Integrand size = 24, antiderivative size = 326 \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {3 A e^3 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 \sqrt [4]{a} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (5 \sqrt {a} B-9 A \sqrt {c}\right ) e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}} \]
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Time = 0.24 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {833, 847, 856, 854, 1212, 226, 1210} \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {\sqrt [4]{a} e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B-9 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 c^{9/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {3 \sqrt [4]{a} A e^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {3 A e^3 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2} \]
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Rule 226
Rule 833
Rule 847
Rule 854
Rule 856
Rule 1210
Rule 1212
Rubi steps \begin{align*} \text {integral}& = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {\int \frac {\sqrt {e x} \left (\frac {3}{2} a A e^2+\frac {5}{2} a B e^2 x\right )}{\sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {2 \int \frac {-\frac {5}{4} a^2 B e^3+\frac {9}{4} a A c e^3 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{3 a c^2} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {\left (2 \sqrt {x}\right ) \int \frac {-\frac {5}{4} a^2 B e^3+\frac {9}{4} a A c e^3 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{3 a c^2 \sqrt {e x}} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {\left (4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {5}{4} a^2 B e^3+\frac {9}{4} a A c e^3 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 a c^2 \sqrt {e x}} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}-\frac {\left (\sqrt {a} \left (5 \sqrt {a} B-9 A \sqrt {c}\right ) e^3 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{3 c^2 \sqrt {e x}}-\frac {\left (3 \sqrt {a} A e^3 \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 )}{c^{3/2} \sqrt {e x}} \\ & = -\frac {e (e x)^{3/2} (A+B x)}{c \sqrt {a+c x^2}}+\frac {5 B e^2 \sqrt {e x} \sqrt {a+c x^2}}{3 c^2}+\frac {3 A e^3 x \sqrt {a+c x^2}}{c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {3 \sqrt [4]{a} A e^3 \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 )}{c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (5 \sqrt {a} B-9 A \sqrt {c}\right ) e^3 \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 )}{6 c^{9/4} \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 = 10.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.37 \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {e^2 \sqrt {e x} \left (5 a B-3 A c x+2 B c x^2-5 a B \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )+3 A c x \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{3 c^2 \sqrt {a+c x^2}} \]
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Time = 1.56 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {e^{2} \sqrt {e x}\, \left (9 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c -18 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c +5 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a -4 B \,c^{2} x^{3}+6 A \,c^{2} x^{2}-10 a B c x \right )}{6 x \sqrt {c \,x^{2}+a}\, c^{3}}\) | \(308\) |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (-\frac {2 x e c \left (\frac {e^{2} A x}{2 c^{2}}-\frac {e^{2} B a}{2 c^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}+\frac {2 B \,e^{2} \sqrt {c e \,x^{3}+a e x}}{3 c^{2}}-\frac {5 B a \,e^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{6 c^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {3 A \,e^{3} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(377\) |
risch | \(\frac {2 B x \sqrt {c \,x^{2}+a}\, e^{3}}{3 c^{2} \sqrt {e x}}+\frac {\left (-\frac {4 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 {3 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}}-3 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 ) e^{3} \sqrt {\left (c \,x^{2}+a \right ) e x}}{3 c^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(647\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.44 \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {5 \, {\left (B a c e^{2} x^{2} + B a^{2} e^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 9 \, {\left (A c^{2} e^{2} x^{2} + A a c e^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (2 \, B c^{2} e^{2} x^{2} - 3 \, A c^{2} e^{2} x + 5 \, B a c e^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{3 \, {\left (c^{4} x^{2} + a c^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 46.87 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.29 \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {A e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} + \frac {B e^{\frac {5}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{5/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
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