Optimal. Leaf size=310 \[ -\frac {(b+a c)^2 \left (c+d x^2\right )^3}{a b^2 d^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{16 a^4 d^3}-\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right )^2 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{24 a^3 b d^3}+\frac {\left (7 b^2+12 a b c+6 a^2 c^2\right ) \left (c+d x^2\right )^3 \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{6 a^2 b^2 d^3}-\frac {b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{16 a^{9/2} d^3} \]
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Rubi [A]
time = 0.34, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1985, 1981,
1980, 473, 393, 205, 214} \begin {gather*} \frac {\left (6 a^2 c^2+12 a b c+7 b^2\right ) \left (c+d x^2\right )^3 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{6 a^2 b^2 d^3}-\frac {b \left (24 a^2 c^2+60 a b c+35 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{16 a^{9/2} d^3}+\frac {\left (24 a^2 c^2+60 a b c+35 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{16 a^4 d^3}-\frac {\left (24 a^2 c^2+60 a b c+35 b^2\right ) \left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{24 a^3 b d^3}-\frac {(a c+b)^2 \left (c+d x^2\right )^3}{a b^2 d^3 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 214
Rule 393
Rule 473
Rule 1980
Rule 1981
Rule 1985
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^5 \left (c+d x^2\right )^{3/2}}{\left (b+a \left (c+d x^2\right )\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^5 \left (c+d x^2\right )^{3/2}}{\left (b+a c+a d x^2\right )^{3/2}} \, dx}{\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\sqrt {b+a \left (c+d x^2\right )} \text {Subst}\left (\int \frac {x^2 (c+d x)^{3/2}}{(b+a c+a d x)^{3/2}} \, dx,x,x^2\right )}{2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\sqrt {b+a \left (c+d x^2\right )} \text {Subst}\left (\int \frac {(c+d x)^{3/2} \left (\frac {1}{2} (b+a c) (5 b+4 a c) d-\frac {1}{2} a b d^2 x\right )}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{a^2 b d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (\left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{12 a^2 b d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (\left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{16 a^3 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^4 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{32 a^4 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^4 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^2}} \, dx,x,\sqrt {c+d x^2}\right )}{16 a^4 d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^4 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{16 a^4 d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^4 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt {b+a \left (c+d x^2\right )} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{16 a^{9/2} d^3 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 183, normalized size = 0.59 \begin {gather*} \frac {\frac {\sqrt {a} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (105 b^3+5 a b^2 \left (43 c+7 d x^2\right )+2 a^2 b \left (59 c^2+16 c d x^2-7 d^2 x^4\right )+8 a^3 \left (c^3+d^3 x^6\right )\right )}{b+a \left (c+d x^2\right )}-3 b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{48 a^{9/2} d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1239\) vs.
\(2(288)=576\).
time = 0.12, size = 1240, normalized size = 4.00
method | result | size |
risch | \(\frac {\left (8 d^{2} a^{2} x^{4}-8 a^{2} c d \,x^{2}-22 a b d \,x^{2}+8 a^{2} c^{2}+62 a b c +57 b^{2}\right ) \left (a d \,x^{2}+a c +b \right )}{48 d^{3} a^{4} \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}+\frac {\left (-\frac {3 b \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right ) c^{2}}{4 a^{2} d^{2} \sqrt {a \,d^{2}}}-\frac {15 b^{2} \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right ) c}{8 a^{3} d^{2} \sqrt {a \,d^{2}}}-\frac {35 b^{3} \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{32 a^{4} d^{2} \sqrt {a \,d^{2}}}+\frac {b \,x^{2} c^{2}}{a^{2} d^{2} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {2 b^{2} x^{2} c}{a^{3} d^{2} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {b^{3} x^{2}}{a^{4} d^{2} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {b \,c^{3}}{a^{2} d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {2 b^{2} c^{2}}{a^{3} d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}+\frac {b^{3} c}{a^{4} d^{3} \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(656\) |
default | \(\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-48 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, a^{3} c \,d^{2} x^{4}-60 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, a^{2} b \,d^{2} x^{4}-72 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{3} b \,c^{2} d^{2} x^{2}-48 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, a^{3} c^{2} d \,x^{2}-180 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} b^{2} c \,d^{2} x^{2}+16 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a^{2} d \,x^{2}-72 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{3} b \,c^{3} d -105 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a \,b^{3} d^{2} x^{2}+54 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, a \,b^{2} d \,x^{2}-252 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} b^{2} c^{2} d +16 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a^{2} c +108 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, a^{2} b \,c^{2}-285 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a \,b^{3} c d +96 \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a^{2} b \,c^{2}+16 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a b +222 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, a \,b^{2} c -105 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) b^{4} d +192 \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,b^{2} c +114 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, b^{3}+96 \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b^{3}\right )}{96 a^{4} d^{3} \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, \left (a d \,x^{2}+a c +b \right )}\) | \(1240\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 389, normalized size = 1.25 \begin {gather*} \frac {48 \, a^{5} b c^{2} + 96 \, a^{4} b^{2} c + 48 \, a^{3} b^{3} - \frac {3 \, {\left (24 \, a^{2} b c^{2} + 60 \, a b^{2} c + 35 \, b^{3}\right )} {\left (a d x^{2} + a c + b\right )}^{3}}{{\left (d x^{2} + c\right )}^{3}} + \frac {8 \, {\left (24 \, a^{3} b c^{2} + 60 \, a^{2} b^{2} c + 35 \, a b^{3}\right )} {\left (a d x^{2} + a c + b\right )}^{2}}{{\left (d x^{2} + c\right )}^{2}} - \frac {3 \, {\left (56 \, a^{4} b c^{2} + 132 \, a^{3} b^{2} c + 77 \, a^{2} b^{3}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}}{48 \, {\left (a^{7} d^{3} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - 3 \, a^{6} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, a^{5} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - a^{4} d^{3} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {7}{2}}\right )}} + \frac {{\left (24 \, a^{2} c^{2} + 60 \, a b c + 35 \, b^{2}\right )} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{32 \, a^{\frac {9}{2}} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.48, size = 675, normalized size = 2.18 \begin {gather*} \left [\frac {3 \, {\left (24 \, a^{3} b c^{3} + 84 \, a^{2} b^{2} c^{2} + 95 \, a b^{3} c + 35 \, b^{4} + {\left (24 \, a^{3} b c^{2} + 60 \, a^{2} b^{2} c + 35 \, a b^{3}\right )} d x^{2}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} - 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (8 \, a^{4} d^{4} x^{8} + 2 \, {\left (4 \, a^{4} c - 7 \, a^{3} b\right )} d^{3} x^{6} + 8 \, a^{4} c^{4} + 118 \, a^{3} b c^{3} + {\left (18 \, a^{3} b c + 35 \, a^{2} b^{2}\right )} d^{2} x^{4} + 215 \, a^{2} b^{2} c^{2} + 105 \, a b^{3} c + {\left (8 \, a^{4} c^{3} + 150 \, a^{3} b c^{2} + 250 \, a^{2} b^{2} c + 105 \, a b^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, {\left (a^{6} d^{4} x^{2} + {\left (a^{6} c + a^{5} b\right )} d^{3}\right )}}, \frac {3 \, {\left (24 \, a^{3} b c^{3} + 84 \, a^{2} b^{2} c^{2} + 95 \, a b^{3} c + 35 \, b^{4} + {\left (24 \, a^{3} b c^{2} + 60 \, a^{2} b^{2} c + 35 \, a b^{3}\right )} d x^{2}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) + 2 \, {\left (8 \, a^{4} d^{4} x^{8} + 2 \, {\left (4 \, a^{4} c - 7 \, a^{3} b\right )} d^{3} x^{6} + 8 \, a^{4} c^{4} + 118 \, a^{3} b c^{3} + {\left (18 \, a^{3} b c + 35 \, a^{2} b^{2}\right )} d^{2} x^{4} + 215 \, a^{2} b^{2} c^{2} + 105 \, a b^{3} c + {\left (8 \, a^{4} c^{3} + 150 \, a^{3} b c^{2} + 250 \, a^{2} b^{2} c + 105 \, a b^{3}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, {\left (a^{6} d^{4} x^{2} + {\left (a^{6} c + a^{5} b\right )} d^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 597 vs.
\(2 (288) = 576\).
time = 4.07, size = 597, normalized size = 1.93 \begin {gather*} \frac {1}{48} \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (2 \, x^{2} {\left (\frac {4 \, x^{2}}{a^{2} d \mathrm {sgn}\left (d x^{2} + c\right )} - \frac {4 \, a^{11} c d^{6} \mathrm {sgn}\left (d x^{2} + c\right ) + 11 \, a^{10} b d^{6} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{13} d^{8}}\right )} + \frac {8 \, a^{11} c^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) + 62 \, a^{10} b c d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) + 57 \, a^{9} b^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{13} d^{8}}\right )} + \frac {{\left (24 \, a^{\frac {5}{2}} b c^{2} + 60 \, a^{\frac {3}{2}} b^{2} c + 35 \, \sqrt {a} b^{3}\right )} \log \left ({\left | -2 \, a^{\frac {7}{2}} c^{3} d - 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{3} c^{2} {\left | d \right |} - 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {5}{2}} c d - 5 \, a^{\frac {5}{2}} b c^{2} d - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a^{2} {\left | d \right |} - 10 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} b c {\left | d \right |} - 5 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {3}{2}} b d - 4 \, a^{\frac {3}{2}} b^{2} c d - 4 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b^{2} {\left | d \right |} - \sqrt {a} b^{3} d \right |}\right )}{96 \, a^{5} d^{2} {\left | d \right |} \mathrm {sgn}\left (d x^{2} + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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