Integrand size = 24, antiderivative size = 376 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=-\frac {8 a c (63 A-25 B x) \sqrt {a+c x^2}}{105 e^3 \sqrt {e x}}+\frac {48 a A c^{3/2} x \sqrt {a+c x^2}}{5 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 (25 a B-21 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{3/2}}-\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}-\frac {48 a^{5/4} A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^3 \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^{5/4} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 e^3 \sqrt {e x} \sqrt {a+c x^2}} \]
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Time = 0.27 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {827, 856, 854, 1212, 226, 1210} \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\frac {8 a^{5/4} c^{3/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (25 \sqrt {a} B+63 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {48 a^{5/4} A c^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^3 \sqrt {e x} \sqrt {a+c x^2}}-\frac {8 a c \sqrt {a+c x^2} (63 A-25 B x)}{105 e^3 \sqrt {e x}}-\frac {4 \left (a+c x^2\right )^{3/2} (25 a B-21 A c x)}{105 e^2 (e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2} (7 A-5 B x)}{35 e (e x)^{5/2}}+\frac {48 a A c^{3/2} x \sqrt {a+c x^2}}{5 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )} \]
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Rule 226
Rule 827
Rule 854
Rule 856
Rule 1210
Rule 1212
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}-\frac {2 \int \frac {(-5 a B e-7 A c e x) \left (a+c x^2\right )^{3/2}}{(e x)^{5/2}} \, dx}{7 e^2} \\ & = -\frac {4 (25 a B-21 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{3/2}}-\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}+\frac {4 \int \frac {\left (21 a A c e^2+25 a B c e^2 x\right ) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx}{35 e^4} \\ & = -\frac {8 a c (63 A-25 B x) \sqrt {a+c x^2}}{105 e^3 \sqrt {e x}}-\frac {4 (25 a B-21 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{3/2}}-\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}-\frac {8 \int \frac {-25 a^2 B c e^3-63 a A c^2 e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{105 e^6} \\ & = -\frac {8 a c (63 A-25 B x) \sqrt {a+c x^2}}{105 e^3 \sqrt {e x}}-\frac {4 (25 a B-21 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{3/2}}-\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}-\frac {\left (8 \sqrt {x}\right ) \int \frac {-25 a^2 B c e^3-63 a A c^2 e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{105 e^6 \sqrt {e x}} \\ & = -\frac {8 a c (63 A-25 B x) \sqrt {a+c x^2}}{105 e^3 \sqrt {e x}}-\frac {4 (25 a B-21 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{3/2}}-\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}-\frac {\left (16 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-25 a^2 B c e^3-63 a A c^2 e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{105 e^6 \sqrt {e x}} \\ & = -\frac {8 a c (63 A-25 B x) \sqrt {a+c x^2}}{105 e^3 \sqrt {e x}}-\frac {4 (25 a B-21 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{3/2}}-\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}+\frac {\left (16 a^{3/2} \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 )}{105 e^3 \sqrt {e x}}-\frac {\left (48 a^{3/2} 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 e^3 \sqrt {e x}} \\ & = -\frac {8 a c (63 A-25 B x) \sqrt {a+c x^2}}{105 e^3 \sqrt {e x}}+\frac {48 a A c^{3/2} x \sqrt {a+c x^2}}{5 e^3 \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {4 (25 a B-21 A c x) \left (a+c x^2\right )^{3/2}}{105 e^2 (e x)^{3/2}}-\frac {2 (7 A-5 B x) \left (a+c x^2\right )^{5/2}}{35 e (e x)^{5/2}}-\frac {48 a^{5/4} 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 e^3 \sqrt {e x} \sqrt {a+c x^2}}+\frac {8 a^{5/4} \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 )}{105 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 = 10.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.23 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=-\frac {2 a^2 x \sqrt {a+c x^2} \left (3 A \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {5}{4},-\frac {1}{4},-\frac {c x^2}{a}\right )+5 B x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},-\frac {3}{4},\frac {1}{4},-\frac {c x^2}{a}\right )\right )}{15 (e x)^{7/2} \sqrt {1+\frac {c x^2}{a}}} \]
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Time = 0.72 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\frac {48 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,x^{2}}{5}-\frac {24 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,x^{2}}{5}+\frac {40 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}}{21}+\frac {2 B \,c^{3} x^{7}}{7}+\frac {2 A \,c^{3} x^{6}}{5}+\frac {38 a B \,c^{2} x^{5}}{21}-\frac {22 a A \,c^{2} x^{4}}{5}+\frac {6 a^{2} B c \,x^{3}}{7}-\frac {26 a^{2} A c \,x^{2}}{5}-\frac {2 a^{3} B x}{3}-\frac {2 A \,a^{3}}{5}}{x^{2} \sqrt {c \,x^{2}+a}\, e^{3} \sqrt {e x}}\) | \(367\) |
risch | \(-\frac {2 \sqrt {c \,x^{2}+a}\, \left (-15 B \,c^{2} x^{5}-21 A \,c^{2} x^{4}-80 a B c \,x^{3}+252 a A c \,x^{2}+35 a^{2} B x +21 A \,a^{2}\right )}{105 x^{2} e^{3} \sqrt {e x}}+\frac {8 a c \left (\frac {25 B a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {63 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{\sqrt {c e \,x^{3}+a e x}}\right ) \sqrt {\left (c \,x^{2}+a \right ) e x}}{105 e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(376\) |
elliptic | \(\frac {\sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 a^{2} A \sqrt {c e \,x^{3}+a e x}}{5 e^{4} x^{3}}-\frac {2 a^{2} B \sqrt {c e \,x^{3}+a e x}}{3 e^{4} x^{2}}-\frac {24 \left (c e \,x^{2}+a e \right ) A a c}{5 e^{4} \sqrt {x \left (c e \,x^{2}+a e \right )}}+\frac {2 B \,c^{2} x^{2} \sqrt {c e \,x^{3}+a e x}}{7 e^{4}}+\frac {2 A \,c^{2} x \sqrt {c e \,x^{3}+a e x}}{5 e^{4}}+\frac {32 B a c \sqrt {c e \,x^{3}+a e x}}{21 e^{4}}+\frac {40 B \,a^{2} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{21 e^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {24 A a c \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{5 e^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(457\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.33 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\frac {2 \, {\left (200 \, \sqrt {c e} B a^{2} x^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 504 \, \sqrt {c e} A a c x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (15 \, B c^{2} x^{5} + 21 \, A c^{2} x^{4} + 80 \, B a c x^{3} - 252 \, A a c x^{2} - 35 \, B a^{2} x - 21 \, A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{105 \, e^{4} x^{3}} \]
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Result contains complex when optimal does not.
Time = 42.81 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\frac {A a^{\frac {5}{2}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {A a^{\frac {3}{2}} c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {A \sqrt {a} c^{2} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B a^{\frac {5}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B a^{\frac {3}{2}} c \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{e^{\frac {7}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} c^{2} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (B x + A\right )}}{\left (e x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (e\,x\right )}^{7/2}} \,d x \]
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