Integrand size = 28, antiderivative size = 245 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e \sqrt {e x} \sqrt {c+d x^2}}{21 c d^3}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}+\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt {c+d x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 245, 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 = {474, 470, 327, 335, 226} \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{42 \sqrt [4]{c} d^{13/4} \sqrt {c+d x^2}}-\frac {e \sqrt {e x} \sqrt {c+d x^2} \left (21 a^2 d^2-70 a b c d+45 b^2 c^2\right )}{21 c d^3}+\frac {(e x)^{5/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e} \]
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Rule 226
Rule 327
Rule 335
Rule 470
Rule 474
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {(e x)^{3/2} \left (\frac {1}{2} \left (-2 a^2 d^2+5 (b c-a d)^2\right )-b^2 c d x^2\right )}{\sqrt {c+d x^2}} \, dx}{c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) \int \frac {(e x)^{3/2}}{\sqrt {c+d x^2}} \, dx}{14 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e \sqrt {e x} \sqrt {c+d x^2}}{21 c d^3}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}+\frac {\left (\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e^2\right ) \int \frac {1}{\sqrt {e x} \sqrt {c+d x^2}} \, dx}{42 d^3} \\ & = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e \sqrt {e x} \sqrt {c+d x^2}}{21 c d^3}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}+\frac {\left (\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 d^3} \\ & = \frac {(b c-a d)^2 (e x)^{5/2}}{c d^2 e \sqrt {c+d x^2}}-\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e \sqrt {e x} \sqrt {c+d x^2}}{21 c d^3}+\frac {2 b^2 (e x)^{5/2} \sqrt {c+d x^2}}{7 d^2 e}+\frac {\left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) e^{3/2} \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 )}{42 \sqrt [4]{c} d^{13/4} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.78 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {e \sqrt {e x} \left (\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} \left (-21 a^2 d^2+14 a b d \left (5 c+2 d x^2\right )-3 b^2 \left (15 c^2+6 c d x^2-2 d^2 x^4\right )\right )+i \left (45 b^2 c^2-70 a b c d+21 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )\right )}{21 \sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^3 \sqrt {c+d x^2}} \]
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Time = 3.71 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.35
method | result | size |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (-\frac {e^{2} x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e \,x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d^{2}}+\frac {2 \left (\frac {b \left (2 a d -b c \right ) e^{2}}{d^{2}}-\frac {5 b^{2} e^{2} c}{7 d^{2}}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e^{2}}{2 d^{3}}-\frac {\left (\frac {b \left (2 a d -b c \right ) e^{2}}{d^{2}}-\frac {5 b^{2} e^{2} c}{7 d^{2}}\right ) c}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(331\) |
default | \(\frac {e \sqrt {e x}\, \left (21 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2}-70 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d +45 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}+12 b^{2} d^{3} x^{5}+56 a b \,d^{3} x^{3}-36 b^{2} c \,d^{2} x^{3}-42 x \,a^{2} d^{3}+140 x a b c \,d^{2}-90 x \,b^{2} c^{2} d \right )}{42 x \sqrt {d \,x^{2}+c}\, d^{4}}\) | \(363\) |
risch | \(\frac {2 b \left (3 b d \,x^{2}+14 a d -12 b c \right ) x \sqrt {d \,x^{2}+c}\, e^{2}}{21 d^{3} \sqrt {e x}}+\frac {\left (\frac {21 a^{2} d \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}+\frac {33 b^{2} c^{2} \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}-\frac {56 a b c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {d e \,x^{3}+c e x}}-21 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{2} \sqrt {e x \left (d \,x^{2}+c \right )}}{21 d^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(572\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.71 \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (45 \, b^{2} c^{2} d - 70 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} e x^{2} + {\left (45 \, b^{2} c^{3} - 70 \, a b c^{2} d + 21 \, a^{2} c d^{2}\right )} e\right )} \sqrt {d e} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (6 \, b^{2} d^{3} e x^{4} - 2 \, {\left (9 \, b^{2} c d^{2} - 14 \, a b d^{3}\right )} e x^{2} - {\left (45 \, b^{2} c^{2} d - 70 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} e\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{21 \, {\left (d^{5} x^{2} + c d^{4}\right )}} \]
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\[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (e x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
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