Integrand size = 26, antiderivative size = 187 \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b d^3}-\frac {(5 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d}-\frac {(b c-a d)^2 (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{3/2} d^{7/2}} \]
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Time = 0.14 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {457, 81, 52, 65, 223, 212} \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=-\frac {(b c-a d)^2 (a d+5 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{3/2} d^{7/2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d) (a d+5 b c)}{16 b d^3}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (a d+5 b c)}{24 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d} \]
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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)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d}-\frac {(5 b c+a d) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{12 b d} \\ & = -\frac {(5 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d}+\frac {((b c-a d) (5 b c+a d)) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{16 b d^2} \\ & = \frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b d^3}-\frac {(5 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d)^2 (5 b c+a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{32 b d^3} \\ & = \frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b d^3}-\frac {(5 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d)^2 (5 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 )}{16 b^2 d^3} \\ & = \frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b d^3}-\frac {(5 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d)^2 (5 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 )}{16 b^2 d^3} \\ & = \frac {(b c-a d) (5 b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b d^3}-\frac {(5 b c+a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b d^2}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{6 b d}-\frac {(b c-a d)^2 (5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{3/2} d^{7/2}} \\ \end{align*}
Time = 1.58 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (3 a^2 d^2+2 a b d \left (-11 c+7 d x^2\right )+b^2 \left (15 c^2-10 c d x^2+8 d^2 x^4\right )\right )}{48 b d^3}-\frac {(b c-a d)^2 (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{16 b^{3/2} d^{7/2}} \]
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Time = 3.20 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {\left (8 b^{2} d^{2} x^{4}+14 x^{2} a b \,d^{2}-10 x^{2} b^{2} c d +3 a^{2} d^{2}-22 a b c d +15 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{48 b \,d^{3}}-\frac {\left (a^{3} d^{3}+3 a^{2} b c \,d^{2}-9 a \,b^{2} c^{2} d +5 b^{3} c^{3}\right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{32 b \,d^{3} \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(212\) |
default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-16 b^{2} d^{2} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}-28 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b \,d^{2} x^{2}+20 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{2} c d \,x^{2}+3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{3}+9 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c \,d^{2}-27 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2} d +15 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3}-6 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} d^{2}+44 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b c d -30 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{2} c^{2}\right )}{96 b \,d^{3} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}}\) | \(455\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {b \,x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{6 d}+\frac {7 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a}{24 d}-\frac {5 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} c}{24 d^{2}}+\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2}}{16 b d}-\frac {11 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a c}{24 d^{2}}+\frac {5 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{2}}{16 d^{3}}-\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{3}}{32 b \sqrt {b d}}-\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2} c}{32 d \sqrt {b d}}+\frac {9 b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a \,c^{2}}{32 d^{2} \sqrt {b d}}-\frac {5 b^{2} \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{3}}{32 d^{3} \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(473\) |
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Time = 0.27 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.35 \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\left [\frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} - 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) + 4 \, {\left (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{192 \, b^{2} d^{4}}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 22 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{96 \, b^{2} d^{4}}\right ] \]
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\[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{3} \left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.20 \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (\frac {4 \, {\left (b x^{2} + a\right )}}{b^{2} d} - \frac {5 \, b^{3} c d^{3} + a b^{2} d^{4}}{b^{4} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )}}{b^{4} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b d^{3}}\right )} b}{48 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^3 \left (a+b x^2\right )^{3/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^3\,{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {d\,x^2+c}} \,d x \]
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