Integrand size = 28, antiderivative size = 162 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 d (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+\frac {40}{3} c d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {20 \left (b^2-4 a c\right )^{5/4} d^{7/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 \sqrt {a+b x+c x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {700, 706, 705, 703, 227} \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {20 d^{7/2} \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 \sqrt {a+b x+c x^2}}+\frac {40}{3} c d^3 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}-\frac {2 d (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}} \]
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Rule 227
Rule 700
Rule 703
Rule 705
Rule 706
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+\left (10 c d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx \\ & = -\frac {2 d (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+\frac {40}{3} c d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {1}{3} \left (10 c \left (b^2-4 a c\right ) 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}}{\sqrt {a+b x+c x^2}}+\frac {40}{3} c d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {\left (10 c \left (b^2-4 a c\right ) 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}}{\sqrt {a+b x+c x^2}}+\frac {40}{3} c d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {\left (20 \left (b^2-4 a c\right ) 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}}{\sqrt {a+b x+c x^2}}+\frac {40}{3} c d^3 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {20 \left (b^2-4 a c\right )^{5/4} 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 )}{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 = 0.75 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 d^3 \sqrt {d (b+2 c x)} \left (-3 b^2+8 b c x+4 c \left (5 a+2 c x^2\right )+10 \left (b^2-4 a c\right ) \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 \sqrt {a+x (b+c x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(365\) vs. \(2(136)=272\).
Time = 5.94 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.26
method | result | size |
default | \(-\frac {2 \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d^{3} \left (20 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) a c -5 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-b -2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, F\left (\frac {\sqrt {\frac {b +2 c x +\sqrt {-4 a c +b^{2}}}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {2}}{2}, \sqrt {2}\right ) b^{2}-16 c^{3} x^{3}-24 b \,c^{2} x^{2}-40 a \,c^{2} x -2 b^{2} c x -20 a b c +3 b^{3}\right )}{3 \left (2 c^{2} x^{3}+3 c b \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(366\) |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \sqrt {d \left (2 c x +b \right )}\, \left (\frac {2 \left (2 c^{2} d x +b c d \right ) \left (4 a c -b^{2}\right ) d^{3}}{c \sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (2 c^{2} d x +b c d \right )}}+\frac {16 d^{3} c \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{3}+\frac {2 \left (-8 c \left (2 a c -b^{2}\right ) d^{4}+2 d^{4} c \left (4 a c -b^{2}\right )-\frac {16 d^{3} c \left (a c d +\frac {1}{2} b^{2} d \right )}{3}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}\right )}{\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d}\) | \(572\) |
risch | \(\frac {16 \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, c \,d^{4}}{3 \sqrt {d \left (2 c x +b \right )}}+\frac {\left (-\frac {32 c \left (4 a c -b^{2}\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 )}{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}}-\frac {\left (-48 a^{2} c^{2}+24 a \,b^{2} c -3 b^{4}\right ) \left (\frac {4 c^{2} d x +2 b c d}{\left (4 a c -b^{2}\right ) d c \sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (2 c^{2} d x +b c d \right )}}+\frac {4 c \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 )}{\left (4 a c -b^{2}\right ) \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 )}{3}\right ) d^{4} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(921\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (5 \, \sqrt {2} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} x^{2} + {\left (b^{3} - 4 \, a b c\right )} d^{3} x + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (8 \, c^{3} d^{3} x^{2} + 8 \, b c^{2} d^{3} x - {\left (3 \, b^{2} c - 20 \, a c^{2}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{2} + b c x + a c\right )}} \]
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\[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d \left (b + 2 c x\right )\right )^{\frac {7}{2}}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]
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