Integrand size = 26, antiderivative size = 209 \[ \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{5/2} d^{7/2}} \]
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Time = 0.18 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {457, 92, 81, 52, 65, 223, 212} \[ \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=-\frac {(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{5/2} d^{7/2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (a^2 d^2+2 a b c d+5 b^2 c^2\right )}{16 b^2 d^3}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (3 a d+5 b c)}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d} \]
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Rule 52
Rule 65
Rule 81
Rule 92
Rule 212
Rule 223
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2 \sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x} \left (-a c-\frac {1}{2} (5 b c+3 a d) x\right )}{\sqrt {c+d x}} \, dx,x,x^2\right )}{6 b d} \\ & = -\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}+\frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{16 b^2 d^2} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{32 b^2 d^3} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\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^3 d^3} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {\left ((b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right )\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^3 d^3} \\ & = \frac {\left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{16 b^2 d^3}-\frac {(5 b c+3 a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{24 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{6 b d}-\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{16 b^{5/2} d^{7/2}} \\ \end{align*}
Time = 1.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.89 \[ \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {-b \sqrt {d} \sqrt {a+b x^2} \left (c+d x^2\right ) \left (3 a^2 d^2-2 a b d \left (-2 c+d x^2\right )+b^2 \left (-15 c^2+10 c d x^2-8 d^2 x^4\right )\right )-3 (b c-a d)^{3/2} \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{48 b^3 d^{7/2} \sqrt {c+d x^2}} \]
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Time = 2.13 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.01
method | result | size |
risch | \(-\frac {\left (-8 b^{2} d^{2} x^{4}-2 x^{2} a b \,d^{2}+10 x^{2} b^{2} c d +3 a^{2} d^{2}+4 a b c d -15 b^{2} c^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{48 b^{2} d^{3}}+\frac {\left (a^{3} d^{3}+a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{32 b^{2} d^{3} \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(211\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (16 b^{2} d^{2} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+4 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b \,d^{2} x^{2}-20 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{2} c d \,x^{2}+3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{3}+3 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c \,d^{2}+9 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2} d -15 \ln \left (\frac {2 b d \,x^{2}+2 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3}-6 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} d^{2}-8 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a b c d +30 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{2} c^{2}\right )}{96 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, d^{3} b^{2} \sqrt {b d}}\) | \(455\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{6 d}+\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a}{24 b d}-\frac {5 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} c}{24 d^{2}}-\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2}}{16 b^{2} d}-\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a c}{12 b \,d^{2}}+\frac {5 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{2}}{16 d^{3}}+\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{3}}{32 b^{2} \sqrt {b d}}+\frac {\ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a^{2} c}{32 b d \sqrt {b d}}+\frac {3 \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) a \,c^{2}}{32 d^{2} \sqrt {b d}}-\frac {5 b \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) c^{3}}{32 d^{3} \sqrt {b d}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(476\) |
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Time = 0.30 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.11 \[ \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {b d}\right ) - 4 \, {\left (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{192 \, b^{3} d^{4}}, \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{4} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (8 \, b^{3} d^{3} x^{4} + 15 \, b^{3} c^{2} d - 4 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3} - 2 \, {\left (5 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{96 \, b^{3} d^{4}}\right ] \]
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\[ \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{5} \sqrt {a + b x^{2}}}{\sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.08 \[ \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (\frac {4 \, {\left (b x^{2} + a\right )}}{b^{3} d} - \frac {5 \, b^{7} c d^{3} + 7 \, a b^{6} d^{4}}{b^{9} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{2} d^{2} + 2 \, a b^{7} c d^{3} + a^{2} b^{6} d^{4}\right )}}{b^{9} d^{5}}\right )} + \frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}}\right )} b}{48 \, {\left | b \right |}} \]
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Time = 67.88 (sec) , antiderivative size = 993, normalized size of antiderivative = 4.75 \[ \int \frac {x^5 \sqrt {a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {d}\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2+2\,a\,b\,c\,d+5\,b^2\,c^2\right )}{8\,b^{5/2}\,d^{7/2}}-\frac {\frac {\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )\,\left (\frac {a^3\,b^3\,d^3}{8}+\frac {a^2\,b^4\,c\,d^2}{8}+\frac {3\,a\,b^5\,c^2\,d}{8}-\frac {5\,b^6\,c^3}{8}\right )}{d^9\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^5\,\left (\frac {19\,a^3\,b\,d^3}{4}+\frac {275\,a^2\,b^2\,c\,d^2}{4}+\frac {313\,a\,b^3\,c^2\,d}{4}+\frac {33\,b^4\,c^3}{4}\right )}{d^7\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^5}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^7\,\left (\frac {19\,a^3\,d^3}{4}+\frac {275\,a^2\,b\,c\,d^2}{4}+\frac {313\,a\,b^2\,c^2\,d}{4}+\frac {33\,b^3\,c^3}{4}\right )}{d^6\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^7}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3\,\left (\frac {17\,a^3\,b^2\,d^3}{24}+\frac {91\,a^2\,b^3\,c\,d^2}{8}+\frac {17\,a\,b^4\,c^2\,d}{8}-\frac {85\,b^5\,c^3}{24}\right )}{d^8\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^3}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{11}\,\left (\frac {a^3\,d^3}{8}+\frac {a^2\,b\,c\,d^2}{8}+\frac {3\,a\,b^2\,c^2\,d}{8}-\frac {5\,b^3\,c^3}{8}\right )}{b^2\,d^4\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^{11}}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^9\,\left (\frac {17\,a^3\,d^3}{24}+\frac {91\,a^2\,b\,c\,d^2}{8}+\frac {17\,a\,b^2\,c^2\,d}{8}-\frac {85\,b^3\,c^3}{24}\right )}{b\,d^5\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^9}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^8\,\left (16\,d\,a^2+48\,b\,c\,a\right )}{d^4\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^8}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4\,\left (16\,d\,a^2\,b^2+48\,c\,a\,b^3\right )}{d^6\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}+\frac {\sqrt {a}\,\sqrt {c}\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6\,\left (32\,a^2\,b\,d^2+\frac {352\,a\,b^2\,c\,d}{3}+64\,b^3\,c^2\right )}{d^6\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^6}}{\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{12}}{{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^{12}}+\frac {b^6}{d^6}-\frac {6\,b^5\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2}{d^5\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}+\frac {15\,b^4\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^4}{d^4\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^4}-\frac {20\,b^3\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^6}{d^3\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^6}+\frac {15\,b^2\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^8}{d^2\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^8}-\frac {6\,b\,{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^{10}}{d\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^{10}}} \]
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