Integrand size = 21, antiderivative size = 156 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^{5/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b^3}+\frac {\sqrt {d} \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {427, 542, 537, 223, 212, 385, 211} \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {\sqrt {d} \left (8 a^2 d^2-20 a b c d+15 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3}+\frac {(b c-a d)^{5/2} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b^3}+\frac {d x \sqrt {c+d x^2} (7 b c-4 a d)}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b} \]
[In]
[Out]
Rule 211
Rule 212
Rule 223
Rule 385
Rule 427
Rule 537
Rule 542
Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {\int \frac {\sqrt {c+d x^2} \left (c (4 b c-a d)+d (7 b c-4 a d) x^2\right )}{a+b x^2} \, dx}{4 b} \\ & = \frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {\int \frac {c \left (8 b^2 c^2-9 a b c d+4 a^2 d^2\right )+d \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{8 b^2} \\ & = \frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^3 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{b^3}+\frac {\left (d \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx}{8 b^3} \\ & = \frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^3 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{b^3}+\frac {\left (d \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{8 b^3} \\ & = \frac {d (7 b c-4 a d) x \sqrt {c+d x^2}}{8 b^2}+\frac {d x \left (c+d x^2\right )^{3/2}}{4 b}+\frac {(b c-a d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} b^3}+\frac {\sqrt {d} \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{8 b^3} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\frac {b d x \sqrt {c+d x^2} \left (9 b c-4 a d+2 b d x^2\right )-\frac {8 (b c-a d)^{5/2} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a}}-\sqrt {d} \left (15 b^2 c^2-20 a b c d+8 a^2 d^2\right ) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right )}{8 b^3} \]
[In]
[Out]
Time = 3.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(-\frac {\frac {d \left (b \sqrt {d \,x^{2}+c}\, \left (-2 b d \,x^{2}+4 a d -9 b c \right ) x -\frac {\left (8 a^{2} d^{2}-20 a b c d +15 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{\sqrt {d}}\right )}{4}+\frac {2 \left (a d -b c \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{2 b^{3}}\) | \(135\) |
risch | \(-\frac {x d \left (-2 b d \,x^{2}+4 a d -9 b c \right ) \sqrt {d \,x^{2}+c}}{8 b^{2}}+\frac {\frac {\sqrt {d}\, \left (8 a^{2} d^{2}-20 a b c d +15 b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{b}-\frac {\left (4 a^{3} d^{3}-12 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (-4 a^{3} d^{3}+12 a^{2} b c \,d^{2}-12 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\sqrt {-a b}\, b \sqrt {-\frac {a d -b c}{b}}}}{8 b^{2}}\) | \(470\) |
default | \(\text {Expression too large to display}\) | \(2074\) |
[In]
[Out]
none
Time = 0.85 (sec) , antiderivative size = 931, normalized size of antiderivative = 5.97 \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\left [\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3}}, -\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3}}, \frac {8 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + {\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{16 \, b^{3}}, -\frac {{\left (15 \, b^{2} c^{2} - 20 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) - {\left (2 \, b^{2} d^{2} x^{3} + {\left (9 \, b^{2} c d - 4 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c}}{8 \, b^{3}}\right ] \]
[In]
[Out]
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{a + b x^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}}}{b x^{2} + a} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{5/2}}{b\,x^2+a} \,d x \]
[In]
[Out]