Integrand size = 28, antiderivative size = 274 \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (b^2-4 a c\right )^{17/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{462 c^3 \sqrt {a+b x+c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {699, 706, 705, 703, 227} \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {d^{7/2} \left (b^2-4 a c\right )^{17/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{462 c^3 \sqrt {a+b x+c x^2}}+\frac {d^3 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{231 c^2}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} (b d+2 c d x)^{9/2}}{110 c^2 d}+\frac {d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} (b d+2 c d x)^{5/2}}{385 c^2}+\frac {\left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{9/2}}{15 c d} \]
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Rule 227
Rule 699
Rule 703
Rule 705
Rule 706
Rubi steps \begin{align*} \text {integral}& = \frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}-\frac {\left (b^2-4 a c\right ) \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx}{10 c} \\ & = -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {(b d+2 c d x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx}{220 c^2} \\ & = \frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx}{308 c^2} \\ & = \frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (\left (b^2-4 a c\right )^4 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx}{924 c^2} \\ & = \frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (\left (b^2-4 a c\right )^4 d^4 \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}{924 c^2 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (\left (b^2-4 a c\right )^4 d^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 )}{462 c^3 \sqrt {a+b x+c x^2}} \\ & = \frac {\left (b^2-4 a c\right )^3 d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{231 c^2}+\frac {\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}}{385 c^2}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{9/2} \sqrt {a+b x+c x^2}}{110 c^2 d}+\frac {(b d+2 c d x)^{9/2} \left (a+b x+c x^2\right )^{3/2}}{15 c d}+\frac {\left (b^2-4 a c\right )^{17/4} d^{7/2} \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 )}{462 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 = 10.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.59 \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {4 (d (b+2 c x))^{7/2} \sqrt {a+x (b+c x)} \left (11 (b+2 c x)^2 (a+x (b+c x))^2-10 \left (a-\frac {b^2}{4 c}\right ) c \left (2 (a+x (b+c x))^2-\frac {\left (b^2-4 a c\right )^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{16 c^2 \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}}\right )\right )}{165 (b+2 c x)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(641\) vs. \(2(234)=468\).
Time = 3.77 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.34
method | result | size |
risch | \(-\frac {\left (-2464 c^{6} x^{6}-7392 b \,c^{5} x^{5}-3808 a \,c^{5} x^{4}-8288 b^{2} c^{4} x^{4}-7616 a b \,c^{4} x^{3}-4256 x^{3} b^{3} c^{3}-384 a^{2} c^{4} x^{2}-5520 a \,b^{2} c^{3} x^{2}-906 x^{2} b^{4} c^{2}-384 a^{2} b \,c^{3} x -1712 x a \,b^{3} c^{2}-10 x \,b^{5} c +640 c^{3} a^{3}-576 a^{2} b^{2} c^{2}-70 a \,b^{4} c +5 b^{6}\right ) \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d^{4}}{2310 c^{2} \sqrt {d \left (2 c x +b \right )}}+\frac {\left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{4} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{462 c^{2} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(642\) |
default | \(\text {Expression too large to display}\) | \(1057\) |
elliptic | \(\text {Expression too large to display}\) | \(5816\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.07 \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {5 \, \sqrt {2} {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c^{2} d} d^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (2464 \, c^{8} d^{3} x^{6} + 7392 \, b c^{7} d^{3} x^{5} + 224 \, {\left (37 \, b^{2} c^{6} + 17 \, a c^{7}\right )} d^{3} x^{4} + 224 \, {\left (19 \, b^{3} c^{5} + 34 \, a b c^{6}\right )} d^{3} x^{3} + 6 \, {\left (151 \, b^{4} c^{4} + 920 \, a b^{2} c^{5} + 64 \, a^{2} c^{6}\right )} d^{3} x^{2} + 2 \, {\left (5 \, b^{5} c^{3} + 856 \, a b^{3} c^{4} + 192 \, a^{2} b c^{5}\right )} d^{3} x - {\left (5 \, b^{6} c^{2} - 70 \, a b^{4} c^{3} - 576 \, a^{2} b^{2} c^{4} + 640 \, a^{3} c^{5}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{4620 \, c^{4}} \]
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\[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int \left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int { {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{3/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
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