Integrand size = 28, antiderivative size = 325 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {a+b x+c x^2}}+\frac {56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {a+b x+c x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {701, 707, 705, 704, 313, 227, 1213, 435} \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt {a+b x+c x^2}}-\frac {56 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{d^{3/2} \left (b^2-4 a c\right )^{9/4} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 \sqrt {b d+2 c d x}}+\frac {28 c}{3 d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \sqrt {b d+2 c d x}} \]
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Rule 227
Rule 313
Rule 435
Rule 701
Rule 704
Rule 705
Rule 707
Rule 1213
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}-\frac {(14 c) \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {\left (28 c^2\right ) \int \frac {1}{(b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {\left (28 c^2\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^3 d^2} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {\left (28 c^2 \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}{\left (b^2-4 a c\right )^3 d^2 \sqrt {a+b x+c x^2}} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {\left (56 c \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 )}{\left (b^2-4 a c\right )^3 d^3 \sqrt {a+b x+c x^2}} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}+\frac {\left (56 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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt {a+b x+c x^2}}-\frac {\left (56 c \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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt {a+b x+c x^2}} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}+\frac {56 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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {a+b x+c x^2}}-\frac {\left (56 c \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 )}{\left (b^2-4 a c\right )^{5/2} d^2 \sqrt {a+b x+c x^2}} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}+\frac {28 c}{3 \left (b^2-4 a c\right )^2 d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}+\frac {112 c^2 \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d \sqrt {b d+2 c d x}}-\frac {56 c \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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \sqrt {a+b x+c x^2}}+\frac {56 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 )}{\left (b^2-4 a c\right )^{9/4} d^{3/2} \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 = 10.05 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.30 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {32 c \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {5}{2},\frac {3}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{\left (b^2-4 a c\right )^2 d \sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(876\) vs. \(2(281)=562\).
Time = 5.40 (sec) , antiderivative size = 877, normalized size of antiderivative = 2.70
method | result | size |
default | \(\frac {2 \left (168 \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}}}}\, 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 ) a \,c^{3} x^{2}-42 \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}}}}\, 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 ) b^{2} c^{2} x^{2}+168 \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}}}}\, 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 ) a b \,c^{2} x -42 \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}}}}\, 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 ) b^{3} c x +168 \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}}}}\, 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 ) a^{2} c^{2}-42 \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}}}}\, 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 ) a \,b^{2} c -168 c^{4} x^{4}-336 b \,c^{3} x^{3}-280 x^{2} c^{3} a -182 b^{2} c^{2} x^{2}-280 a b \,c^{2} x -14 b^{3} c x -96 a^{2} c^{2}-22 a \,b^{2} c +b^{4}\right ) \sqrt {d \left (2 c x +b \right )}}{3 d^{2} \left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(877\) |
elliptic | \(\text {Expression too large to display}\) | \(1220\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.58 \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (84 \, \sqrt {2} {\left (2 \, c^{4} x^{5} + 5 \, b c^{3} x^{4} + a^{2} b c + 4 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{3} + {\left (b^{3} c + 6 \, a b c^{2}\right )} x^{2} + 2 \, {\left (a b^{2} c + a^{2} c^{2}\right )} x\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (168 \, c^{4} x^{4} + 336 \, b c^{3} x^{3} - b^{4} + 22 \, a b^{2} c + 96 \, a^{2} c^{2} + 14 \, {\left (13 \, b^{2} c^{2} + 20 \, a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 20 \, a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (2 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{2} x^{5} + 5 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{2} x^{4} + 4 \, {\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} d^{2} x^{3} + {\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} d^{2} x^{2} + 2 \, {\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} d^{2} x + {\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} d^{2}\right )}} \]
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\[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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