Integrand size = 28, antiderivative size = 196 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {40 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{\sqrt {a+b x+c x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {700, 706, 705, 703, 227} \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {40 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}-\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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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)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}+\left (6 c d^2\right ) \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)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+\left (60 c^2 d^4\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)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\left (20 c^2 \left (b^2-4 a c\right ) d^6\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)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {\left (20 c^2 \left (b^2-4 a c\right ) d^6 \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}{\sqrt {a+b x+c x^2}} \\ & = -\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {\left (40 c \left (b^2-4 a c\right ) d^5 \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 )}{\sqrt {a+b x+c x^2}} \\ & = -\frac {2 d (b d+2 c d x)^{9/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {12 c d^3 (b d+2 c d x)^{5/2}}{\sqrt {a+b x+c x^2}}+80 c^2 d^5 \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {40 c \left (b^2-4 a c\right )^{5/4} d^{11/2} \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 )}{\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 = 10.14 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.91 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 d^5 \sqrt {d (b+2 c x)} \left (-b^4-26 b^3 c x+6 b^2 c \left (-3 a+c x^2\right )+8 b c^2 x \left (21 a+8 c x^2\right )+8 c^2 \left (15 a^2+21 a c x^2+4 c^2 x^4\right )+60 c \left (b^2-4 a c\right ) (a+x (b+c x)) \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 (a+x (b+c x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(656\) vs. \(2(168)=336\).
Time = 9.59 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.35
method | result | size |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \sqrt {d \left (2 c x +b \right )}\, \left (-\frac {2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) d^{5} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{3 c^{2} \left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right )^{2}}+\frac {52 \left (2 c^{2} d x +b c d \right ) d^{5} \left (4 a c -b^{2}\right )}{3 \sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (2 c^{2} d x +b c d \right )}}+\frac {64 d^{5} c^{2} \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}{3}+\frac {2 \left (-16 c^{2} \left (8 a c -3 b^{2}\right ) d^{6}+\frac {52 d^{6} c^{2} \left (4 a c -b^{2}\right )}{3}-\frac {64 d^{5} c^{2} \left (a c d +\frac {1}{2} b^{2} d \right )}{3}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, F\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +a b d}}\right )}{\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d}\) | \(657\) |
default | \(-\frac {2 \left (120 \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}}\, a \,c^{3} x^{2}-30 \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^{2} c^{2} x^{2}+120 \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}}\, a b \,c^{2} x -30 \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^{3} c x -64 c^{5} x^{5}+120 \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}}\, a^{2} c^{2}-30 \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}}\, a \,b^{2} c -160 b \,x^{4} c^{4}-336 a \,c^{4} x^{3}-76 b^{2} c^{3} x^{3}-504 a b \,c^{3} x^{2}+46 x^{2} b^{3} c^{2}-240 a^{2} c^{3} x -132 a \,c^{2} b^{2} x +28 c x \,b^{4}-120 a^{2} b \,c^{2}+18 a \,b^{3} c +b^{5}\right ) d^{5} \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}}}\) | \(958\) |
risch | \(\text {Expression too large to display}\) | \(1470\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.27 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.52 \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (30 \, \sqrt {2} {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{4} + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} d^{5} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} d^{5} x + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{5}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (32 \, c^{4} d^{5} x^{4} + 64 \, b c^{3} d^{5} x^{3} + 6 \, {\left (b^{2} c^{2} + 28 \, a c^{3}\right )} d^{5} x^{2} - 2 \, {\left (13 \, b^{3} c - 84 \, a b c^{2}\right )} d^{5} x - {\left (b^{4} + 18 \, a b^{2} c - 120 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {11}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {11}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{11/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]
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