Integrand size = 26, antiderivative size = 340 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{256 b^{5/2} d^{11/2}} \]
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Time = 0.28 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {457, 92, 81, 52, 65, 223, 212} \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=-\frac {(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{256 b^{5/2} d^{11/2}}+\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{256 b^2 d^5}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{384 b^2 d^4}+\frac {\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{480 b^2 d^3}-\frac {3 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2} (a d+3 b c)}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d} \]
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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 (a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}+\frac {\text {Subst}\left (\int \frac {(a+b x)^{5/2} \left (-a c-\frac {3}{2} (3 b c+a d) x\right )}{\sqrt {c+d x}} \, dx,x,x^2\right )}{10 b d} \\ & = -\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \text {Subst}\left (\int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{160 b^2 d^2} \\ & = \frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {\left ((b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{192 b^2 d^3} \\ & = -\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}+\frac {\left ((b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{256 b^2 d^4} \\ & = \frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 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 )}{512 b^2 d^5} \\ & = \frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 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 )}{256 b^3 d^5} \\ & = \frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {\left ((b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 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 )}{256 b^3 d^5} \\ & = \frac {(b c-a d)^2 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{256 b^2 d^5}-\frac {(b c-a d) \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{384 b^2 d^4}+\frac {\left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}}{480 b^2 d^3}-\frac {3 (3 b c+a d) \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{80 b^2 d^2}+\frac {x^2 \left (a+b x^2\right )^{7/2} \sqrt {c+d x^2}}{10 b d}-\frac {(b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{256 b^{5/2} d^{11/2}} \\ \end{align*}
Time = 3.84 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.80 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (-\frac {24 (3 b c+a d) \left (a+b x^2\right )^4}{b d}+64 x^2 \left (a+b x^2\right )^4+\frac {5 (b c-a d)^3 \left (63 b^2 c^2+14 a b c d+3 a^2 d^2\right ) \left (\frac {2 d \left (a+b x^2\right )}{b c-a d}-\frac {4 d^2 \left (a+b x^2\right )^2}{3 (b c-a d)^2}+\frac {16 d^3 \left (a+b x^2\right )^3}{15 (b c-a d)^3}-\frac {2 \sqrt {d} \sqrt {a+b x^2} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}}}\right )}{4 b d^5}\right )}{640 b d \sqrt {a+b x^2}} \]
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Time = 3.22 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.08
method | result | size |
risch | \(-\frac {\left (-384 b^{4} x^{8} d^{4}-1008 a \,b^{3} d^{4} x^{6}+432 b^{4} c \,d^{3} x^{6}-744 a^{2} b^{2} d^{4} x^{4}+1184 a \,b^{3} c \,d^{3} x^{4}-504 b^{4} c^{2} d^{2} x^{4}-30 a^{3} b \,d^{4} x^{2}+962 a^{2} b^{2} c \,d^{3} x^{2}-1498 a \,b^{3} c^{2} d^{2} x^{2}+630 b^{4} c^{3} d \,x^{2}+45 a^{4} d^{4}+90 a^{3} b c \,d^{3}-1564 a^{2} b^{2} c^{2} d^{2}+2310 a \,b^{3} c^{3} d -945 b^{4} c^{4}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{3840 b^{2} d^{5}}+\frac {\left (3 a^{5} d^{5}+5 a^{4} b c \,d^{4}+30 a^{3} b^{2} c^{2} d^{3}-150 a^{2} b^{3} c^{3} d^{2}+175 a \,b^{4} c^{4} d -63 b^{5} c^{5}\right ) \ln \left (\frac {\frac {1}{2} a d +\frac {1}{2} b c +b d \,x^{2}}{\sqrt {b d}}+\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{512 b^{2} d^{5} \sqrt {b d}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(368\) |
default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (768 b^{4} d^{4} x^{8} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+2016 a \,b^{3} d^{4} x^{6} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}-864 b^{4} c \,d^{3} x^{6} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+1488 a^{2} b^{2} d^{4} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}-2368 a \,b^{3} c \,d^{3} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+1008 b^{4} c^{2} d^{2} x^{4} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}+60 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{3} b \,d^{4} x^{2}-1924 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} b^{2} c \,d^{3} x^{2}+2996 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a \,b^{3} c^{2} d^{2} x^{2}-1260 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{4} c^{3} d \,x^{2}+45 \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^{5} d^{5}+75 \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} b c \,d^{4}+450 \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^{2} c^{2} d^{3}-2250 \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^{3} c^{3} d^{2}+2625 \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^{4} c^{4} d -945 \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^{5} c^{5}-90 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{4} d^{4}-180 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{3} b c \,d^{3}+3128 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a^{2} b^{2} c^{2} d^{2}-4620 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, a \,b^{3} c^{3} d +1890 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}\, b^{4} c^{4}\right )}{7680 b^{2} d^{5} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \sqrt {b d}}\) | \(900\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {21 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} c^{3}}{128 d^{4}}+\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a^{3}}{128 b d}+\frac {175 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^{4} a}{512 d^{4} \sqrt {b d}}+\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} c}{512 b d \sqrt {b d}}-\frac {37 b \,x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a c}{120 d^{2}}-\frac {75 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^{3}}{256 d^{3} \sqrt {b d}}+\frac {749 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a \,c^{2}}{1920 d^{3}}-\frac {77 b \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a \,c^{3}}{128 d^{4}}-\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{3} c}{128 b \,d^{2}}+\frac {21 b \,x^{6} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a}{80 d}-\frac {9 b^{2} x^{6} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c}{80 d^{2}}+\frac {21 b^{2} x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{2}}{160 d^{3}}+\frac {15 \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} c^{2}}{256 d^{2} \sqrt {b d}}-\frac {481 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, x^{2} a^{2} c}{1920 d^{2}}-\frac {63 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^{5}}{512 d^{5} \sqrt {b d}}+\frac {31 x^{4} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2}}{160 d}+\frac {63 b^{2} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, c^{4}}{256 d^{5}}+\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^{5}}{512 b^{2} \sqrt {b d}}-\frac {3 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{4}}{256 b^{2} d}+\frac {b^{2} x^{8} \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{10 d}+\frac {391 \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}\, a^{2} c^{2}}{960 d^{3}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(932\) |
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Time = 0.34 (sec) , antiderivative size = 734, normalized size of antiderivative = 2.16 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\left [-\frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{8} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{6} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{4} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15360 \, b^{3} d^{6}}, \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\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 (384 \, b^{5} d^{5} x^{8} + 945 \, b^{5} c^{4} d - 2310 \, a b^{4} c^{3} d^{2} + 1564 \, a^{2} b^{3} c^{2} d^{3} - 90 \, a^{3} b^{2} c d^{4} - 45 \, a^{4} b d^{5} - 144 \, {\left (3 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{6} + 8 \, {\left (63 \, b^{5} c^{2} d^{3} - 148 \, a b^{4} c d^{4} + 93 \, a^{2} b^{3} d^{5}\right )} x^{4} - 2 \, {\left (315 \, b^{5} c^{3} d^{2} - 749 \, a b^{4} c^{2} d^{3} + 481 \, a^{2} b^{3} c d^{4} - 15 \, a^{3} b^{2} d^{5}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{7680 \, b^{3} d^{6}}\right ] \]
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\[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^{5} \left (a + b x^{2}\right )^{\frac {5}{2}}}{\sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.33 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.17 \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\frac {{\left (\sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \sqrt {b x^{2} + a} {\left (2 \, {\left (b x^{2} + a\right )} {\left (4 \, {\left (b x^{2} + a\right )} {\left (6 \, {\left (b x^{2} + a\right )} {\left (\frac {8 \, {\left (b x^{2} + a\right )}}{b^{3} d} - \frac {9 \, b^{7} c d^{7} + 11 \, a b^{6} d^{8}}{b^{9} d^{9}}\right )} + \frac {63 \, b^{8} c^{2} d^{6} + 14 \, a b^{7} c d^{7} + 3 \, a^{2} b^{6} d^{8}}{b^{9} d^{9}}\right )} - \frac {5 \, {\left (63 \, b^{9} c^{3} d^{5} - 49 \, a b^{8} c^{2} d^{6} - 11 \, a^{2} b^{7} c d^{7} - 3 \, a^{3} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} + \frac {15 \, {\left (63 \, b^{10} c^{4} d^{4} - 112 \, a b^{9} c^{3} d^{5} + 38 \, a^{2} b^{8} c^{2} d^{6} + 8 \, a^{3} b^{7} c d^{7} + 3 \, a^{4} b^{6} d^{8}\right )}}{b^{9} d^{9}}\right )} + \frac {15 \, {\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left ({\left | -\sqrt {b x^{2} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{5}}\right )} b}{3840 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^5 \left (a+b x^2\right )^{5/2}}{\sqrt {c+d x^2}} \, dx=\int \frac {x^5\,{\left (b\,x^2+a\right )}^{5/2}}{\sqrt {d\,x^2+c}} \,d x \]
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