Integrand size = 26, antiderivative size = 386 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{11/2}} \, dx=-\frac {2 \left (15 b^2 c^2+a d (6 b c-a d)\right ) \sqrt {c+d x^2}}{15 c^2 \sqrt {x}}+\frac {4 \sqrt {d} \left (15 b^2 c^2+a d (6 b c-a d)\right ) \sqrt {x} \sqrt {c+d x^2}}{15 c^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}-\frac {2 a (6 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^{5/2}}-\frac {4 \sqrt [4]{d} \left (15 b^2 c^2+a d (6 b c-a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {c+d x^2}}+\frac {2 \sqrt [4]{d} \left (15 b^2 c^2+a d (6 b c-a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{15 c^{7/4} \sqrt {c+d x^2}} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {473, 464, 283, 335, 311, 226, 1210} \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{11/2}} \, dx=-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}+\frac {2 \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (6 b c-a d)+15 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{15 c^{7/4} \sqrt {c+d x^2}}-\frac {4 \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \left (a d (6 b c-a d)+15 b^2 c^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{15 c^{7/4} \sqrt {c+d x^2}}-\frac {2 \sqrt {c+d x^2} \left (\frac {a d (6 b c-a d)}{c^2}+15 b^2\right )}{15 \sqrt {x}}+\frac {4 \sqrt {d} \sqrt {x} \sqrt {c+d x^2} \left (a d (6 b c-a d)+15 b^2 c^2\right )}{15 c^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a \left (c+d x^2\right )^{3/2} (6 b c-a d)}{15 c^2 x^{5/2}} \]
[In]
[Out]
Rule 226
Rule 283
Rule 311
Rule 335
Rule 464
Rule 473
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}+\frac {2 \int \frac {\left (\frac {3}{2} a (6 b c-a d)+\frac {9}{2} b^2 c x^2\right ) \sqrt {c+d x^2}}{x^{7/2}} \, dx}{9 c} \\ & = -\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}-\frac {2 a (6 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^{5/2}}+\frac {1}{15} \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right ) \int \frac {\sqrt {c+d x^2}}{x^{3/2}} \, dx \\ & = -\frac {2 \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}}{15 \sqrt {x}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}-\frac {2 a (6 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^{5/2}}+\frac {1}{15} \left (2 d \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right )\right ) \int \frac {\sqrt {x}}{\sqrt {c+d x^2}} \, dx \\ & = -\frac {2 \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}}{15 \sqrt {x}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}-\frac {2 a (6 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^{5/2}}+\frac {1}{15} \left (4 d \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c+d x^4}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}}{15 \sqrt {x}}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}-\frac {2 a (6 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^{5/2}}+\frac {1}{15} \left (4 \sqrt {c} \sqrt {d} \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x^4}} \, dx,x,\sqrt {x}\right )-\frac {1}{15} \left (4 \sqrt {c} \sqrt {d} \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {d} x^2}{\sqrt {c}}}{\sqrt {c+d x^4}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right ) \sqrt {c+d x^2}}{15 \sqrt {x}}+\frac {4 \sqrt {d} \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right ) \sqrt {x} \sqrt {c+d x^2}}{15 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {2 a^2 \left (c+d x^2\right )^{3/2}}{9 c x^{9/2}}-\frac {2 a (6 b c-a d) \left (c+d x^2\right )^{3/2}}{15 c^2 x^{5/2}}-\frac {4 \sqrt [4]{c} \sqrt [4]{d} \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right ) \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 {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{15 \sqrt {c+d x^2}}+\frac {2 \sqrt [4]{c} \sqrt [4]{d} \left (15 b^2+\frac {a d (6 b c-a d)}{c^2}\right ) \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 {x}}{\sqrt [4]{c}}\right )|\frac {1}{2}\right )}{15 \sqrt {c+d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{11/2}} \, dx=\frac {-2 \left (c+d x^2\right ) \left (45 b^2 c^2 x^4+18 a b c x^2 \left (c+2 d x^2\right )+a^2 \left (5 c^2+2 c d x^2-6 d^2 x^4\right )\right )+12 d \left (15 b^2 c^2+6 a b c d-a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x^6 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )}{45 c^2 x^{9/2} \sqrt {c+d x^2}} \]
[In]
[Out]
Time = 3.16 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-6 a^{2} d^{2} x^{4}+36 x^{4} a b c d +45 b^{2} c^{2} x^{4}+2 a^{2} c d \,x^{2}+18 a b \,c^{2} x^{2}+5 a^{2} c^{2}\right )}{45 x^{\frac {9}{2}} c^{2}}-\frac {2 \left (a^{2} d^{2}-6 a b c d -15 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right ) \sqrt {x \left (d \,x^{2}+c \right )}}{15 c^{2} \sqrt {d \,x^{3}+c x}\, \sqrt {x}\, \sqrt {d \,x^{2}+c}}\) | \(282\) |
elliptic | \(\frac {\sqrt {x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 a^{2} \sqrt {d \,x^{3}+c x}}{9 x^{5}}-\frac {4 a \left (a d +9 b c \right ) \sqrt {d \,x^{3}+c x}}{45 c \,x^{3}}+\frac {2 \left (d \,x^{2}+c \right ) \left (2 a^{2} d^{2}-12 a b c d -15 b^{2} c^{2}\right )}{15 c^{2} \sqrt {x \left (d \,x^{2}+c \right )}}+\frac {\left (b^{2} d -\frac {d \left (2 a^{2} d^{2}-12 a b c d -15 b^{2} c^{2}\right )}{15 c^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d \,x^{3}+c x}}\right )}{\sqrt {x}\, \sqrt {d \,x^{2}+c}}\) | \(311\) |
default | \(-\frac {2 \left (6 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x^{4}-36 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \,x^{4}-90 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x^{4}-3 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x^{4}+18 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d \,x^{4}+45 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x^{4}-6 a^{2} d^{3} x^{6}+36 x^{6} d^{2} a b c +45 b^{2} c^{2} d \,x^{6}-4 a^{2} c \,d^{2} x^{4}+54 a b \,c^{2} d \,x^{4}+45 b^{2} c^{3} x^{4}+7 a^{2} c^{2} d \,x^{2}+18 a b \,c^{3} x^{2}+5 a^{2} c^{3}\right )}{45 \sqrt {d \,x^{2}+c}\, x^{\frac {9}{2}} c^{2}}\) | \(659\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.33 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{11/2}} \, dx=-\frac {2 \, {\left (6 \, {\left (15 \, b^{2} c^{2} + 6 \, a b c d - a^{2} d^{2}\right )} \sqrt {d} x^{5} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (3 \, {\left (15 \, b^{2} c^{2} + 12 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{4} + 5 \, a^{2} c^{2} + 2 \, {\left (9 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )}}{45 \, c^{2} x^{5}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 29.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.39 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{11/2}} \, dx=\frac {a^{2} \sqrt {c} \Gamma \left (- \frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {9}{4}, - \frac {1}{2} \\ - \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 x^{\frac {9}{2}} \Gamma \left (- \frac {5}{4}\right )} + \frac {a b \sqrt {c} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {b^{2} \sqrt {c} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
[In]
[Out]
\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{x^{\frac {11}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{11/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c}}{x^{\frac {11}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^{11/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}}{x^{11/2}} \,d x \]
[In]
[Out]