Integrand size = 26, antiderivative size = 335 \[ \int \frac {x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=-\frac {\left (13 a c-\frac {8 b c^2}{d}-\frac {3 a^2 d}{b}\right ) x \sqrt {a+b x^2}}{15 d \sqrt {c+d x^2}}-\frac {2 (2 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d^2}+\frac {b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\sqrt {c} \left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {2 c^{3/2} (2 b c-3 a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.26 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {488, 596, 545, 429, 506, 422} \[ \int \frac {x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c} \sqrt {a+b x^2} \left (3 a^2 d^2-13 a b c d+8 b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \sqrt {a+b x^2} \left (-\frac {3 a^2 d}{b}+13 a c-\frac {8 b c^2}{d}\right )}{15 d \sqrt {c+d x^2}}+\frac {2 c^{3/2} \sqrt {a+b x^2} (2 b c-3 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{5/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 x \sqrt {a+b x^2} \sqrt {c+d x^2} (2 b c-3 a d)}{15 d^2}+\frac {b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d} \]
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Rule 422
Rule 429
Rule 488
Rule 506
Rule 545
Rule 596
Rubi steps \begin{align*} \text {integral}& = \frac {b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}+\frac {\int \frac {x^2 \left (-a (3 b c-5 a d)-2 b (2 b c-3 a d) x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{5 d} \\ & = -\frac {2 (2 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d^2}+\frac {b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\int \frac {-2 a b c (2 b c-3 a d)-b \left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 b d^2} \\ & = -\frac {2 (2 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d^2}+\frac {b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}+\frac {(2 a c (2 b c-3 a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 d^2}+\frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 d^2} \\ & = \frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{15 b d^2 \sqrt {c+d x^2}}-\frac {2 (2 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d^2}+\frac {b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}+\frac {2 c^{3/2} (2 b c-3 a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (c \left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{15 b d^2} \\ & = \frac {\left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) x \sqrt {a+b x^2}}{15 b d^2 \sqrt {c+d x^2}}-\frac {2 (2 b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d^2}+\frac {b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\sqrt {c} \left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {2 c^{3/2} (2 b c-3 a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.18 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-4 b c+6 a d+3 b d x^2\right )-i c \left (8 b^2 c^2-13 a b c d+3 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (8 b^2 c^2-17 a b c d+9 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
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Time = 7.64 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.23
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5 d}+\frac {\left (2 a b -\frac {b \left (4 a d +4 b c \right )}{5 d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}-\frac {\left (2 a b -\frac {b \left (4 a d +4 b c \right )}{5 d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (a^{2}-\frac {3 b a c}{5 d}-\frac {\left (2 a b -\frac {b \left (4 a d +4 b c \right )}{5 d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(411\) |
risch | \(\frac {x \left (3 b d \,x^{2}+6 a d -4 b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{15 d^{2}}-\frac {\left (\frac {6 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {4 b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {\left (3 a^{2} d^{2}-13 a b c d +8 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{15 d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(412\) |
default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}-9 \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}-6 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}+4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+9 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-17 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+13 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-6 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x +4 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{15 d^{3} \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {-\frac {b}{a}}}\) | \(544\) |
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Time = 0.09 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=-\frac {{\left (8 \, b^{2} c^{3} - 13 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, b^{2} c^{3} - 13 \, a b c^{2} d - 6 \, a^{2} d^{3} + {\left (3 \, a^{2} + 4 \, a b\right )} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} d^{3} x^{4} + 8 \, b^{2} c^{2} d - 13 \, a b c d^{2} + 3 \, a^{2} d^{3} - 2 \, {\left (2 \, b^{2} c d^{2} - 3 \, a b d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b d^{4} x} \]
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\[ \int \frac {x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{2} \left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{2}}{\sqrt {d x^{2} + c}} \,d x } \]
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\[ \int \frac {x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} x^{2}}{\sqrt {d x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^2\,{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {d\,x^2+c}} \,d x \]
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