Integrand size = 24, antiderivative size = 393 \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx=\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {21 A c^{3/2} x \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {21 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (25 \sqrt {a} B+63 A \sqrt {c}\right ) c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{30 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}} \]
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Time = 0.33 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {837, 849, 856, 854, 1212, 226, 1210} \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx=-\frac {c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{30 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}+\frac {21 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {21 A c^{3/2} x \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}} \]
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Rule 226
Rule 837
Rule 849
Rule 854
Rule 856
Rule 1210
Rule 1212
Rubi steps \begin{align*} \text {integral}& = \frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {7}{2} a A c e^2-\frac {5}{2} a B c e^2 x}{(e x)^{7/2} \sqrt {a+c x^2}} \, dx}{a^2 c e^2} \\ & = \frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}+\frac {2 \int \frac {\frac {25}{4} a^2 B c e^3-\frac {21}{4} a A c^2 e^3 x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx}{5 a^3 c e^4} \\ & = \frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}-\frac {4 \int \frac {\frac {63}{8} a^2 A c^2 e^4+\frac {25}{8} a^2 B c^2 e^4 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{15 a^4 c e^6} \\ & = \frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}+\frac {8 \int \frac {-\frac {25}{16} a^3 B c^2 e^5-\frac {63}{16} a^2 A c^3 e^5 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{15 a^5 c e^8} \\ & = \frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}+\frac {\left (8 \sqrt {x}\right ) \int \frac {-\frac {25}{16} a^3 B c^2 e^5-\frac {63}{16} a^2 A c^3 e^5 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{15 a^5 c e^8 \sqrt {e x}} \\ & = \frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}+\frac {\left (16 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {25}{16} a^3 B c^2 e^5-\frac {63}{16} a^2 A c^3 e^5 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 a^5 c e^8 \sqrt {e x}} \\ & = \frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {\left (\left (25 \sqrt {a} B+63 A \sqrt {c}\right ) c \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15 a^{5/2} e^3 \sqrt {e x}}+\frac {\left (21 A c^{3/2} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{5 a^{5/2} e^3 \sqrt {e x}} \\ & = \frac {A+B x}{a e (e x)^{5/2} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac {5 B \sqrt {a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac {21 A c \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {21 A c^{3/2} x \sqrt {a+c x^2}}{5 a^3 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {21 A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (25 \sqrt {a} B+63 A \sqrt {c}\right ) c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{30 a^{11/4} e^3 \sqrt {e x} \sqrt {a+c x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.27 \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx=\frac {x \left (15 (A+B x)-21 A \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{2},-\frac {1}{4},-\frac {c x^2}{a}\right )-25 B x \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {c x^2}{a}\right )\right )}{15 a (e x)^{7/2} \sqrt {a+c x^2}} \]
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Time = 1.64 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {63 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}-126 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}-25 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,x^{2}+126 A \,c^{2} x^{4}-50 a B c \,x^{3}+84 a A c \,x^{2}-20 a^{2} B x -12 A \,a^{2}}{30 x^{2} \sqrt {c \,x^{2}+a}\, e^{3} \sqrt {e x}\, a^{3}}\) | \(331\) |
elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 x e c \left (-\frac {A c x}{2 a^{3} e^{4}}+\frac {B}{2 a^{2} e^{4}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {2 A \sqrt {c e \,x^{3}+a e x}}{5 a^{2} e^{4} x^{3}}-\frac {2 B \sqrt {c e \,x^{3}+a e x}}{3 a^{2} e^{4} x^{2}}+\frac {16 \left (c e \,x^{2}+a e \right ) A c}{5 a^{3} e^{4} \sqrt {x \left (c e \,x^{2}+a e \right )}}-\frac {5 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{6 a^{2} e^{3} \sqrt {c e \,x^{3}+a e x}}-\frac {21 c A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{10 a^{3} e^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(433\) |
risch | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (-24 A c \,x^{2}+5 a B x +3 a A \right )}{15 a^{3} x^{2} e^{3} \sqrt {e x}}-\frac {c \left (\frac {5 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {24 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}-15 a \left (-\frac {2 c e x \left (-\frac {A x}{2 a e}+\frac {B}{2 e c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{2 c \sqrt {c e \,x^{3}+a e x}}-\frac {A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 a \sqrt {c e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{15 a^{3} e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(666\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.39 \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx=-\frac {25 \, {\left (B a c x^{5} + B a^{2} x^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 63 \, {\left (A c^{2} x^{5} + A a c x^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (63 \, A c^{2} x^{4} - 25 \, B a c x^{3} + 42 \, A a c x^{2} - 10 \, B a^{2} x - 6 \, A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{15 \, {\left (a^{3} c e^{4} x^{5} + a^{4} e^{4} x^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 73.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.26 \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} \]
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\[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (e\,x\right )}^{7/2}\,{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
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