Integrand size = 28, antiderivative size = 165 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d (b d+2 c d x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {20 c d^3 \sqrt {b d+2 c d x}}{3 \sqrt {a+b x+c x^2}}+\frac {40 c \sqrt [4]{b^2-4 a c} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 \sqrt {a+b x+c x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {700, 705, 703, 227} \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {40 c d^{7/2} \sqrt [4]{b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 \sqrt {a+b x+c x^2}}-\frac {20 c d^3 \sqrt {b d+2 c d x}}{3 \sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rule 227
Rule 700
Rule 703
Rule 705
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d (b d+2 c d x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac {1}{3} \left (10 c d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx \\ & = -\frac {2 d (b d+2 c d x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {20 c d^3 \sqrt {b d+2 c d x}}{3 \sqrt {a+b x+c x^2}}+\frac {1}{3} \left (20 c^2 d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d (b d+2 c d x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {20 c d^3 \sqrt {b d+2 c d x}}{3 \sqrt {a+b x+c x^2}}+\frac {\left (20 c^2 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}{3 \sqrt {a+b x+c x^2}} \\ & = -\frac {2 d (b d+2 c d x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {20 c d^3 \sqrt {b d+2 c d x}}{3 \sqrt {a+b x+c x^2}}+\frac {\left (40 c 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 )}{3 \sqrt {a+b x+c x^2}} \\ & = -\frac {2 d (b d+2 c d x)^{5/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {20 c d^3 \sqrt {b d+2 c d x}}{3 \sqrt {a+b x+c x^2}}+\frac {40 c \sqrt [4]{b^2-4 a c} 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 )}{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 = 10.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.74 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^3 \sqrt {d (b+2 c x)} \left (b^2+14 b c x+2 c \left (5 a+7 c x^2\right )-20 c (a+x (b+c x)) \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 (a+x (b+c x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs. \(2(137)=274\).
Time = 5.67 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.90
method | result | size |
default | \(\frac {2 \left (10 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, c^{2} x^{2}+10 \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) \sqrt {-4 a c +b^{2}}\, b c x +10 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a c -28 c^{3} x^{3}-42 b \,c^{2} x^{2}-20 a \,c^{2} x -16 b^{2} c x -10 a b c -b^{3}\right ) d^{3} \sqrt {d \left (2 c x +b \right )}}{3 \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(479\) |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \sqrt {d \left (2 c x +b \right )}\, \left (\frac {2 \left (4 a c -b^{2}\right ) d^{3} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{3 c^{2} \left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right )^{2}}-\frac {28 \left (2 c^{2} d x +b c d \right ) d^{3}}{3 \sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (2 c^{2} d x +b c d \right )}}+\frac {40 c^{2} d^{4} \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{3 \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}\right )}{\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d}\) | \(544\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.12 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (10 \, \sqrt {2} {\left (c^{2} d^{3} x^{4} + 2 \, b c d^{3} x^{3} + 2 \, a b d^{3} x + {\left (b^{2} + 2 \, a c\right )} d^{3} x^{2} + a^{2} d^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - {\left (14 \, c^{2} d^{3} x^{2} + 14 \, b c d^{3} x + {\left (b^{2} + 10 \, a c\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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