Integrand size = 28, antiderivative size = 442 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{10 c d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt {c+d x^2}} \]
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Time = 0.27 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {474, 470, 294, 335, 311, 226, 1210} \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{20 c^{3/4} d^{15/4} \sqrt {c+d x^2}}+\frac {e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{10 c d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {e (e x)^{3/2} \left (5 a^2 d^2-70 a b c d+77 b^2 c^2\right )}{30 c d^3 \sqrt {c+d x^2}}+\frac {(e x)^{7/2} (b c-a d)^2}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}} \]
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Rule 226
Rule 294
Rule 311
Rule 335
Rule 470
Rule 474
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}-\frac {\int \frac {(e x)^{5/2} \left (\frac {1}{2} \left (-6 a^2 d^2+7 (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)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) \int \frac {(e x)^{5/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{30 c d^2} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{20 c d^3} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{10 c d^3} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{10 \sqrt {c} d^{7/2}}+\frac {\left (\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{10 \sqrt {c} d^{7/2}} \\ & = \frac {(b c-a d)^2 (e x)^{7/2}}{3 c d^2 e \left (c+d x^2\right )^{3/2}}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e (e x)^{3/2}}{30 c d^3 \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{7/2}}{5 d^2 e \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{10 c d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{10 c^{3/4} d^{15/4} \sqrt {c+d x^2}}-\frac {\left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) e^{5/2} \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 )}{20 c^{3/4} d^{15/4} \sqrt {c+d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.35 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {e (e x)^{3/2} \left (5 a^2 d^2 \left (c+3 d x^2\right )-10 a b c d \left (7 c+9 d x^2\right )+b^2 c \left (77 c^2+99 c d x^2+12 d^2 x^4\right )-3 \left (77 b^2 c^2-70 a b c d+5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \left (c+d x^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{30 c d^3 \left (c+d x^2\right )^{3/2}} \]
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Time = 4.73 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.88
method | result | size |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (-\frac {e^{2} x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d^{5} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {e^{3} x^{2} \left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right )}{2 d^{3} c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} e^{2} x \sqrt {d e \,x^{3}+c e x}}{5 d^{3}}+\frac {\left (\frac {2 \left (a d -b c \right ) b \,e^{3}}{d^{3}}-\frac {e^{3} \left (a^{2} d^{2}-6 a b c d +5 b^{2} c^{2}\right )}{4 d^{3} c}-\frac {3 b^{2} e^{3} c}{5 d^{3}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(387\) |
risch | \(\frac {2 b^{2} x^{2} \sqrt {d \,x^{2}+c}\, e^{3}}{5 d^{3} \sqrt {e x}}+\frac {\left (\frac {b \left (10 a d -13 b c \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}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\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}\right )}{d \sqrt {d e \,x^{3}+c e x}}+\left (5 a^{2} d^{2}-20 a b c d +15 b^{2} c^{2}\right ) \left (\frac {x^{2}}{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}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\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}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )-5 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x \sqrt {d e \,x^{3}+c e x}}{3 c e \,d^{2} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {x^{2}}{2 c^{2} \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}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\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}\right )}{4 c^{2} d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e^{3} \sqrt {e x \left (d \,x^{2}+c \right )}}{5 d^{3} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(691\) |
default | \(\text {Expression too large to display}\) | \(1191\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.59 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {3 \, {\left ({\left (77 \, b^{2} c^{2} d^{2} - 70 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} e^{2} x^{4} + 2 \, {\left (77 \, b^{2} c^{3} d - 70 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} e^{2} x^{2} + {\left (77 \, b^{2} c^{4} - 70 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} e^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (12 \, b^{2} c d^{3} e^{2} x^{5} + 3 \, {\left (33 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} e^{2} x^{3} + {\left (77 \, b^{2} c^{3} d - 70 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} e^{2} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{30 \, {\left (c d^{6} x^{4} + 2 \, c^{2} d^{5} x^{2} + c^{3} d^{4}\right )}} \]
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Timed out. \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]
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