Integrand size = 28, antiderivative size = 326 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{130 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{130 c^3 \sqrt {a+b x+c x^2}} \]
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Time = 0.21 (sec) , antiderivative size = 326, 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 = {699, 706, 705, 704, 313, 227, 1213, 435} \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{130 c^3 \sqrt {a+b x+c x^2}}+\frac {d^{5/2} \left (b^2-4 a c\right )^{15/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )\right |-1\right )}{130 c^3 \sqrt {a+b x+c x^2}}+\frac {d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{3/2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{7/2}}{78 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{7/2}}{13 c d} \]
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Rule 227
Rule 313
Rule 435
Rule 699
Rule 704
Rule 705
Rule 706
Rule 1213
Rubi steps \begin{align*} \text {integral}& = \frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2} \, dx}{26 c} \\ & = -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {(b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}} \, dx}{156 c^2} \\ & = \frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\right ) \int \frac {\sqrt {b d+2 c d x}}{\sqrt {a+b x+c x^2}} \, dx}{260 c^2} \\ & = \frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d^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}{260 c^2 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d \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 )}{130 c^3 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}-\frac {\left (\left (b^2-4 a c\right )^{7/2} d^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 )}{130 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (\left (b^2-4 a c\right )^{7/2} d^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 )}{130 c^3 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}-\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \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 )}{130 c^3 \sqrt {a+b x+c x^2}}+\frac {\left (\left (b^2-4 a c\right )^{7/2} d^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 )}{130 c^3 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt {a+b x+c x^2}}{195 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2}}{78 c^2 d}+\frac {(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2}}{13 c d}+\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \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 )}{130 c^3 \sqrt {a+b x+c x^2}}-\frac {\left (b^2-4 a c\right )^{15/4} d^{5/2} \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 )}{130 c^3 \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 = 10.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.36 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2}{13} d (d (b+2 c x))^{3/2} \sqrt {a+x (b+c x)} \left (2 (a+x (b+c x))^2-\frac {\left (b^2-4 a c\right )^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{16 c^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(937\) vs. \(2(276)=552\).
Time = 3.00 (sec) , antiderivative size = 938, normalized size of antiderivative = 2.88
method | result | size |
default | \(-\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{2} \left (-2560 a \,c^{7} x^{6}-1856 a^{2} c^{6} x^{4}-256 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}-4544 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 -312 x^{3} b^{5} c^{3}-736 a^{2} c^{3} b^{3} x -1056 x^{2} a \,b^{4} c^{3}-64 a^{3} b^{2} c^{3}-2592 x^{2} a^{2} b^{2} c^{4}-1916 b^{4} c^{4} x^{4}-6080 b^{2} c^{6} x^{6}+10 b^{6} c^{2} x^{2}-4800 b^{3} c^{5} x^{5}-3840 b \,c^{7} x^{7}-960 c^{8} x^{8}-256 a^{3} b \,c^{4} x -64 a \,b^{5} c^{2} x +6 a \,b^{6} c +6 b^{7} c x -7680 a b \,c^{6} x^{5}-8672 a \,b^{2} c^{5} x^{4}-3712 a^{2} b \,c^{5} x^{3}\right )}{780 c^{3} \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(938\) |
risch | \(\text {Expression too large to display}\) | \(2234\) |
elliptic | \(\text {Expression too large to display}\) | \(3580\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.75 \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {3 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} d^{2} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) - {\left (240 \, c^{6} d^{2} x^{5} + 600 \, b c^{5} d^{2} x^{4} + 100 \, {\left (5 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{2} x^{3} + 150 \, {\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{2} x^{2} + 4 \, {\left (b^{4} c^{2} + 67 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} x - {\left (3 \, b^{5} c - 34 \, a b^{3} c^{2} - 32 \, a^{2} b c^{3}\right )} d^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{390 \, c^{3}} \]
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\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int \left (d \left (b + 2 c x\right )\right )^{\frac {5}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
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