Integrand size = 28, antiderivative size = 430 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {2 \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d^3}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {2 c \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{195 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 c^{5/4} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}}-\frac {c^{5/4} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {475, 470, 327, 335, 311, 226, 1210} \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=-\frac {c^{5/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}}+\frac {2 c^{5/4} e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{15/4} \sqrt {c+d x^2}}-\frac {2 c e^2 \sqrt {e x} \sqrt {c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{195 d^{7/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 e (e x)^{3/2} \sqrt {c+d x^2} \left (117 a^2 d^2+7 b c (11 b c-26 a d)\right )}{585 d^3}-\frac {2 b (e x)^{7/2} \sqrt {c+d x^2} (11 b c-26 a d)}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3} \]
[In]
[Out]
Rule 226
Rule 311
Rule 327
Rule 335
Rule 470
Rule 475
Rule 1210
Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}+\frac {2 \int \frac {(e x)^{5/2} \left (\frac {13 a^2 d}{2}-\frac {1}{2} b (11 b c-26 a d) x^2\right )}{\sqrt {c+d x^2}} \, dx}{13 d} \\ & = -\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {1}{117} \left (-117 a^2-\frac {7 b c (11 b c-26 a d)}{d^2}\right ) \int \frac {(e x)^{5/2}}{\sqrt {c+d x^2}} \, dx \\ & = \frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\right ) \int \frac {\sqrt {e x}}{\sqrt {c+d x^2}} \, dx}{195 d} \\ & = \frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (2 c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d} \\ & = \frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {\left (2 c^{3/2} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d^{3/2}}+\frac {\left (2 c^{3/2} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c+\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{195 d^{3/2}} \\ & = \frac {2 \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e (e x)^{3/2} \sqrt {c+d x^2}}{585 d}-\frac {2 b (11 b c-26 a d) (e x)^{7/2} \sqrt {c+d x^2}}{117 d^2 e}+\frac {2 b^2 (e x)^{11/2} \sqrt {c+d x^2}}{13 d e^3}-\frac {2 c \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^2 \sqrt {e x} \sqrt {c+d x^2}}{195 d^{3/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {2 c^{5/4} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} \sqrt {c+d x^2}}-\frac {c^{5/4} \left (117 a^2+\frac {7 b c (11 b c-26 a d)}{d^2}\right ) e^{5/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{195 d^{7/4} \sqrt {c+d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.33 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {2 e (e x)^{3/2} \left (\left (c+d x^2\right ) \left (117 a^2 d^2+26 a b d \left (-7 c+5 d x^2\right )+b^2 \left (77 c^2-55 c d x^2+45 d^2 x^4\right )\right )-3 c \left (77 b^2 c^2-182 a b c d+117 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{585 d^3 \sqrt {c+d x^2}} \]
[In]
[Out]
Time = 3.11 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {2 x^{2} \left (45 b^{2} d^{2} x^{4}+130 x^{2} a b \,d^{2}-55 x^{2} b^{2} c d +117 a^{2} d^{2}-182 a b c d +77 b^{2} c^{2}\right ) \sqrt {d \,x^{2}+c}\, e^{3}}{585 d^{3} \sqrt {e x}}-\frac {c \left (117 a^{2} d^{2}-182 a b c d +77 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) e^{3} \sqrt {e x \left (d \,x^{2}+c \right )}}{195 d^{4} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(294\) |
elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} e^{2} x^{5} \sqrt {d e \,x^{3}+c e x}}{13 d}+\frac {2 \left (2 a b \,e^{3}-\frac {11 b^{2} e^{3} c}{13 d}\right ) x^{3} \sqrt {d e \,x^{3}+c e x}}{9 d e}+\frac {2 \left (a^{2} e^{3}-\frac {7 \left (2 a b \,e^{3}-\frac {11 b^{2} e^{3} c}{13 d}\right ) c}{9 d}\right ) x \sqrt {d e \,x^{3}+c e x}}{5 d e}-\frac {3 \left (a^{2} e^{3}-\frac {7 \left (2 a b \,e^{3}-\frac {11 b^{2} e^{3} c}{13 d}\right ) c}{9 d}\right ) c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{5 d^{2} \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(357\) |
default | \(-\frac {e^{2} \sqrt {e x}\, \left (-90 b^{2} d^{4} x^{8}-260 a b \,d^{4} x^{6}+20 b^{2} c \,d^{3} x^{6}+702 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}-1092 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d +462 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}-351 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}+546 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d -231 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}-234 a^{2} d^{4} x^{4}+104 c a b \,x^{4} d^{3}-44 b^{2} c^{2} d^{2} x^{4}-234 a^{2} c \,d^{3} x^{2}+364 a b \,c^{2} d^{2} x^{2}-154 b^{2} c^{3} d \,x^{2}\right )}{585 x \sqrt {d \,x^{2}+c}\, d^{4}}\) | \(661\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.34 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {2 \, {\left (3 \, {\left (77 \, b^{2} c^{3} - 182 \, a b c^{2} d + 117 \, a^{2} c d^{2}\right )} \sqrt {d e} e^{2} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (45 \, b^{2} d^{3} e^{2} x^{5} - 5 \, {\left (11 \, b^{2} c d^{2} - 26 \, a b d^{3}\right )} e^{2} x^{3} + {\left (77 \, b^{2} c^{2} d - 182 \, a b c d^{2} + 117 \, a^{2} d^{3}\right )} e^{2} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{585 \, d^{4}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 26.17 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.33 \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\frac {a^{2} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {11}{4}\right )} + \frac {a b e^{\frac {5}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\sqrt {c} \Gamma \left (\frac {15}{4}\right )} + \frac {b^{2} e^{\frac {5}{2}} x^{\frac {15}{2}} \Gamma \left (\frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {15}{4} \\ \frac {19}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {19}{4}\right )} \]
[In]
[Out]
\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{\sqrt {d x^{2} + c}} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \left (e x\right )^{\frac {5}{2}}}{\sqrt {d x^{2} + c}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{5/2} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,{\left (b\,x^2+a\right )}^2}{\sqrt {d\,x^2+c}} \,d x \]
[In]
[Out]