Integrand size = 24, antiderivative size = 427 \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 a^2 e^3 \sqrt {e x} (325 A+539 B x) \sqrt {a+c x^2}}{15015 c^2}+\frac {28 a^3 B e^4 x \sqrt {a+c x^2}}{195 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {28 a^{13/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 a^{11/4} \left (539 \sqrt {a} B+325 A \sqrt {c}\right ) e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15015 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}} \]
[Out]
Time = 0.38 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {847, 829, 856, 854, 1212, 226, 1210} \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 a^{11/4} e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (539 \sqrt {a} B+325 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15015 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {28 a^{13/4} B e^4 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{195 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {28 a^3 B e^4 x \sqrt {a+c x^2}}{195 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 a^2 e^3 \sqrt {e x} \sqrt {a+c x^2} (325 A+539 B x)}{15015 c^2}-\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c} \]
[In]
[Out]
Rule 226
Rule 829
Rule 847
Rule 854
Rule 856
Rule 1210
Rule 1212
Rubi steps \begin{align*} \text {integral}& = \frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac {2 \int (e x)^{5/2} \left (-\frac {7}{2} a B e+\frac {13}{2} A c e x\right ) \sqrt {a+c x^2} \, dx}{13 c} \\ & = \frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac {4 \int (e x)^{3/2} \left (-\frac {65}{4} a A c e^2-\frac {77}{4} a B c e^2 x\right ) \sqrt {a+c x^2} \, dx}{143 c^2} \\ & = -\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac {8 \int \sqrt {e x} \left (\frac {231}{8} a^2 B c e^3-\frac {585}{8} a A c^2 e^3 x\right ) \sqrt {a+c x^2} \, dx}{1287 c^3} \\ & = -\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac {16 \int \frac {\left (\frac {585}{16} a^2 A c^2 e^4+\frac {1617}{16} a^2 B c^2 e^4 x\right ) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx}{9009 c^4} \\ & = \frac {2 a^2 e^3 \sqrt {e x} (325 A+539 B x) \sqrt {a+c x^2}}{15015 c^2}-\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac {64 \int \frac {\frac {2925}{32} a^3 A c^3 e^6+\frac {4851}{32} a^3 B c^3 e^6 x}{\sqrt {e x} \sqrt {a+c x^2}} \, dx}{135135 c^5 e^2} \\ & = \frac {2 a^2 e^3 \sqrt {e x} (325 A+539 B x) \sqrt {a+c x^2}}{15015 c^2}-\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac {\left (64 \sqrt {x}\right ) \int \frac {\frac {2925}{32} a^3 A c^3 e^6+\frac {4851}{32} a^3 B c^3 e^6 x}{\sqrt {x} \sqrt {a+c x^2}} \, dx}{135135 c^5 e^2 \sqrt {e x}} \\ & = \frac {2 a^2 e^3 \sqrt {e x} (325 A+539 B x) \sqrt {a+c x^2}}{15015 c^2}-\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}+\frac {\left (128 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {2925}{32} a^3 A c^3 e^6+\frac {4851}{32} a^3 B c^3 e^6 x^2}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{135135 c^5 e^2 \sqrt {e x}} \\ & = \frac {2 a^2 e^3 \sqrt {e x} (325 A+539 B x) \sqrt {a+c x^2}}{15015 c^2}-\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {\left (28 a^{7/2} B e^4 \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 )}{195 c^{5/2} \sqrt {e x}}+\frac {\left (4 a^3 \left (539 \sqrt {a} B+325 A \sqrt {c}\right ) e^4 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^4}} \, dx,x,\sqrt {x}\right )}{15015 c^{5/2} \sqrt {e x}} \\ & = \frac {2 a^2 e^3 \sqrt {e x} (325 A+539 B x) \sqrt {a+c x^2}}{15015 c^2}+\frac {28 a^3 B e^4 x \sqrt {a+c x^2}}{195 c^{5/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {10 a A e^3 \sqrt {e x} \left (a+c x^2\right )^{3/2}}{77 c^2}-\frac {14 a B e^2 (e x)^{3/2} \left (a+c x^2\right )^{3/2}}{117 c^2}+\frac {2 A e (e x)^{5/2} \left (a+c x^2\right )^{3/2}}{11 c}+\frac {2 B (e x)^{7/2} \left (a+c x^2\right )^{3/2}}{13 c}-\frac {28 a^{13/4} B e^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 )}{195 c^{11/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {2 a^{11/4} \left (539 \sqrt {a} B+325 A \sqrt {c}\right ) e^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 )}{15015 c^{11/4} \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 = 10.11 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.33 \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 e^3 \sqrt {e x} \sqrt {a+c x^2} \left (-\left (\left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \left (-63 c x^2 (13 A+11 B x)+a (585 A+539 B x)\right )\right )+585 a^2 A \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {c x^2}{a}\right )+539 a^2 B x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{9009 c^2 \sqrt {1+\frac {c x^2}{a}}} \]
[In]
[Out]
Time = 1.04 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {2 e^{3} \sqrt {e x}\, \left (3465 B \,c^{4} x^{8}+4095 A \,c^{4} x^{7}+4235 a B \,c^{3} x^{6}+975 A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{3}+5265 a A \,c^{3} x^{5}+3234 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{4}-1617 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{4}-308 a^{2} B \,c^{2} x^{4}-780 a^{2} A \,c^{2} x^{3}-1078 a^{3} B c \,x^{2}-1950 a^{3} A c x \right )}{45045 x \sqrt {c \,x^{2}+a}\, c^{3}}\) | \(368\) |
risch | \(-\frac {2 \left (-3465 B \,c^{2} x^{5}-4095 A \,c^{2} x^{4}-770 a B c \,x^{3}-1170 a A c \,x^{2}+1078 a^{2} B x +1950 A \,a^{2}\right ) x \sqrt {c \,x^{2}+a}\, e^{4}}{45045 c^{2} \sqrt {e x}}+\frac {2 a^{3} \left (\frac {325 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {539 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{4} \sqrt {\left (c \,x^{2}+a \right ) e x}}{15015 c^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(383\) |
elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B \,e^{3} x^{5} \sqrt {c e \,x^{3}+a e x}}{13}+\frac {2 A \,e^{3} x^{4} \sqrt {c e \,x^{3}+a e x}}{11}+\frac {4 B a \,e^{3} x^{3} \sqrt {c e \,x^{3}+a e x}}{117 c}+\frac {4 a A \,e^{3} x^{2} \sqrt {c e \,x^{3}+a e x}}{77 c}-\frac {28 B \,a^{2} e^{3} x \sqrt {c e \,x^{3}+a e x}}{585 c^{2}}-\frac {20 a^{2} A \,e^{3} \sqrt {c e \,x^{3}+a e x}}{231 c^{2}}+\frac {10 a^{3} A \,e^{4} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{231 c^{3} \sqrt {c e \,x^{3}+a e x}}+\frac {14 B \,a^{3} e^{4} \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{195 c^{3} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(466\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.34 \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {2 \, {\left (1950 \, \sqrt {c e} A a^{3} e^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 3234 \, \sqrt {c e} B a^{3} e^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (3465 \, B c^{3} e^{3} x^{5} + 4095 \, A c^{3} e^{3} x^{4} + 770 \, B a c^{2} e^{3} x^{3} + 1170 \, A a c^{2} e^{3} x^{2} - 1078 \, B a^{2} c e^{3} x - 1950 \, A a^{2} c e^{3}\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{45045 \, c^{3}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 43.64 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.23 \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\frac {A \sqrt {a} e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} + \frac {B \sqrt {a} e^{\frac {7}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {15}{4}\right )} \]
[In]
[Out]
\[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}} \,d x } \]
[In]
[Out]
\[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\int { \sqrt {c x^{2} + a} {\left (B x + A\right )} \left (e x\right )^{\frac {7}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (e x)^{7/2} (A+B x) \sqrt {a+c x^2} \, dx=\int {\left (e\,x\right )}^{7/2}\,\sqrt {c\,x^2+a}\,\left (A+B\,x\right ) \,d x \]
[In]
[Out]