Optimal. Leaf size=187 \[ \frac {3 b (4 a c+5 b) \tanh ^{-1}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{8 a^{7/2} d^2}-\frac {(4 a c+7 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{8 a^3 d^2}-\frac {b (a c+b)}{a^3 d^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}+\frac {\left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a^2 d^2} \]
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Rubi [A] time = 0.57, antiderivative size = 242, normalized size of antiderivative = 1.29, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6722, 1975, 446, 78, 50, 63, 217, 206} \[ \frac {(4 a c+5 b) \left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}{4 a^2 b d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 (4 a c+5 b) \left (a \left (c+d x^2\right )+b\right )}{8 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {3 b (4 a c+5 b) \sqrt {a \left (c+d x^2\right )+b} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {a \left (c+d x^2\right )+b}}\right )}{8 a^{7/2} d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}-\frac {(a c+b) \left (c+d x^2\right )^2}{a b d^2 \sqrt {a+\frac {b}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rule 446
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int \frac {x^3}{\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^3 \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^3 \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 )} \operatorname {Subst}\left (\int \frac {x (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) \left (c+d x^2\right )^2}{a b d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left ((5 b+4 a c) \sqrt {b+a \left (c+d x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^{3/2}}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{2 a b d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(b+a c) \left (c+d x^2\right )^2}{a b d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(5 b+4 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a^2 b d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left (3 (5 b+4 a c) \sqrt {b+a \left (c+d x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{8 a^2 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(b+a c) \left (c+d x^2\right )^2}{a b d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 (5 b+4 a c) \left (b+a \left (c+d x^2\right )\right )}{8 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(5 b+4 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a^2 b d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b (5 b+4 a c) \sqrt {b+a \left (c+d x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+d x} \sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{16 a^3 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(b+a c) \left (c+d x^2\right )^2}{a b d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 (5 b+4 a c) \left (b+a \left (c+d x^2\right )\right )}{8 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(5 b+4 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a^2 b d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b (5 b+4 a c) \sqrt {b+a \left (c+d x^2\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^2}} \, dx,x,\sqrt {c+d x^2}\right )}{8 a^3 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(b+a c) \left (c+d x^2\right )^2}{a b d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 (5 b+4 a c) \left (b+a \left (c+d x^2\right )\right )}{8 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(5 b+4 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a^2 b d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (3 b (5 b+4 a c) \sqrt {b+a \left (c+d x^2\right )}\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 )}{8 a^3 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(b+a c) \left (c+d x^2\right )^2}{a b d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {3 (5 b+4 a c) \left (b+a \left (c+d x^2\right )\right )}{8 a^3 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {(5 b+4 a c) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a^2 b d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {3 b (5 b+4 a c) \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 )}{8 a^{7/2} d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 133, normalized size = 0.71 \[ \frac {3 b (4 a c+5 b) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )-\sqrt {a} \left (2 a^2 \left (c^2-d^2 x^4\right )+a b \left (17 c+5 d x^2\right )+15 b^2\right )}{8 a^{7/2} d^2 \sqrt {a+\frac {b}{c+d x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 541, normalized size = 2.89 \[ \left [\frac {3 \, {\left (4 \, a^{2} b c^{2} + 9 \, a b^{2} c + {\left (4 \, a^{2} b c + 5 \, a b^{2}\right )} d x^{2} + 5 \, b^{3}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (2 \, a^{3} d^{3} x^{6} + {\left (2 \, a^{3} c - 5 \, a^{2} b\right )} d^{2} x^{4} - 2 \, a^{3} c^{3} - 17 \, a^{2} b c^{2} - 15 \, a b^{2} c - {\left (2 \, a^{3} c^{2} + 22 \, a^{2} b c + 15 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, {\left (a^{5} d^{3} x^{2} + {\left (a^{5} c + a^{4} b\right )} d^{2}\right )}}, -\frac {3 \, {\left (4 \, a^{2} b c^{2} + 9 \, a b^{2} c + {\left (4 \, a^{2} b c + 5 \, a b^{2}\right )} d x^{2} + 5 \, b^{3}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \, {\left (2 \, a^{3} d^{3} x^{6} + {\left (2 \, a^{3} c - 5 \, a^{2} b\right )} d^{2} x^{4} - 2 \, a^{3} c^{3} - 17 \, a^{2} b c^{2} - 15 \, a b^{2} c - {\left (2 \, a^{3} c^{2} + 22 \, a^{2} b c + 15 \, a b^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, {\left (a^{5} d^{3} x^{2} + {\left (a^{5} c + a^{4} b\right )} d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.57, size = 510, normalized size = 2.73 \[ \frac {1}{8} \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (\frac {2 \, x^{2}}{a^{2} d \mathrm {sgn}\left (d x^{2} + c\right )} - \frac {2 \, a^{6} c d^{2} + 7 \, a^{5} b d^{2}}{a^{8} d^{4} \mathrm {sgn}\left (d x^{2} + c\right )}\right )} - \frac {{\left (4 \, a^{\frac {3}{2}} b c + 5 \, \sqrt {a} b^{2}\right )} \log \left ({\left | -2 \, a^{\frac {7}{2}} c^{3} d - 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{3} c^{2} {\left | d \right |} - 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {5}{2}} c d - 5 \, a^{\frac {5}{2}} b c^{2} d - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} a^{2} {\left | d \right |} - 10 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{2} b c {\left | d \right |} - 5 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a^{\frac {3}{2}} b d - 4 \, a^{\frac {3}{2}} b^{2} c d - 4 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a b^{2} {\left | d \right |} - \sqrt {a} b^{3} d \right |}\right )}{16 \, a^{4} d {\left | d \right |} \mathrm {sgn}\left (d x^{2} + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 783, normalized size = 4.19 \[ -\frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (-12 a^{2} b c \,d^{2} x^{2} \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-4 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a^{2} d^{2} x^{4}-15 a \,b^{2} d^{2} x^{2} \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-12 a^{2} b \,c^{2} d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-27 a \,b^{2} c d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+10 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b d \,x^{2}-15 b^{3} d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+4 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a^{2} c^{2}+16 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, a b c +18 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a b c +16 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, b^{2}+14 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b^{2}\right )}{16 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, \left (a d \,x^{2}+a c +b \right ) a^{3} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.31, size = 262, normalized size = 1.40 \[ -\frac {8 \, a^{3} b c + 8 \, a^{2} b^{2} + \frac {3 \, {\left (a d x^{2} + a c + b\right )}^{2} {\left (4 \, a b c + 5 \, b^{2}\right )}}{{\left (d x^{2} + c\right )}^{2}} - \frac {5 \, {\left (4 \, a^{2} b c + 5 \, a b^{2}\right )} {\left (a d x^{2} + a c + b\right )}}{d x^{2} + c}}{8 \, {\left (a^{5} d^{2} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - 2 \, a^{4} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} + a^{3} d^{2} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {5}{2}}\right )}} - \frac {3 \, {\left (4 \, a c + 5 \, b\right )} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{16 \, a^{\frac {7}{2}} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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