Integrand size = 28, antiderivative size = 302 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {474, 470, 294, 327, 335, 226} \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {5 e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}}-\frac {5 e^3 \sqrt {e x} \sqrt {c+d x^2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{42 c d^4}+\frac {e (e x)^{5/2} \left (7 a^2 d^2-42 a b c d+39 b^2 c^2\right )}{14 c d^3 \sqrt {c+d x^2}}+\frac {(e x)^{9/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}} \]
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Rule 226
Rule 294
Rule 327
Rule 335
Rule 470
Rule 474
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {(e x)^{7/2} \left (-\frac {3}{2} \left (2 a^2 d^2-3 (b c-a d)^2\right )-3 b^2 c d x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx}{3 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) \int \frac {(e x)^{7/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{14 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx}{28 c d^3} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^4\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{84 d^4} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {\left (5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{42 d^4} \\ & = \frac {(b c-a d)^2 (e x)^{9/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e (e x)^{5/2}}{14 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{9/2}}{7 d^2 e \sqrt {c+d x^2}}-\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^3 \sqrt {e x} \sqrt {c+d x^2}}{42 c d^4}+\frac {5 \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) e^{7/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{84 \sqrt [4]{c} d^{17/4} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.74 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(e x)^{7/2} \left (\frac {\sqrt {x} \left (-7 a^2 d^2 \left (5 c+7 d x^2\right )+14 a b d \left (15 c^2+21 c d x^2+4 d^2 x^4\right )-b^2 \left (195 c^3+273 c^2 d x^2+52 c d^2 x^4-12 d^3 x^6\right )\right )}{d^4 \left (c+d x^2\right )}+\frac {5 i \left (39 b^2 c^2-42 a b c d+7 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^4}\right )}{42 x^{7/2} \sqrt {c+d x^2}} \]
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Time = 4.54 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.39
method | result | size |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {c \,e^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d^{6} \left (x^{2}+\frac {c}{d}\right )^{2}}-\frac {e^{4} x \left (7 a^{2} d^{2}-26 a b c d +19 b^{2} c^{2}\right )}{6 d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e^{3} x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d^{3}}+\frac {2 \left (\frac {2 b \left (a d -b c \right ) e^{4}}{d^{3}}-\frac {5 b^{2} e^{4} c}{7 d^{3}}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (\frac {\left (a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}\right ) e^{4}}{d^{4}}-\frac {e^{4} \left (7 a^{2} d^{2}-26 a b c d +19 b^{2} c^{2}\right )}{12 d^{4}}-\frac {\left (\frac {2 b \left (a d -b c \right ) e^{4}}{d^{3}}-\frac {5 b^{2} e^{4} c}{7 d^{3}}\right ) c}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(420\) |
default | \(\frac {\left (35 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a^{2} d^{3} x^{2}-210 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, a b c \,d^{2} x^{2}+195 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-c d}\, b^{2} c^{2} d \,x^{2}+24 b^{2} d^{4} x^{7}+35 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-210 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +195 \sqrt {-c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}+112 a b \,d^{4} x^{5}-104 b^{2} c \,d^{3} x^{5}-98 a^{2} d^{4} x^{3}+588 x^{3} d^{3} b a c -546 b^{2} c^{2} d^{2} x^{3}-70 a^{2} c \,d^{3} x +420 a b \,c^{2} d^{2} x -390 b^{2} d x \,c^{3}\right ) e^{3} \sqrt {e x}}{84 x \,d^{5} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(696\) |
risch | \(\frac {2 b \left (3 b d \,x^{2}+14 a d -19 b c \right ) x \sqrt {d \,x^{2}+c}\, e^{4}}{21 d^{4} \sqrt {e x}}+\frac {\left (\frac {21 a^{2} d \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}+\frac {82 b^{2} c^{2} \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}-\frac {98 a b c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}-42 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right ) \left (\frac {x}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )+21 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {d e \,x^{3}+c e x}}{3 c e \,d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {5 x}{6 c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {5 \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{12 c^{2} d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{4} \sqrt {e x \left (d \,x^{2}+c \right )}}{21 d^{4} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(769\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.91 \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {5 \, {\left ({\left (39 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} e^{3} x^{4} + 2 \, {\left (39 \, b^{2} c^{3} d - 42 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} e^{3} x^{2} + {\left (39 \, b^{2} c^{4} - 42 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} e^{3}\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (12 \, b^{2} d^{4} e^{3} x^{6} - 4 \, {\left (13 \, b^{2} c d^{3} - 14 \, a b d^{4}\right )} e^{3} x^{4} - 7 \, {\left (39 \, b^{2} c^{2} d^{2} - 42 \, a b c d^{3} + 7 \, a^{2} d^{4}\right )} e^{3} x^{2} - 5 \, {\left (39 \, b^{2} c^{3} d - 42 \, a b c^{2} d^{2} + 7 \, a^{2} c d^{3}\right )} e^{3}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{42 \, {\left (d^{7} x^{4} + 2 \, c d^{6} x^{2} + c^{2} d^{5}\right )}} \]
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Timed out. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {7}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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