Integrand size = 28, antiderivative size = 229 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=\frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{308 c^4 \sqrt {d} \sqrt {a+b x+c x^2}} \]
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Time = 0.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {699, 705, 703, 227} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=-\frac {5 \left (b^2-4 a c\right )^{13/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{308 c^4 \sqrt {d} \sqrt {a+b x+c x^2}}+\frac {5 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}}{154 c^2 d}+\frac {\left (a+b x+c x^2\right )^{5/2} \sqrt {b d+2 c d x}}{11 c d} \]
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Rule 227
Rule 699
Rule 703
Rule 705
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )\right ) \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx}{22 c} \\ & = -\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}+\frac {\left (15 \left (b^2-4 a c\right )^2\right ) \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx}{308 c^2} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {\left (5 \left (b^2-4 a c\right )^3\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{616 c^3} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {\left (5 \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 {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}{616 c^3 \sqrt {a+b x+c x^2}} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {\left (5 \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 {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 )}{308 c^4 d \sqrt {a+b x+c x^2}} \\ & = \frac {5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{308 c^3 d}-\frac {5 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{154 c^2 d}+\frac {\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}}{11 c d}-\frac {5 \left (b^2-4 a c\right )^{13/4} \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 )}{308 c^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 = 10.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.44 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=\frac {\left (b^2-4 a c\right )^2 \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{32 c^3 d \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(195)=390\).
Time = 2.68 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.40
method | result | size |
risch | \(\frac {\left (28 c^{4} x^{4}+56 b \,c^{3} x^{3}+96 x^{2} c^{3} a +18 b^{2} c^{2} x^{2}+96 a b \,c^{2} x -10 b^{3} c x +148 a^{2} c^{2}-50 a \,b^{2} c +5 b^{4}\right ) \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{308 c^{3} \sqrt {d \left (2 c x +b \right )}}+\frac {5 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{308 c^{3} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(549\) |
default | \(\frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {d \left (2 c x +b \right )}\, \left (112 c^{7} x^{7}+392 b \,c^{6} x^{6}+496 a \,c^{6} x^{5}+464 b^{2} c^{5} x^{5}+320 \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}}\, a^{3} c^{3}-240 \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}}\, a^{2} b^{2} c^{2}+60 \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}}\, a \,b^{4} c -5 \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^{6}+1240 a b \,c^{5} x^{4}+180 b^{3} c^{4} x^{4}+976 a^{2} c^{5} x^{3}+752 a \,b^{2} c^{4} x^{3}-4 b^{4} c^{3} x^{3}+1464 a^{2} b \,c^{4} x^{2}-112 a \,b^{3} c^{3} x^{2}+10 b^{5} c^{2} x^{2}+592 a^{3} c^{4} x +288 a^{2} b^{2} c^{3} x -100 c^{2} a \,b^{4} x +10 b^{6} c x +296 a^{3} c^{3} b -100 a^{2} c^{2} b^{3}+10 a \,b^{5} c \right )}{616 d \,c^{4} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(798\) |
elliptic | \(\text {Expression too large to display}\) | \(1885\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=-\frac {5 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (28 \, c^{6} x^{4} + 56 \, b c^{5} x^{3} + 5 \, b^{4} c^{2} - 50 \, a b^{2} c^{3} + 148 \, a^{2} c^{4} + 6 \, {\left (3 \, b^{2} c^{4} + 16 \, a c^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 48 \, a b c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{616 \, c^{5} d} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\sqrt {d \left (b + 2 c x\right )}}\, dx \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{\sqrt {2 \, c d x + b d}} \,d x } \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{\sqrt {2 \, c d x + b d}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{\sqrt {b\,d+2\,c\,d\,x}} \,d x \]
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