16 d^{2} c^{2}}-\frac {4 a c -b^{2}}{48 c^{2} d^{2}}-\frac {a c d +\frac {1}{2} b^{2} d}{12 d^{3} c^{2}}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(573\) |
default | \(\frac {\left (8 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a \,c^{2} x -2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{2} c x +4 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a b c -\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{3}+4 c^{4} x^{4}+8 b \,c^{3} x^{3}+6 b^{2} c^{2} x^{2}+2 b^{3} c x -4 a^{2} c^{2}+2 a \,b^{2} c \right ) \sqrt {d \left (2 c x +b \right )}}{12 d^{3} \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right )^{2} c^{3}}\) | \(626\) |
risch | \(\text {Expression too large to display}\) | \(1287\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx=-\frac {\sqrt {2} {\left (b^{4} - 4 \, a b^{2} c + 4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (2 \, c^{4} x^{2} + 2 \, b c^{3} x + b^{2} c^{2} - 2 \, a c^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{12 \, {\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{\left (d \left (b + 2 c x\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^{5/2}} \,d x \]
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