Integrand size = 32, antiderivative size = 245 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=-\frac {a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt {a+b x^2}}{256 b^5}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{11/2}} \]
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Time = 0.18 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1823, 1281, 470, 327, 223, 212} \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {x^5 \sqrt {a+b x^2} \left (63 a^2 f-70 a b e+80 b^2 d\right )}{480 b^3}+\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{256 b^{11/2}}-\frac {a x \sqrt {a+b x^2} \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{256 b^5}+\frac {x^3 \sqrt {a+b x^2} \left (-63 a^3 f+70 a^2 b e-80 a b^2 d+96 b^3 c\right )}{384 b^4}+\frac {x^7 \sqrt {a+b x^2} (10 b e-9 a f)}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b} \]
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Rule 212
Rule 223
Rule 327
Rule 470
Rule 1281
Rule 1823
Rubi steps \begin{align*} \text {integral}& = \frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\int \frac {x^4 \left (10 b c+10 b d x^2+(10 b e-9 a f) x^4\right )}{\sqrt {a+b x^2}} \, dx}{10 b} \\ & = \frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\int \frac {x^4 \left (80 b^2 c+\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx}{80 b^2} \\ & = \frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) \int \frac {x^4}{\sqrt {a+b x^2}} \, dx}{96 b^3} \\ & = \frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}-\frac {\left (a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{128 b^4} \\ & = -\frac {a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt {a+b x^2}}{256 b^5}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\left (a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{256 b^5} \\ & = -\frac {a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt {a+b x^2}}{256 b^5}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {\left (a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{256 b^5} \\ & = -\frac {a \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x \sqrt {a+b x^2}}{256 b^5}+\frac {\left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) x^3 \sqrt {a+b x^2}}{384 b^4}+\frac {\left (80 b^2 d-70 a b e+63 a^2 f\right ) x^5 \sqrt {a+b x^2}}{480 b^3}+\frac {(10 b e-9 a f) x^7 \sqrt {a+b x^2}}{80 b^2}+\frac {f x^9 \sqrt {a+b x^2}}{10 b}+\frac {a^2 \left (96 b^3 c-80 a b^2 d+70 a^2 b e-63 a^3 f\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{11/2}} \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {x \sqrt {a+b x^2} \left (-1440 a b^3 c+1200 a^2 b^2 d-1050 a^3 b e+945 a^4 f+960 b^4 c x^2-800 a b^3 d x^2+700 a^2 b^2 e x^2-630 a^3 b f x^2+640 b^4 d x^4-560 a b^3 e x^4+504 a^2 b^2 f x^4+480 b^4 e x^6-432 a b^3 f x^6+384 b^4 f x^8\right )}{3840 b^5}-\frac {a^2 \left (-96 b^3 c+80 a b^2 d-70 a^2 b e+63 a^3 f\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{128 b^{11/2}} \]
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Time = 3.70 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(-\frac {63 \left (a^{2} \left (f \,a^{3}-\frac {10}{9} a^{2} b e +\frac {80}{63} a \,b^{2} d -\frac {32}{21} b^{3} c \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )-\left (-\frac {32 \left (\frac {3}{10} f \,x^{6}+\frac {7}{18} e \,x^{4}+\frac {5}{9} d \,x^{2}+c \right ) a \,b^{\frac {7}{2}}}{21}+\frac {64 x^{2} \left (\frac {2}{5} f \,x^{6}+\frac {1}{2} e \,x^{4}+\frac {2}{3} d \,x^{2}+c \right ) b^{\frac {9}{2}}}{63}+\left (\left (\frac {8}{15} f \,x^{4}+\frac {20}{27} e \,x^{2}+\frac {80}{63} d \right ) b^{\frac {5}{2}}+\left (\left (-\frac {2 f \,x^{2}}{3}-\frac {10 e}{9}\right ) b^{\frac {3}{2}}+a f \sqrt {b}\right ) a \right ) a^{2}\right ) \sqrt {b \,x^{2}+a}\, x \right )}{256 b^{\frac {11}{2}}}\) | \(169\) |
risch | \(\frac {x \left (384 f \,x^{8} b^{4}-432 a \,b^{3} f \,x^{6}+480 b^{4} e \,x^{6}+504 a^{2} b^{2} f \,x^{4}-560 a \,b^{3} e \,x^{4}+640 b^{4} d \,x^{4}-630 a^{3} b f \,x^{2}+700 a^{2} b^{2} e \,x^{2}-800 a \,b^{3} d \,x^{2}+960 b^{4} c \,x^{2}+945 a^{4} f -1050 a^{3} b e +1200 a^{2} b^{2} d -1440 a \,b^{3} c \right ) \sqrt {b \,x^{2}+a}}{3840 b^{5}}-\frac {a^{2} \left (63 f \,a^{3}-70 a^{2} b e +80 a \,b^{2} d -96 b^{3} c \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {11}{2}}}\) | \(198\) |
default | \(e \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )+d \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )+c \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )+f \left (\frac {x^{9} \sqrt {b \,x^{2}+a}}{10 b}-\frac {9 a \left (\frac {x^{7} \sqrt {b \,x^{2}+a}}{8 b}-\frac {7 a \left (\frac {x^{5} \sqrt {b \,x^{2}+a}}{6 b}-\frac {5 a \left (\frac {x^{3} \sqrt {b \,x^{2}+a}}{4 b}-\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2 b}-\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}\right )}{6 b}\right )}{8 b}\right )}{10 b}\right )\) | \(402\) |
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Time = 0.36 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.69 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\left [-\frac {15 \, {\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d + 70 \, a^{4} b e - 63 \, a^{5} f\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (384 \, b^{5} f x^{9} + 48 \, {\left (10 \, b^{5} e - 9 \, a b^{4} f\right )} x^{7} + 8 \, {\left (80 \, b^{5} d - 70 \, a b^{4} e + 63 \, a^{2} b^{3} f\right )} x^{5} + 10 \, {\left (96 \, b^{5} c - 80 \, a b^{4} d + 70 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{3} - 15 \, {\left (96 \, a b^{4} c - 80 \, a^{2} b^{3} d + 70 \, a^{3} b^{2} e - 63 \, a^{4} b f\right )} x\right )} \sqrt {b x^{2} + a}}{7680 \, b^{6}}, -\frac {15 \, {\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d + 70 \, a^{4} b e - 63 \, a^{5} f\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (384 \, b^{5} f x^{9} + 48 \, {\left (10 \, b^{5} e - 9 \, a b^{4} f\right )} x^{7} + 8 \, {\left (80 \, b^{5} d - 70 \, a b^{4} e + 63 \, a^{2} b^{3} f\right )} x^{5} + 10 \, {\left (96 \, b^{5} c - 80 \, a b^{4} d + 70 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{3} - 15 \, {\left (96 \, a b^{4} c - 80 \, a^{2} b^{3} d + 70 \, a^{3} b^{2} e - 63 \, a^{4} b f\right )} x\right )} \sqrt {b x^{2} + a}}{3840 \, b^{6}}\right ] \]
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Time = 0.61 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.99 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\begin {cases} \frac {3 a^{2} \left (- \frac {5 a \left (- \frac {7 a \left (- \frac {9 a f}{10 b} + e\right )}{8 b} + d\right )}{6 b} + c\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b^{2}} + \sqrt {a + b x^{2}} \left (- \frac {3 a x \left (- \frac {5 a \left (- \frac {7 a \left (- \frac {9 a f}{10 b} + e\right )}{8 b} + d\right )}{6 b} + c\right )}{8 b^{2}} + \frac {f x^{9}}{10 b} + \frac {x^{7} \left (- \frac {9 a f}{10 b} + e\right )}{8 b} + \frac {x^{5} \left (- \frac {7 a \left (- \frac {9 a f}{10 b} + e\right )}{8 b} + d\right )}{6 b} + \frac {x^{3} \left (- \frac {5 a \left (- \frac {7 a \left (- \frac {9 a f}{10 b} + e\right )}{8 b} + d\right )}{6 b} + c\right )}{4 b}\right ) & \text {for}\: b \neq 0 \\\frac {\frac {c x^{5}}{5} + \frac {d x^{7}}{7} + \frac {e x^{9}}{9} + \frac {f x^{11}}{11}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.38 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} f x^{9}}{10 \, b} + \frac {\sqrt {b x^{2} + a} e x^{7}}{8 \, b} - \frac {9 \, \sqrt {b x^{2} + a} a f x^{7}}{80 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x^{5}}{6 \, b} - \frac {7 \, \sqrt {b x^{2} + a} a e x^{5}}{48 \, b^{2}} + \frac {21 \, \sqrt {b x^{2} + a} a^{2} f x^{5}}{160 \, b^{3}} + \frac {\sqrt {b x^{2} + a} c x^{3}}{4 \, b} - \frac {5 \, \sqrt {b x^{2} + a} a d x^{3}}{24 \, b^{2}} + \frac {35 \, \sqrt {b x^{2} + a} a^{2} e x^{3}}{192 \, b^{3}} - \frac {21 \, \sqrt {b x^{2} + a} a^{3} f x^{3}}{128 \, b^{4}} - \frac {3 \, \sqrt {b x^{2} + a} a c x}{8 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a} a^{2} d x}{16 \, b^{3}} - \frac {35 \, \sqrt {b x^{2} + a} a^{3} e x}{128 \, b^{4}} + \frac {63 \, \sqrt {b x^{2} + a} a^{4} f x}{256 \, b^{5}} + \frac {3 \, a^{2} c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {5 \, a^{3} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {7}{2}}} + \frac {35 \, a^{4} e \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {9}{2}}} - \frac {63 \, a^{5} f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {11}{2}}} \]
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Time = 0.32 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, f x^{2}}{b} + \frac {10 \, b^{8} e - 9 \, a b^{7} f}{b^{9}}\right )} x^{2} + \frac {80 \, b^{8} d - 70 \, a b^{7} e + 63 \, a^{2} b^{6} f}{b^{9}}\right )} x^{2} + \frac {5 \, {\left (96 \, b^{8} c - 80 \, a b^{7} d + 70 \, a^{2} b^{6} e - 63 \, a^{3} b^{5} f\right )}}{b^{9}}\right )} x^{2} - \frac {15 \, {\left (96 \, a b^{7} c - 80 \, a^{2} b^{6} d + 70 \, a^{3} b^{5} e - 63 \, a^{4} b^{4} f\right )}}{b^{9}}\right )} \sqrt {b x^{2} + a} x - \frac {{\left (96 \, a^{2} b^{3} c - 80 \, a^{3} b^{2} d + 70 \, a^{4} b e - 63 \, a^{5} f\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {11}{2}}} \]
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Timed out. \[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx=\int \frac {x^4\,\left (f\,x^6+e\,x^4+d\,x^2+c\right )}{\sqrt {b\,x^2+a}} \,d x \]
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