Integrand size = 23, antiderivative size = 315 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {(b c-a d) x \sqrt {c+d x^2}}{5 a b \left (a+b x^2\right )^{5/2}}+\frac {2 (2 b c+a d) x \sqrt {c+d x^2}}{15 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} b^{3/2} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \sqrt {d} (4 b c-a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a^3 b (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]
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Time = 0.22 (sec) , antiderivative size = 315, 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 = {424, 541, 539, 429, 422} \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{7/2}} \, dx=-\frac {c^{3/2} \sqrt {d} \sqrt {a+b x^2} (4 b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a^3 b \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 x \sqrt {c+d x^2} (a d+2 b c)}{15 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {\sqrt {c+d x^2} \left (-2 a^2 d^2-3 a b c d+8 b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} b^{3/2} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {x \sqrt {c+d x^2} (b c-a d)}{5 a b \left (a+b x^2\right )^{5/2}} \]
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Rule 422
Rule 424
Rule 429
Rule 539
Rule 541
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) x \sqrt {c+d x^2}}{5 a b \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {c (4 b c+a d)+d (3 b c+2 a d) x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx}{5 a b} \\ & = \frac {(b c-a d) x \sqrt {c+d x^2}}{5 a b \left (a+b x^2\right )^{5/2}}+\frac {2 (2 b c+a d) x \sqrt {c+d x^2}}{15 a^2 b \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-c (b c-a d) (8 b c+a d)-2 d (b c-a d) (2 b c+a d) x^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx}{15 a^2 b (b c-a d)} \\ & = \frac {(b c-a d) x \sqrt {c+d x^2}}{5 a b \left (a+b x^2\right )^{5/2}}+\frac {2 (2 b c+a d) x \sqrt {c+d x^2}}{15 a^2 b \left (a+b x^2\right )^{3/2}}-\frac {(c d (4 b c-a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{15 a^2 b (b c-a d)}+\frac {\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx}{15 a^2 b (b c-a d)} \\ & = \frac {(b c-a d) x \sqrt {c+d x^2}}{5 a b \left (a+b x^2\right )^{5/2}}+\frac {2 (2 b c+a d) x \sqrt {c+d x^2}}{15 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{5/2} b^{3/2} (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {c^{3/2} \sqrt {d} (4 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 )}{15 a^3 b (b c-a 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 = 4.59 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{7/2}} \, dx=\frac {\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (3 a^2 (b c-a d)^2+2 a (b c-a d) (2 b c+a d) \left (a+b x^2\right )+\left (8 b^2 c^2-3 a b c d-2 a^2 d^2\right ) \left (a+b x^2\right )^2\right )-i c \left (a+b x^2\right )^2 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (\left (-8 b^2 c^2+3 a b c d+2 a^2 d^2\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+\left (8 b^2 c^2-7 a b c d-a^2 d^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 a^4 \left (\frac {b}{a}\right )^{3/2} (b c-a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \]
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Time = 3.41 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.75
method | result | size |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\left (a d -b c \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{5 a \,b^{4} \left (x^{2}+\frac {a}{b}\right )^{3}}+\frac {2 \left (a d +2 b c \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{15 a^{2} b^{3} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {\left (b d \,x^{2}+b c \right ) x \left (2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}\right )}{15 b^{2} a^{3} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {2 d \left (a d +2 b c \right )}{15 a^{2} b^{2}}-\frac {2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}}{15 b^{2} a^{3}}-\frac {c \left (2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}\right )}{15 b \,a^{3} \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {\left (2 a^{2} d^{2}+3 a b c d -8 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{15 b \left (a d -b c \right ) a^{3} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(552\) |
default | \(\text {Expression too large to display}\) | \(1410\) |
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Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (297) = 594\).
Time = 0.10 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.02 \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{7/2}} \, dx=-\frac {{\left (8 \, a^{3} b^{3} c^{2} - 3 \, a^{4} b^{2} c d - 2 \, a^{5} b d^{2} + {\left (8 \, b^{6} c^{2} - 3 \, a b^{5} c d - 2 \, a^{2} b^{4} d^{2}\right )} x^{6} + 3 \, {\left (8 \, a b^{5} c^{2} - 3 \, a^{2} b^{4} c d - 2 \, a^{3} b^{3} d^{2}\right )} x^{4} + 3 \, {\left (8 \, a^{2} b^{4} c^{2} - 3 \, a^{3} b^{3} c d - 2 \, a^{4} b^{2} d^{2}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, a^{3} b^{3} c^{2} + {\left (8 \, b^{6} c^{2} + {\left (4 \, a^{2} b^{4} - 3 \, a b^{5}\right )} c d - {\left (a^{3} b^{3} + 2 \, a^{2} b^{4}\right )} d^{2}\right )} x^{6} + 3 \, {\left (8 \, a b^{5} c^{2} + {\left (4 \, a^{3} b^{3} - 3 \, a^{2} b^{4}\right )} c d - {\left (a^{4} b^{2} + 2 \, a^{3} b^{3}\right )} d^{2}\right )} x^{4} + {\left (4 \, a^{5} b - 3 \, a^{4} b^{2}\right )} c d - {\left (a^{6} + 2 \, a^{5} b\right )} d^{2} + 3 \, {\left (8 \, a^{2} b^{4} c^{2} + {\left (4 \, a^{4} b^{2} - 3 \, a^{3} b^{3}\right )} c d - {\left (a^{5} b + 2 \, a^{4} b^{2}\right )} d^{2}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (8 \, a b^{5} c^{2} - 3 \, a^{2} b^{4} c d - 2 \, a^{3} b^{3} d^{2}\right )} x^{5} + 2 \, {\left (10 \, a^{2} b^{4} c^{2} - 4 \, a^{3} b^{3} c d - 3 \, a^{4} b^{2} d^{2}\right )} x^{3} + {\left (15 \, a^{3} b^{3} c^{2} - 11 \, a^{4} b^{2} c d - a^{5} b d^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, {\left (a^{7} b^{3} c - a^{8} b^{2} d + {\left (a^{4} b^{6} c - a^{5} b^{5} d\right )} x^{6} + 3 \, {\left (a^{5} b^{5} c - a^{6} b^{4} d\right )} x^{4} + 3 \, {\left (a^{6} b^{4} c - a^{7} b^{3} d\right )} x^{2}\right )}} \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{7/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{7/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{7/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}}{{\left (b\,x^2+a\right )}^{7/2}} \,d x \]
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