Integrand size = 23, antiderivative size = 328 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\frac {\left (7 a c-\frac {2 b c^2}{d}+\frac {3 a^2 d}{b}\right ) x \sqrt {a+b x^2}}{15 \sqrt {c+d x^2}}-\frac {2 (b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d}+\frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}+\frac {\sqrt {c} \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} (b c-9 a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.20 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {427, 542, 545, 429, 506, 422} \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\frac {\sqrt {c} \sqrt {a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {x \sqrt {a+b x^2} \left (\frac {3 a^2 d}{b}+7 a c-\frac {2 b c^2}{d}\right )}{15 \sqrt {c+d x^2}}-\frac {c^{3/2} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {2 x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-3 a d)}{15 d} \]
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Rule 422
Rule 427
Rule 429
Rule 506
Rule 542
Rule 545
Rubi steps \begin{align*} \text {integral}& = \frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}+\frac {\int \frac {\sqrt {c+d x^2} \left (-a (b c-5 a d)-2 b (b c-3 a d) x^2\right )}{\sqrt {a+b x^2}} \, dx}{5 d} \\ & = -\frac {2 (b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d}+\frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}+\frac {\int \frac {-a b c (b c-9 a d)-b \left (2 b^2 c^2-7 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} \\ & = -\frac {2 (b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d}+\frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {(a c (b c-9 a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 d}-\frac {\left (2 b^2 c^2-7 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} \\ & = \frac {\left (7 a c-\frac {2 b c^2}{d}+\frac {3 a^2 d}{b}\right ) x \sqrt {a+b x^2}}{15 \sqrt {c+d x^2}}-\frac {2 (b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d}+\frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac {c^{3/2} (b c-9 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^{3/2} \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 (2 b^2 c^2-7 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} \\ & = \frac {\left (7 a c-\frac {2 b c^2}{d}+\frac {3 a^2 d}{b}\right ) x \sqrt {a+b x^2}}{15 \sqrt {c+d x^2}}-\frac {2 (b c-3 a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 d}+\frac {b x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}+\frac {\sqrt {c} \left (2 b^2 c^2-7 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^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} (b c-9 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^{3/2} \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 = 2.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.74 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a d+b \left (c+3 d x^2\right )\right )-i c \left (-2 b^2 c^2+7 a b c d+3 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-2 i c \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
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Time = 4.06 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.26
method | result | size |
risch | \(\frac {x \left (3 b d \,x^{2}+6 a d +b c \right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{15 d}+\frac {\left (\frac {9 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {b \,c^{2} a \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (3 a^{2} d^{2}+7 a b c d -2 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{15 d \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(412\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5}+\frac {\left (2 a b d +b^{2} c -\frac {b \left (4 a d +4 b c \right )}{5}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b d}+\frac {\left (a^{2} c -\frac {\left (2 a b d +b^{2} c -\frac {b \left (4 a d +4 b c \right )}{5}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (a^{2} d +\frac {7 a b c}{5}-\frac {\left (2 a b d +b^{2} c -\frac {b \left (4 a d +4 b c \right )}{5}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(423\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+9 \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}+4 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+6 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}+10 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+6 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x +\sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{15 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) d^{2} \sqrt {-\frac {b}{a}}}\) | \(543\) |
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Time = 0.10 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.71 \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\frac {{\left (2 \, b^{2} c^{3} - 7 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{2} c^{3} - 7 \, a b c^{2} d - 9 \, a^{2} d^{3} - {\left (3 \, a^{2} - a b\right )} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, b^{2} d^{3} x^{4} - 2 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 3 \, a^{2} d^{3} + {\left (b^{2} c d^{2} + 6 \, a b d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b d^{3} x} \]
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\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\int \left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}\, dx \]
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\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \,d x } \]
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\[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} \,d x } \]
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Timed out. \[ \int \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \, dx=\int {\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c} \,d x \]
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