Integrand size = 24, antiderivative size = 373 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {7 A \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{11/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}} \]
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Time = 0.29 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {837, 849, 856, 854, 1212, 226, 1210} \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{11/4} e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {7 A \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}+\frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}} \]
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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}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-\frac {7}{2} a A c e^2-\frac {5}{2} a B c e^2 x}{(e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx}{3 a^2 c e^2} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\int \frac {\frac {21}{4} a^2 A c^2 e^4+\frac {5}{4} a^2 B c^2 e^4 x}{(e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 a^4 c^2 e^4} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}-\frac {2 \int \frac {-\frac {5}{8} a^3 B c^2 e^5-\frac {21}{8} a^2 A c^3 e^5 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{3 a^5 c^2 e^6} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}-\frac {\left (2 \sqrt {x}\right ) \int \frac {-\frac {5}{8} a^3 B c^2 e^5-\frac {21}{8} a^2 A c^3 e^5 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{3 a^5 c^2 e^6 \sqrt {e x}} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}-\frac {\left (4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {5}{8} a^3 B c^2 e^5-\frac {21}{8} a^2 A c^3 e^5 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a^5 c^2 e^6 \sqrt {e x}} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {\left (\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{6 a^{5/2} e \sqrt {e x}}-\frac {\left (7 A \sqrt {c} \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 )}{2 a^{5/2} e \sqrt {e x}} \\ & = \frac {A+B x}{3 a e \sqrt {e x} \left (a+c x^2\right )^{3/2}}+\frac {7 A+5 B x}{6 a^2 e \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 A \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x}}+\frac {7 A \sqrt {c} x \sqrt {a+c x^2}}{2 a^3 e \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 A \sqrt [4]{c} \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 )}{2 a^{11/4} e \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (5 \sqrt {a} B+21 A \sqrt {c}\right ) \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 )}{12 a^{11/4} \sqrt [4]{c} e \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 = 10.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.37 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {x \left (9 a A+7 a B x+7 A c x^2+5 B c x^3-21 A \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c x^2}{a}\right )+5 B x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )\right )}{6 a^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}} \]
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Time = 2.32 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.17
method | result | size |
elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (-\frac {A x}{3 a^{2} e^{2} c}+\frac {B}{3 a \,e^{2} c^{2}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (\frac {3 A x}{4 a^{3} e^{2}}-\frac {5 B}{12 a^{2} e^{2} c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {2 \left (c e \,x^{2}+a e \right ) A}{a^{3} e^{2} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {5 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{12 a^{2} e c \sqrt {c e \,x^{3}+a e x}}+\frac {7 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 a^{3} e \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(437\) |
default | \(-\frac {21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-5 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -5 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+42 A \,c^{3} x^{4}-10 a B \,c^{2} x^{3}+70 a A \,c^{2} x^{2}-14 a^{2} B c x +24 A \,a^{2} c}{12 a^{3} e \sqrt {e x}\, c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(605\) |
risch | \(-\frac {2 A \sqrt {c \,x^{2}+a}}{a^{3} e \sqrt {e x}}+\frac {\left (\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 )}{\sqrt {c e \,x^{3}+a e x}}-A a c \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\frac {\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 c \sqrt {c e \,x^{3}+a e x}}\right )-a^{2} \left (\frac {\left (\frac {A x}{3 a e c}-\frac {B}{3 e \,c^{2}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 c e x \left (-\frac {A x}{4 a^{2} e}+\frac {5 B}{12 a e c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}-\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 )}{12 a 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 )}{4 a^{2} \sqrt {c e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{a^{3} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(779\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {5 \, {\left (B a c^{2} x^{5} + 2 \, B a^{2} c x^{3} + B a^{3} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 21 \, {\left (A c^{3} x^{5} + 2 \, A a c^{2} x^{3} + A a^{2} c x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (21 \, A c^{3} x^{4} - 5 \, B a c^{2} x^{3} + 35 \, A a c^{2} x^{2} - 7 \, B a^{2} c x + 12 \, A a^{2} c\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{6 \, {\left (a^{3} c^{3} e^{2} x^{5} + 2 \, a^{4} c^{2} e^{2} x^{3} + a^{5} c e^{2} x\right )}} \]
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Result contains complex when optimal does not.
Time = 105.88 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.26 \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (e\,x\right )}^{3/2}\,{\left (c\,x^2+a\right )}^{5/2}} \,d x \]
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