Integrand size = 24, antiderivative size = 368 \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {7 A e^5 x \sqrt {a+c x^2}}{2 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 \sqrt [4]{a} A e^5 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (5 \sqrt {a} B-7 A \sqrt {c}\right ) e^5 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 c^{13/4} \sqrt {e x} \sqrt {a+c x^2}} \]
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Time = 0.29 (sec) , antiderivative size = 368, 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 = {833, 847, 856, 854, 1212, 226, 1210} \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {\sqrt [4]{a} e^5 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (5 \sqrt {a} B-7 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 c^{13/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {7 \sqrt [4]{a} A e^5 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}-\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {7 A e^5 x \sqrt {a+c x^2}}{2 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3} \]
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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)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {\int \frac {(e x)^{5/2} \left (\frac {7}{2} a A e^2+\frac {9}{2} a B e^2 x\right )}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {\int \frac {\sqrt {e x} \left (\frac {21}{4} a^2 A e^4+\frac {45}{4} a^2 B e^4 x\right )}{\sqrt {a+c x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {2 \int \frac {-\frac {45}{8} a^3 B e^5+\frac {63}{8} a^2 A c e^5 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{9 a^2 c^3} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {\left (2 \sqrt {x}\right ) \int \frac {-\frac {45}{8} a^3 B e^5+\frac {63}{8} a^2 A c e^5 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{9 a^2 c^3 \sqrt {e x}} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {\left (4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {-\frac {45}{8} a^3 B e^5+\frac {63}{8} a^2 A c e^5 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{9 a^2 c^3 \sqrt {e x}} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}-\frac {\left (\sqrt {a} \left (5 \sqrt {a} B-7 A \sqrt {c}\right ) e^5 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{2 c^3 \sqrt {e x}}-\frac {\left (7 \sqrt {a} A e^5 \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 c^{5/2} \sqrt {e x}} \\ & = -\frac {e (e x)^{7/2} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}-\frac {e^3 (e x)^{3/2} (7 A+9 B x)}{6 c^2 \sqrt {a+c x^2}}+\frac {5 B e^4 \sqrt {e x} \sqrt {a+c x^2}}{2 c^3}+\frac {7 A e^5 x \sqrt {a+c x^2}}{2 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {7 \sqrt [4]{a} A e^5 \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 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\sqrt [4]{a} \left (5 \sqrt {a} B-7 A \sqrt {c}\right ) e^5 \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 )}{4 c^{13/4} \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 = 10.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.42 \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {e^4 \sqrt {e x} \left (15 a^2 B-7 a A c x+21 a B c x^2-9 A c^2 x^3+4 B c^2 x^4-15 a B \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )+7 A c x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{6 c^3 \left (a+c x^2\right )^{3/2}} \]
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Time = 2.06 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.16
method | result | size |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (\frac {a \,e^{4} A x}{3 c^{4}}-\frac {a^{2} e^{4} B}{3 c^{5}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (\frac {3 e^{4} A x}{4 c^{3}}-\frac {13 e^{4} B a}{12 c^{4}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}+\frac {2 B \,e^{4} \sqrt {c e \,x^{3}+a e x}}{3 c^{3}}-\frac {5 B a \,e^{5} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{4 c^{4} \sqrt {c e \,x^{3}+a e x}}+\frac {7 A \,e^{5} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 c^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(426\) |
default | \(-\frac {\left (21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}+15 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+21 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -42 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c +15 B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-8 B \,c^{3} x^{5}+18 A \,c^{3} x^{4}-42 a B \,c^{2} x^{3}+14 a A \,c^{2} x^{2}-30 a^{2} B c x \right ) e^{4} \sqrt {e x}}{12 x \,c^{4} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(607\) |
risch | \(\text {Expression too large to display}\) | \(1019\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.58 \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {15 \, {\left (B a c^{2} e^{4} x^{4} + 2 \, B a^{2} c e^{4} x^{2} + B a^{3} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 21 \, {\left (A c^{3} e^{4} x^{4} + 2 \, A a c^{2} e^{4} x^{2} + A a^{2} c e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (4 \, B c^{3} e^{4} x^{4} - 9 \, A c^{3} e^{4} x^{3} + 21 \, B a c^{2} e^{4} x^{2} - 7 \, A a c^{2} e^{4} x + 15 \, B a^{2} c e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{6 \, {\left (c^{6} x^{4} + 2 \, a c^{5} x^{2} + a^{2} c^{4}\right )}} \]
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Timed out. \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {9}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(e x)^{9/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{9/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]
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