Integrand size = 26, antiderivative size = 237 \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=-\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {5 (b c-a d)^3 (7 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{3/2} d^{9/2}} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {457, 81, 52, 65, 223, 212} \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {5 (b c-a d)^3 (a d+7 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{3/2} d^{9/2}}-\frac {5 \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2 (a d+7 b c)}{128 b d^4}+\frac {5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d) (a d+7 b c)}{192 b d^3}-\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} (a d+7 b c)}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d} \]
[In]
[Out]
Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}-\frac {(7 b c+a d) \text {Subst}\left (\int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{16 b d} \\ & = -\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {(5 (b c-a d) (7 b c+a d)) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{96 b d^2} \\ & = \frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}-\frac {\left (5 (b c-a d)^2 (7 b c+a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{128 b d^3} \\ & = -\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left (5 (b c-a d)^3 (7 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{256 b d^4} \\ & = -\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left (5 (b c-a d)^3 (7 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x^2}\right )}{128 b^2 d^4} \\ & = -\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {\left (5 (b c-a d)^3 (7 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{128 b^2 d^4} \\ & = -\frac {5 (b c-a d)^2 (7 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{128 b d^4}+\frac {5 (b c-a d) (7 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{192 b d^3}-\frac {(7 b c+a d) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{48 b d^2}+\frac {\left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{8 b d}+\frac {5 (b c-a d)^3 (7 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{128 b^{3/2} d^{9/2}} \\ \end{align*}
Time = 3.22 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.90 \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {b \sqrt {d} \sqrt {a+b x^2} \left (c+d x^2\right ) \left (15 a^3 d^3+a^2 b d^2 \left (-191 c+118 d x^2\right )+a b^2 d \left (265 c^2-172 c d x^2+136 d^2 x^4\right )+b^3 \left (-105 c^3+70 c^2 d x^2-56 c d^2 x^4+48 d^3 x^6\right )\right )+15 (b c-a d)^{7/2} (7 b c+a d) \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{384 b^2 d^{9/2} \sqrt {c+d x^2}} \]
[In]
[Out]
Time = 3.15 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.19
method | result | size |
risch | \(\frac {\left (48 b^{3} d^{3} x^{6}+136 a \,b^{2} d^{3} x^{4}-56 b^{3} c \,d^{2} x^{4}+118 x^{2} a^{2} b \,d^{3}-172 x^{2} a \,b^{2} c \,d^{2}+70 x^{2} b^{3} c^{2} d +15 a^{3} d^{3}-191 a^{2} b c \,d^{2}+265 a \,b^{2} c^{2} d -105 b^{3} c^{3}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{384 b \,d^{4}}-\frac {5 \left (a^{4} d^{4}+4 a^{3} b c \,d^{3}-18 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d -7 b^{4} c^{4}\right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{256 b \,d^{4} \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(281\) |
default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-96 b^{3} d^{3} x^{6} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}-272 a \,b^{2} d^{3} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+112 b^{3} c \,d^{2} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}-236 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} b \,d^{3} x^{2}+344 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a \,b^{2} c \,d^{2} x^{2}-140 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{3} c^{2} d \,x^{2}+15 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{4} d^{4}+60 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} b c \,d^{3}-270 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b^{2} c^{2} d^{2}+300 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{3} c^{3} d -105 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{4} c^{4}-30 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{3} d^{3}+382 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} b c \,d^{2}-530 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a \,b^{2} c^{2} d +210 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{3} c^{3}\right )}{768 b \,d^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}}\) | \(658\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {59 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a^{2}}{192 d}+\frac {5 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{3}}{128 b d}-\frac {191 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2} c}{384 d^{2}}-\frac {35 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{3}}{128 d^{4}}-\frac {5 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{4}}{256 b \sqrt {b d}}+\frac {b^{2} x^{6} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{8 d}+\frac {17 b \,x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a}{48 d}-\frac {7 b^{2} x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c}{48 d^{2}}+\frac {35 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} c^{2}}{192 d^{3}}+\frac {265 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a \,c^{2}}{384 d^{3}}-\frac {5 a^{3} c \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{64 d \sqrt {b d}}+\frac {35 b^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{4}}{256 d^{4} \sqrt {b d}}-\frac {43 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a c}{96 d^{2}}+\frac {45 b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2} c^{2}}{128 d^{2} \sqrt {b d}}-\frac {25 b^{2} a \,c^{3} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right )}{64 d^{3} \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(685\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.42 \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\left [-\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) - 4 \, {\left (48 \, b^{4} d^{4} x^{6} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{1536 \, b^{2} d^{5}}, -\frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (48 \, b^{4} d^{4} x^{6} - 105 \, b^{4} c^{3} d + 265 \, a b^{3} c^{2} d^{2} - 191 \, a^{2} b^{2} c d^{3} + 15 \, a^{3} b d^{4} - 8 \, {\left (7 \, b^{4} c d^{3} - 17 \, a b^{3} d^{4}\right )} x^{4} + 2 \, {\left (35 \, b^{4} c^{2} d^{2} - 86 \, a b^{3} c d^{3} + 59 \, a^{2} b^{2} d^{4}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{768 \, b^{2} d^{5}}\right ] \]
[In]
[Out]
\[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{3} \left (a + b x^{2}\right )^{\frac {5}{2}}}{\sqrt {c + d x^{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.28 \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (4 \, {\left (b x^{2} + a\right )} {\left (\frac {6 \, {\left (b x^{2} + a\right )}}{b^{2} d} - \frac {7 \, b^{3} c d^{5} + a b^{2} d^{6}}{b^{4} d^{7}}\right )} + \frac {5 \, {\left (7 \, b^{4} c^{2} d^{4} - 6 \, a b^{3} c d^{5} - a^{2} b^{2} d^{6}\right )}}{b^{4} d^{7}}\right )} - \frac {15 \, {\left (7 \, b^{5} c^{3} d^{3} - 13 \, a b^{4} c^{2} d^{4} + 5 \, a^{2} b^{3} c d^{5} + a^{3} b^{2} d^{6}\right )}}{b^{4} d^{7}}\right )} - \frac {15 \, {\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{4}}\right )} b}{384 \, {\left | b \right |}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^3 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^3\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \]
[In]
[Out]