Optimal. Leaf size=148 \[ \frac {b (4 a c+3 b) \tanh ^{-1}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{8 a^{5/2} d^2}-\frac {(4 a c+3 b) \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{8 a^2 d^2}+\frac {\left (c+d x^2\right )^2 \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{4 a d^2} \]
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Rubi [A] time = 0.46, antiderivative size = 189, normalized size of antiderivative = 1.28, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6722, 1975, 446, 80, 50, 63, 217, 206} \[ -\frac {(4 a c+3 b) \left (a \left (c+d x^2\right )+b\right )}{8 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {b (4 a c+3 b) \sqrt {a \left (c+d x^2\right )+b} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {a \left (c+d x^2\right )+b}}\right )}{8 a^{5/2} d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}{4 a d^2 \sqrt {a+\frac {b}{c+d x^2}}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 206
Rule 217
Rule 446
Rule 1975
Rule 6722
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a+\frac {b}{c+d x^2}}} \, dx &=\frac {\sqrt {b+a \left (c+d x^2\right )} \int \frac {x^3 \sqrt {c+d x^2}}{\sqrt {b+a \left (c+d x^2\right )}} \, 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 \sqrt {c+d x^2}}{\sqrt {b+a c+a d x^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 \sqrt {c+d x}}{\sqrt {b+a c+a d x}} \, dx,x,x^2\right )}{2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=\frac {\left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}-\frac {\left ((3 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 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(3 b+4 a c) \left (b+a \left (c+d x^2\right )\right )}{8 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (b (3 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^2 d \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(3 b+4 a c) \left (b+a \left (c+d x^2\right )\right )}{8 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (b (3 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^2 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(3 b+4 a c) \left (b+a \left (c+d x^2\right )\right )}{8 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (b (3 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^2 d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ &=-\frac {(3 b+4 a c) \left (b+a \left (c+d x^2\right )\right )}{8 a^2 d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{4 a d^2 \sqrt {a+\frac {b}{c+d x^2}}}+\frac {b (3 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^{5/2} d^2 \sqrt {c+d x^2} \sqrt {a+\frac {b}{c+d x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 101, normalized size = 0.68 \[ \frac {b (4 a c+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )-\sqrt {a} \left (c+d x^2\right ) \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (2 a \left (c-d x^2\right )+3 b\right )}{8 a^{5/2} d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 333, normalized size = 2.25 \[ \left [\frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt {a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (2 \, a^{2} d^{2} x^{4} - 3 \, a b d x^{2} - 2 \, a^{2} c^{2} - 3 \, a b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{32 \, a^{3} d^{2}}, -\frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \sqrt {-a} \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \, {\left (2 \, a^{2} d^{2} x^{4} - 3 \, a b d x^{2} - 2 \, a^{2} c^{2} - 3 \, a b c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{16 \, a^{3} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 191, normalized size = 1.29 \[ \frac {1}{8} \, \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} {\left (\frac {2 \, x^{2}}{a d \mathrm {sgn}\left (d x^{2} + c\right )} - \frac {2 \, a c d \mathrm {sgn}\left (d x^{2} + c\right ) + 3 \, b d \mathrm {sgn}\left (d x^{2} + c\right )}{a^{2} d^{3}}\right )} - \frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \log \left ({\left | -2 \, a c d - 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} \sqrt {a} {\left | d \right |} - b d \right |}\right )}{16 \, a^{\frac {5}{2}} d {\left | d \right |} \mathrm {sgn}\left (d x^{2} + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 354, normalized size = 2.39 \[ \frac {\sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (4 a b c d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )+4 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a d \,x^{2}+3 b^{2} d \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +b d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}}{2 \sqrt {a \,d^{2}}}\right )-4 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a c -6 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, b \right )}{16 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, a^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.49, size = 223, normalized size = 1.51 \[ -\frac {{\left (4 \, a b c + 3 \, b^{2}\right )} \left (\frac {a d x^{2} + a c + b}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (4 \, a^{2} b c + 5 \, a b^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, {\left (a^{4} d^{2} - \frac {2 \, {\left (a d x^{2} + a c + b\right )} a^{3} d^{2}}{d x^{2} + c} + \frac {{\left (a d x^{2} + a c + b\right )}^{2} a^{2} d^{2}}{{\left (d x^{2} + c\right )}^{2}}\right )}} - \frac {{\left (4 \, a c + 3 \, b\right )} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{16 \, a^{\frac {5}{2}} d^{2}} \]
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 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{d\,x^2+c}}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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