Integrand size = 33, antiderivative size = 214 \[ \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx=\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \sqrt {a+b x^2}}{b^6}-\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) \left (a+b x^2\right )^{3/2}}{3 b^6}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) \left (a+b x^2\right )^{5/2}}{5 b^6}+\frac {\left (b^2 d-4 a b e+10 a^2 f\right ) \left (a+b x^2\right )^{7/2}}{7 b^6}+\frac {(b e-5 a f) \left (a+b x^2\right )^{9/2}}{9 b^6}+\frac {f \left (a+b x^2\right )^{11/2}}{11 b^6} \]
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Time = 0.18 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1825, 1813, 1864} \[ \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx=\frac {\left (a+b x^2\right )^{7/2} \left (10 a^2 f-4 a b e+b^2 d\right )}{7 b^6}+\frac {\left (a+b x^2\right )^{5/2} \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{5 b^6}-\frac {a \left (a+b x^2\right )^{3/2} \left (-5 a^3 f+4 a^2 b e-3 a b^2 d+2 b^3 c\right )}{3 b^6}+\frac {a^2 \sqrt {a+b x^2} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}+\frac {\left (a+b x^2\right )^{9/2} (b e-5 a f)}{9 b^6}+\frac {f \left (a+b x^2\right )^{11/2}}{11 b^6} \]
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Rule 1813
Rule 1825
Rule 1864
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (c x^4+d x^6+e x^8+f x^{10}\right )}{\sqrt {a+b x^2}} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {c x^2+d x^3+e x^4+f x^5}{\sqrt {a+b x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {a^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^5 \sqrt {a+b x}}+\frac {a \left (-2 b^3 c+3 a b^2 d-4 a^2 b e+5 a^3 f\right ) \sqrt {a+b x}}{b^5}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) (a+b x)^{3/2}}{b^5}+\frac {\left (b^2 d-4 a b e+10 a^2 f\right ) (a+b x)^{5/2}}{b^5}+\frac {(b e-5 a f) (a+b x)^{7/2}}{b^5}+\frac {f (a+b x)^{9/2}}{b^5}\right ) \, dx,x,x^2\right ) \\ & = \frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \sqrt {a+b x^2}}{b^6}-\frac {a \left (2 b^3 c-3 a b^2 d+4 a^2 b e-5 a^3 f\right ) \left (a+b x^2\right )^{3/2}}{3 b^6}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) \left (a+b x^2\right )^{5/2}}{5 b^6}+\frac {\left (b^2 d-4 a b e+10 a^2 f\right ) \left (a+b x^2\right )^{7/2}}{7 b^6}+\frac {(b e-5 a f) \left (a+b x^2\right )^{9/2}}{9 b^6}+\frac {f \left (a+b x^2\right )^{11/2}}{11 b^6} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.74 \[ \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (-1280 a^5 f+128 a^4 b \left (11 e+5 f x^2\right )-16 a^3 b^2 \left (99 d+44 e x^2+30 f x^4\right )+8 a^2 b^3 \left (231 c+99 d x^2+66 e x^4+50 f x^6\right )-2 a b^4 x^2 \left (462 c+297 d x^2+220 e x^4+175 f x^6\right )+b^5 x^4 \left (693 c+5 \left (99 d x^2+77 e x^4+63 f x^6\right )\right )\right )}{3465 b^6} \]
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Time = 3.68 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(-\frac {256 \left (-\frac {693 \left (\frac {5}{11} f \,x^{6}+\frac {5}{9} e \,x^{4}+\frac {5}{7} d \,x^{2}+c \right ) x^{4} b^{5}}{1280}+\frac {231 \left (\frac {25}{66} f \,x^{6}+\frac {10}{21} e \,x^{4}+\frac {9}{14} d \,x^{2}+c \right ) x^{2} a \,b^{4}}{320}-\frac {231 \left (\frac {50}{231} f \,x^{6}+\frac {2}{7} e \,x^{4}+\frac {3}{7} d \,x^{2}+c \right ) a^{2} b^{3}}{160}+\frac {99 \left (\frac {10}{33} f \,x^{4}+\frac {4}{9} e \,x^{2}+d \right ) a^{3} b^{2}}{80}-\frac {11 \left (\frac {5 f \,x^{2}}{11}+e \right ) a^{4} b}{10}+f \,a^{5}\right ) \sqrt {b \,x^{2}+a}}{693 b^{6}}\) | \(142\) |
gosper | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-315 f \,x^{10} b^{5}+350 a \,b^{4} f \,x^{8}-385 b^{5} e \,x^{8}-400 a^{2} b^{3} f \,x^{6}+440 a \,b^{4} e \,x^{6}-495 b^{5} d \,x^{6}+480 a^{3} b^{2} f \,x^{4}-528 a^{2} b^{3} e \,x^{4}+594 a \,b^{4} d \,x^{4}-693 b^{5} c \,x^{4}-640 a^{4} b f \,x^{2}+704 a^{3} b^{2} e \,x^{2}-792 a^{2} b^{3} d \,x^{2}+924 a \,b^{4} c \,x^{2}+1280 f \,a^{5}-1408 a^{4} e b +1584 a^{3} d \,b^{2}-1848 a^{2} c \,b^{3}\right )}{3465 b^{6}}\) | \(193\) |
trager | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-315 f \,x^{10} b^{5}+350 a \,b^{4} f \,x^{8}-385 b^{5} e \,x^{8}-400 a^{2} b^{3} f \,x^{6}+440 a \,b^{4} e \,x^{6}-495 b^{5} d \,x^{6}+480 a^{3} b^{2} f \,x^{4}-528 a^{2} b^{3} e \,x^{4}+594 a \,b^{4} d \,x^{4}-693 b^{5} c \,x^{4}-640 a^{4} b f \,x^{2}+704 a^{3} b^{2} e \,x^{2}-792 a^{2} b^{3} d \,x^{2}+924 a \,b^{4} c \,x^{2}+1280 f \,a^{5}-1408 a^{4} e b +1584 a^{3} d \,b^{2}-1848 a^{2} c \,b^{3}\right )}{3465 b^{6}}\) | \(193\) |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-315 f \,x^{10} b^{5}+350 a \,b^{4} f \,x^{8}-385 b^{5} e \,x^{8}-400 a^{2} b^{3} f \,x^{6}+440 a \,b^{4} e \,x^{6}-495 b^{5} d \,x^{6}+480 a^{3} b^{2} f \,x^{4}-528 a^{2} b^{3} e \,x^{4}+594 a \,b^{4} d \,x^{4}-693 b^{5} c \,x^{4}-640 a^{4} b f \,x^{2}+704 a^{3} b^{2} e \,x^{2}-792 a^{2} b^{3} d \,x^{2}+924 a \,b^{4} c \,x^{2}+1280 f \,a^{5}-1408 a^{4} e b +1584 a^{3} d \,b^{2}-1848 a^{2} c \,b^{3}\right )}{3465 b^{6}}\) | \(193\) |
default | \(e \left (\frac {x^{8} \sqrt {b \,x^{2}+a}}{9 b}-\frac {8 a \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )+d \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )+c \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )+f \left (\frac {x^{10} \sqrt {b \,x^{2}+a}}{11 b}-\frac {10 a \left (\frac {x^{8} \sqrt {b \,x^{2}+a}}{9 b}-\frac {8 a \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )}{11 b}\right )\) | \(382\) |
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Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.83 \[ \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx=\frac {{\left (315 \, b^{5} f x^{10} + 35 \, {\left (11 \, b^{5} e - 10 \, a b^{4} f\right )} x^{8} + 5 \, {\left (99 \, b^{5} d - 88 \, a b^{4} e + 80 \, a^{2} b^{3} f\right )} x^{6} + 1848 \, a^{2} b^{3} c - 1584 \, a^{3} b^{2} d + 1408 \, a^{4} b e - 1280 \, a^{5} f + 3 \, {\left (231 \, b^{5} c - 198 \, a b^{4} d + 176 \, a^{2} b^{3} e - 160 \, a^{3} b^{2} f\right )} x^{4} - 4 \, {\left (231 \, a b^{4} c - 198 \, a^{2} b^{3} d + 176 \, a^{3} b^{2} e - 160 \, a^{4} b f\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3465 \, b^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (214) = 428\).
Time = 0.46 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.07 \[ \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx=\begin {cases} - \frac {256 a^{5} f \sqrt {a + b x^{2}}}{693 b^{6}} + \frac {128 a^{4} e \sqrt {a + b x^{2}}}{315 b^{5}} + \frac {128 a^{4} f x^{2} \sqrt {a + b x^{2}}}{693 b^{5}} - \frac {16 a^{3} d \sqrt {a + b x^{2}}}{35 b^{4}} - \frac {64 a^{3} e x^{2} \sqrt {a + b x^{2}}}{315 b^{4}} - \frac {32 a^{3} f x^{4} \sqrt {a + b x^{2}}}{231 b^{4}} + \frac {8 a^{2} c \sqrt {a + b x^{2}}}{15 b^{3}} + \frac {8 a^{2} d x^{2} \sqrt {a + b x^{2}}}{35 b^{3}} + \frac {16 a^{2} e x^{4} \sqrt {a + b x^{2}}}{105 b^{3}} + \frac {80 a^{2} f x^{6} \sqrt {a + b x^{2}}}{693 b^{3}} - \frac {4 a c x^{2} \sqrt {a + b x^{2}}}{15 b^{2}} - \frac {6 a d x^{4} \sqrt {a + b x^{2}}}{35 b^{2}} - \frac {8 a e x^{6} \sqrt {a + b x^{2}}}{63 b^{2}} - \frac {10 a f x^{8} \sqrt {a + b x^{2}}}{99 b^{2}} + \frac {c x^{4} \sqrt {a + b x^{2}}}{5 b} + \frac {d x^{6} \sqrt {a + b x^{2}}}{7 b} + \frac {e x^{8} \sqrt {a + b x^{2}}}{9 b} + \frac {f x^{10} \sqrt {a + b x^{2}}}{11 b} & \text {for}\: b \neq 0 \\\frac {\frac {c x^{6}}{6} + \frac {d x^{8}}{8} + \frac {e x^{10}}{10} + \frac {f x^{12}}{12}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.62 \[ \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} f x^{10}}{11 \, b} + \frac {\sqrt {b x^{2} + a} e x^{8}}{9 \, b} - \frac {10 \, \sqrt {b x^{2} + a} a f x^{8}}{99 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x^{6}}{7 \, b} - \frac {8 \, \sqrt {b x^{2} + a} a e x^{6}}{63 \, b^{2}} + \frac {80 \, \sqrt {b x^{2} + a} a^{2} f x^{6}}{693 \, b^{3}} + \frac {\sqrt {b x^{2} + a} c x^{4}}{5 \, b} - \frac {6 \, \sqrt {b x^{2} + a} a d x^{4}}{35 \, b^{2}} + \frac {16 \, \sqrt {b x^{2} + a} a^{2} e x^{4}}{105 \, b^{3}} - \frac {32 \, \sqrt {b x^{2} + a} a^{3} f x^{4}}{231 \, b^{4}} - \frac {4 \, \sqrt {b x^{2} + a} a c x^{2}}{15 \, b^{2}} + \frac {8 \, \sqrt {b x^{2} + a} a^{2} d x^{2}}{35 \, b^{3}} - \frac {64 \, \sqrt {b x^{2} + a} a^{3} e x^{2}}{315 \, b^{4}} + \frac {128 \, \sqrt {b x^{2} + a} a^{4} f x^{2}}{693 \, b^{5}} + \frac {8 \, \sqrt {b x^{2} + a} a^{2} c}{15 \, b^{3}} - \frac {16 \, \sqrt {b x^{2} + a} a^{3} d}{35 \, b^{4}} + \frac {128 \, \sqrt {b x^{2} + a} a^{4} e}{315 \, b^{5}} - \frac {256 \, \sqrt {b x^{2} + a} a^{5} f}{693 \, b^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.21 \[ \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx=\frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \sqrt {b x^{2} + a}}{b^{6}} + \frac {693 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3} c - 2310 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{3} c + 495 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2} d - 2079 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b^{2} d + 3465 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b^{2} d + 385 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} b e - 1980 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a b e + 4158 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b e - 4620 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b e + 315 \, {\left (b x^{2} + a\right )}^{\frac {11}{2}} f - 1925 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} a f + 4950 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} f - 6930 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3} f + 5775 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4} f}{3465 \, b^{6}} \]
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Time = 6.07 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.87 \[ \int \frac {c x^5+d x^7+e x^9+f x^{11}}{\sqrt {a+b x^2}} \, dx=\sqrt {b\,x^2+a}\,\left (\frac {x^6\,\left (400\,f\,a^2\,b^3-440\,e\,a\,b^4+495\,d\,b^5\right )}{3465\,b^6}-\frac {1280\,f\,a^5-1408\,e\,a^4\,b+1584\,d\,a^3\,b^2-1848\,c\,a^2\,b^3}{3465\,b^6}+\frac {x^4\,\left (-480\,f\,a^3\,b^2+528\,e\,a^2\,b^3-594\,d\,a\,b^4+693\,c\,b^5\right )}{3465\,b^6}+\frac {f\,x^{10}}{11\,b}+\frac {x^8\,\left (385\,b^5\,e-350\,a\,b^4\,f\right )}{3465\,b^6}-\frac {4\,a\,x^2\,\left (-160\,f\,a^3+176\,e\,a^2\,b-198\,d\,a\,b^2+231\,c\,b^3\right )}{3465\,b^5}\right ) \]
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