Integrand size = 23, antiderivative size = 273 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 b^2 \sqrt {c+d x^2}}+\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}-\frac {2 \sqrt {c} \sqrt {d} (2 b c-a d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} (3 b c-a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a b \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.11 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {427, 545, 429, 506, 422} \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=-\frac {2 \sqrt {c} \sqrt {d} \sqrt {a+b x^2} (2 b c-a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {c^{3/2} \sqrt {a+b x^2} (3 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 d x \sqrt {a+b x^2} (2 b c-a d)}{3 b^2 \sqrt {c+d x^2}}+\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b} \]
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Rule 422
Rule 427
Rule 429
Rule 506
Rule 545
Rubi steps \begin{align*} \text {integral}& = \frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}+\frac {\int \frac {c (3 b c-a d)+2 d (2 b c-a d) x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 b} \\ & = \frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}+\frac {(2 d (2 b c-a d)) \int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 b}+\frac {(c (3 b c-a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 b} \\ & = \frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 b^2 \sqrt {c+d x^2}}+\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}+\frac {c^{3/2} (3 b c-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 )}{3 a b \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {(2 c d (2 b c-a d)) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 b^2} \\ & = \frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 b^2 \sqrt {c+d x^2}}+\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b}-\frac {2 \sqrt {c} \sqrt {d} (2 b c-a d) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^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} (3 b c-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 )}{3 a b \sqrt {d} \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 = 1.93 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.73 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right )+2 i c (-2 b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 b \sqrt {\frac {b}{a}} \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
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Time = 4.42 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.14
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {d x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 b}+\frac {\left (c^{2}-\frac {d a c}{3 b}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 c d -\frac {d \left (2 a d +2 b c \right )}{3 b}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(310\) |
default | \(\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {-\frac {b}{a}}\, b \,d^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a \,d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b c d \,x^{3}+a c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) d -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d +4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2}+\sqrt {-\frac {b}{a}}\, a c d x \right )}{3 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) b \sqrt {-\frac {b}{a}}}\) | \(330\) |
risch | \(\frac {d x \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{3 b}-\frac {\left (\frac {a c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {3 b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a \,d^{2}-4 b c d \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 b \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(383\) |
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Time = 0.10 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.61 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=-\frac {2 \, {\left (2 \, b c^{2} - a c d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (4 \, b c^{2} - {\left (2 \, a - 3 \, b\right )} c d - a d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b d^{2} x^{2} + 4 \, b c d - 2 \, a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, b^{2} d x} \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{\sqrt {a + b x^{2}}}\, dx \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\sqrt {b x^{2} + a}} \,d x } \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\sqrt {b x^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}}{\sqrt {b\,x^2+a}} \,d x \]
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