Optimal. Leaf size=249 \[ -\frac {\left (-24 a^2 c^2+60 a b c+5 b^2\right ) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{48 a d^3}-\frac {b \left (-24 a^2 c^2+12 a b c+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{16 a^{3/2} d^3}-\frac {b c^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{d^3}+\frac {\left (c+d x^2\right )^3 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{5/2}}{6 a d^3}-\frac {(12 a c+b) \left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{24 d^3} \]
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Rubi [A] time = 0.73, antiderivative size = 311, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6722, 1975, 446, 89, 80, 50, 63, 217, 206} \[ -\frac {\left (-24 a^2 c^2+12 a b c+b^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )}{24 a b d^3}-\frac {\left (-24 a^2 c^2+12 a b c+b^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{16 a d^3}-\frac {b \left (-24 a^2 c^2+12 a b c+b^2\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {a \left (c+d x^2\right )+b}}\right )}{16 a^{3/2} d^3 \sqrt {a \left (c+d x^2\right )+b}}-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )^2}{b d^3}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (a \left (c+d x^2\right )+b\right )^2}{6 a d^3} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 89
Rule 206
Rule 217
Rule 446
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int x^5 \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^5 \left (b+a \left (c+d x^2\right )\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \int \frac {x^5 \left (b+a c+a d x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}} \, dx}{\sqrt {b+a \left (c+d x^2\right )}}\\ &=\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {x^2 (b+a c+a d x)^{3/2}}{(c+d x)^{3/2}} \, dx,x,x^2\right )}{2 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac {\left (\sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {(b+a c+a d x)^{3/2} \left (-\frac {1}{2} c (b-4 a c) d+\frac {1}{2} b d^2 x\right )}{\sqrt {c+d x}} \, dx,x,x^2\right )}{b d^3 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac {\left (\left (-\frac {3}{2} a c (b-4 a c) d^3-\frac {1}{2} b d^2 \left (\frac {5 a c d}{2}+\frac {1}{2} (b+a c) d\right )\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {(b+a c+a d x)^{3/2}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{3 a b d^5 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac {\left (\left (-\frac {3}{2} a c (b-4 a c) d^3-\frac {1}{2} b d^2 \left (\frac {5 a c d}{2}+\frac {1}{2} (b+a c) d\right )\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a c+a d x}}{\sqrt {c+d x}} \, dx,x,x^2\right )}{4 a d^5 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{16 a d^3}-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac {\left (b \left (-\frac {3}{2} a c (b-4 a c) d^3-\frac {1}{2} b d^2 \left (\frac {5 a c d}{2}+\frac {1}{2} (b+a c) d\right )\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{8 a d^5 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{16 a d^3}-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac {\left (b \left (-\frac {3}{2} a c (b-4 a c) d^3-\frac {1}{2} b d^2 \left (\frac {5 a c d}{2}+\frac {1}{2} (b+a c) d\right )\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^2}} \, dx,x,\sqrt {c+d x^2}\right )}{4 a d^6 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{16 a d^3}-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}--\frac {\left (b \left (-\frac {3}{2} a c (b-4 a c) d^3-\frac {1}{2} b d^2 \left (\frac {5 a c d}{2}+\frac {1}{2} (b+a c) d\right )\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}\right ) \operatorname {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 )}{4 a d^6 \sqrt {b+a \left (c+d x^2\right )}}\\ &=-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{16 a d^3}-\frac {\left (b^2+12 a b c-24 a^2 c^2\right ) \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )}{24 a b d^3}-\frac {c^2 \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{b d^3}+\frac {\left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}} \left (b+a \left (c+d x^2\right )\right )^2}{6 a d^3}-\frac {b \left (b^2+12 a b c-24 a^2 c^2\right ) \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}}\right )}{16 a^{3/2} d^3 \sqrt {b+a \left (c+d x^2\right )}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 142, normalized size = 0.57 \[ \frac {\sqrt {a} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (8 a^2 \left (c^3+d^3 x^6\right )-2 a b \left (47 c^2+16 c d x^2-7 d^2 x^4\right )+3 b^2 \left (c+d x^2\right )\right )-3 b \left (-24 a^2 c^2+12 a b c+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{48 a^{3/2} d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 427, normalized size = 1.71 \[ \left [\frac {3 \, {\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - b^{3}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (8 \, a^{3} d^{3} x^{6} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c - {\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{192 \, a^{2} d^{3}}, -\frac {3 \, {\left (24 \, a^{2} b c^{2} - 12 \, a b^{2} c - b^{3}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \, {\left (8 \, a^{3} d^{3} x^{6} + 14 \, a^{2} b d^{2} x^{4} + 8 \, a^{3} c^{3} - 94 \, a^{2} b c^{2} + 3 \, a b^{2} c - {\left (32 \, a^{2} b c - 3 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{96 \, a^{2} d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.10, size = 527, normalized size = 2.12 \[ \frac {1}{48} \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (2 \, {\left (\frac {4 \, a x^{2} \mathrm {sgn}\left (d x^{2} + c\right )}{d} - \frac {4 \, a^{3} c d^{6} \mathrm {sgn}\left (d x^{2} + c\right ) - 7 \, a^{2} b d^{6} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{8}}\right )} x^{2} + \frac {8 \, a^{3} c^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) - 46 \, a^{2} b c d^{5} \mathrm {sgn}\left (d x^{2} + c\right ) + 3 \, a b^{2} d^{5} \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{8}}\right )} - \frac {{\left (24 \, a^{\frac {5}{2}} b c^{2} \mathrm {sgn}\left (d x^{2} + c\right ) - 12 \, a^{\frac {3}{2}} b^{2} c \mathrm {sgn}\left (d x^{2} + c\right ) - \sqrt {a} b^{3} \mathrm {sgn}\left (d x^{2} + c\right )\right )} \log \left ({\left | -2 \, a^{\frac {5}{2}} c^{3} d - 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} c^{2} {\left | d \right |} - 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {3}{2}} c d - a^{\frac {3}{2}} b c^{2} d - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a {\left | d \right |} - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b c {\left | d \right |} - {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} \sqrt {a} b d \right |}\right )}{96 \, a^{2} d^{2} {\left | d \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1018, normalized size = 4.09 \[ -\frac {\left (-72 a^{2} b \,c^{2} d^{2} x^{2} \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+48 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a^{2} c \,d^{2} x^{4}+36 a \,b^{2} c \,d^{2} x^{2} \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-72 a^{2} b \,c^{3} d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-12 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b \,d^{2} x^{4}+3 b^{3} d^{2} x^{2} \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+48 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a^{2} c^{2} d \,x^{2}+36 a \,b^{2} c^{2} d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+96 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b c d \,x^{2}+3 b^{3} c d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-6 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b^{2} d \,x^{2}+96 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, a b \,c^{2}+108 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b \,c^{2}-16 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a d \,x^{2}-6 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b^{2} c -16 \left (a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{96 \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, a \,d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.71, size = 368, normalized size = 1.48 \[ -\frac {b c^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{d^{3}} - \frac {3 \, {\left (8 \, a^{2} b c^{2} - 20 \, a b^{2} c + b^{3}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}} - 8 \, {\left (6 \, a^{3} b c^{2} - 12 \, a^{2} b^{2} c - a b^{3}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + 3 \, {\left (8 \, a^{4} b c^{2} - 12 \, a^{3} b^{2} c - a^{2} b^{3}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{48 \, {\left (a^{4} d^{3} - \frac {3 \, {\left (a d x^{2} + a c + b\right )} a^{3} d^{3}}{d x^{2} + c} + \frac {3 \, {\left (a d x^{2} + a c + b\right )}^{2} a^{2} d^{3}}{{\left (d x^{2} + c\right )}^{2}} - \frac {{\left (a d x^{2} + a c + b\right )}^{3} a d^{3}}{{\left (d x^{2} + c\right )}^{3}}\right )}} - \frac {{\left (24 \, a^{2} c^{2} - 12 \, a b c - b^{2}\right )} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{32 \, a^{\frac {3}{2}} d^{3}} \]
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 \[ \int x^5\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2} \,d x \]
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 \[ \int x^{5} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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