Optimal. Leaf size=94 \[ -\frac {3 b \sqrt {a+\frac {b}{c+d x^2}}}{2 d}+\frac {\left (c+d x^2\right ) \left (a+\frac {b}{c+d x^2}\right )^{3/2}}{2 d}+\frac {3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1605, 248, 43,
52, 65, 214} \begin {gather*} \frac {\left (c+d x^2\right ) \left (a+\frac {b}{c+d x^2}\right )^{3/2}}{2 d}-\frac {3 b \sqrt {a+\frac {b}{c+d x^2}}}{2 d}+\frac {3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 248
Rule 1605
Rubi steps
\begin {align*} \int x \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int \left (a+\frac {b}{x}\right )^{3/2} \, dx,x,c+d x^2\right )}{2 d}\\ &=-\frac {\text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,\frac {1}{c+d x^2}\right )}{2 d}\\ &=\frac {\left (c+d x^2\right ) \left (a+\frac {b}{c+d x^2}\right )^{3/2}}{2 d}-\frac {(3 b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{c+d x^2}\right )}{4 d}\\ &=-\frac {3 b \sqrt {a+\frac {b}{c+d x^2}}}{2 d}+\frac {\left (c+d x^2\right ) \left (a+\frac {b}{c+d x^2}\right )^{3/2}}{2 d}-\frac {(3 a b) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{c+d x^2}\right )}{4 d}\\ &=-\frac {3 b \sqrt {a+\frac {b}{c+d x^2}}}{2 d}+\frac {\left (c+d x^2\right ) \left (a+\frac {b}{c+d x^2}\right )^{3/2}}{2 d}-\frac {(3 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{c+d x^2}}\right )}{2 d}\\ &=-\frac {3 b \sqrt {a+\frac {b}{c+d x^2}}}{2 d}+\frac {\left (c+d x^2\right ) \left (a+\frac {b}{c+d x^2}\right )^{3/2}}{2 d}+\frac {3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 87, normalized size = 0.93 \begin {gather*} \frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (-2 b+a \left (c+d x^2\right )\right )+3 \sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{2 d} \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. \(335\) vs.
\(2(78)=156\).
time = 0.12, size = 336, normalized size = 3.57
method | result | size |
derivativedivides | \(\frac {\sqrt {\frac {a \left (d \,x^{2}+c \right )+b}{d \,x^{2}+c}}\, \left (3 \ln \left (\frac {2 \sqrt {a \left (d \,x^{2}+c \right )^{2}+b \left (d \,x^{2}+c \right )}\, \sqrt {a}+2 a \left (d \,x^{2}+c \right )+b}{2 \sqrt {a}}\right ) a b \left (d \,x^{2}+c \right )^{2}+6 a^{\frac {3}{2}} \sqrt {a \left (d \,x^{2}+c \right )^{2}+b \left (d \,x^{2}+c \right )}\, \left (d \,x^{2}+c \right )^{2}-4 \left (a \left (d \,x^{2}+c \right )^{2}+b \left (d \,x^{2}+c \right )\right )^{\frac {3}{2}} \sqrt {a}\right )}{4 d \left (d \,x^{2}+c \right ) \sqrt {\left (d \,x^{2}+c \right ) \left (a \left (d \,x^{2}+c \right )+b \right )}\, \sqrt {a}}\) | \(187\) |
risch | \(\frac {\left (d \,x^{2}+c \right ) a \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{2 d}+\frac {\left (\frac {3 b a \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {c^{2} a +b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{4 \sqrt {a \,d^{2}}}-\frac {b \,x^{2} a}{\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {b a c}{d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}-\frac {b^{2}}{d \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{a d \,x^{2}+a c +b}\) | \(289\) |
default | \(\frac {\left (3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b \,d^{2} x^{2}+2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, a d \,x^{2}+3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b c d +2 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+c^{2} a +b c}\, \sqrt {a \,d^{2}}\, a c -4 \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{4 d \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}}\) | \(336\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 156, normalized size = 1.66 \begin {gather*} -\frac {a b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a d - \frac {{\left (a d x^{2} + a c + b\right )} d}{d x^{2} + c}\right )}} - \frac {3 \, \sqrt {a} 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 )}{4 \, d} - \frac {b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 269, normalized size = 2.86 \begin {gather*} \left [\frac {3 \, \sqrt {a} b \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 (a d x^{2} + a c - 2 \, b\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, d}, -\frac {3 \, \sqrt {-a} b \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 (a d x^{2} + a c - 2 \, b\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, d}\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 x \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. 387 vs.
\(2 (78) = 156\).
time = 5.26, size = 387, normalized size = 4.12 \begin {gather*} -\frac {\sqrt {a} b \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 ) \mathrm {sgn}\left (d x^{2} + c\right )}{4 \, {\left | d \right |}} - \frac {\sqrt {a} b {\left | d \right |} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (d x^{2} + c\right )}{4 \, d^{2}} + \frac {\sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} a \mathrm {sgn}\left (d x^{2} + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.74, size = 61, normalized size = 0.65 \begin {gather*} -\frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}\,\left (d\,x^2+c\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {a\,\left (d\,x^2+c\right )}{b}\right )}{d\,{\left (\frac {a\,\left (d\,x^2+c\right )}{b}+1\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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