Optimal. Leaf size=139 \[ -\frac {15 (x+1) (3-x)^3}{256 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac {5 (x+1) (3-x)^2}{64 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac {(x+1) (3-x)}{8 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac {15 (x+1)^{3/2} (3-x)^3 \tanh ^{-1}\left (\frac {\sqrt {x+1}}{2}\right )}{512 \left (x^3-5 x^2+3 x+9\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2067, 2064, 51, 63, 206} \[ -\frac {15 (x+1) (3-x)^3}{256 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac {5 (x+1) (3-x)^2}{64 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac {(x+1) (3-x)}{8 \left (x^3-5 x^2+3 x+9\right )^{3/2}}+\frac {15 (x+1)^{3/2} (3-x)^3 \tanh ^{-1}\left (\frac {\sqrt {x+1}}{2}\right )}{512 \left (x^3-5 x^2+3 x+9\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 206
Rule 2064
Rule 2067
Rubi steps
\begin {align*} \int \frac {1}{\left (9+3 x-5 x^2+x^3\right )^{3/2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (\frac {128}{27}-\frac {16 x}{3}+x^3\right )^{3/2}} \, dx,x,-\frac {5}{3}+x\right )\\ &=\frac {\left (2097152 (3-x)^3 (1+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right )^3 \left (\frac {128}{9}+\frac {16 x}{3}\right )^{3/2}} \, dx,x,-\frac {5}{3}+x\right )}{81 \sqrt {3} \left (9+3 x-5 x^2+x^3\right )^{3/2}}\\ &=\frac {(3-x) (1+x)}{8 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {\left (20480 (3-x)^3 (1+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right )^2 \left (\frac {128}{9}+\frac {16 x}{3}\right )^{3/2}} \, dx,x,-\frac {5}{3}+x\right )}{27 \sqrt {3} \left (9+3 x-5 x^2+x^3\right )^{3/2}}\\ &=\frac {(3-x) (1+x)}{8 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {5 (3-x)^2 (1+x)}{64 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {\left (80 (3-x)^3 (1+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right ) \left (\frac {128}{9}+\frac {16 x}{3}\right )^{3/2}} \, dx,x,-\frac {5}{3}+x\right )}{3 \sqrt {3} \left (9+3 x-5 x^2+x^3\right )^{3/2}}\\ &=\frac {(3-x) (1+x)}{8 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {5 (3-x)^2 (1+x)}{64 \left (9+3 x-5 x^2+x^3\right )^{3/2}}-\frac {15 (3-x)^3 (1+x)}{256 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {\left (5 (3-x)^3 (1+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {128}{9}-\frac {32 x}{3}\right ) \sqrt {\frac {128}{9}+\frac {16 x}{3}}} \, dx,x,-\frac {5}{3}+x\right )}{4 \sqrt {3} \left (9+3 x-5 x^2+x^3\right )^{3/2}}\\ &=\frac {(3-x) (1+x)}{8 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {5 (3-x)^2 (1+x)}{64 \left (9+3 x-5 x^2+x^3\right )^{3/2}}-\frac {15 (3-x)^3 (1+x)}{256 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {\left (5 \sqrt {3} (3-x)^3 (1+x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {128}{3}-2 x^2} \, dx,x,\frac {4 \sqrt {1+x}}{\sqrt {3}}\right )}{32 \left (9+3 x-5 x^2+x^3\right )^{3/2}}\\ &=\frac {(3-x) (1+x)}{8 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {5 (3-x)^2 (1+x)}{64 \left (9+3 x-5 x^2+x^3\right )^{3/2}}-\frac {15 (3-x)^3 (1+x)}{256 \left (9+3 x-5 x^2+x^3\right )^{3/2}}+\frac {15 (3-x)^3 (1+x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+x}}{2}\right )}{512 \left (9+3 x-5 x^2+x^3\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 35, normalized size = 0.25 \[ \frac {(x-3) \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\frac {x+1}{4}\right )}{32 \sqrt {(x-3)^2 (x+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 3.62, size = 67, normalized size = 0.48 \[ \frac {(x-3) \sqrt {x+1} \left (\frac {15 x^2-70 x+43}{256 (x-3)^2 \sqrt {x+1}}-\frac {15}{512} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{2}\right )\right )}{\sqrt {(x-3)^2 (x+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.56, size = 138, normalized size = 0.99 \[ -\frac {15 \, {\left (x^{4} - 8 \, x^{3} + 18 \, x^{2} - 27\right )} \log \left (\frac {2 \, x + \sqrt {x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) - 15 \, {\left (x^{4} - 8 \, x^{3} + 18 \, x^{2} - 27\right )} \log \left (-\frac {2 \, x - \sqrt {x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) - 4 \, \sqrt {x^{3} - 5 \, x^{2} + 3 \, x + 9} {\left (15 \, x^{2} - 70 \, x + 43\right )}}{1024 \, {\left (x^{4} - 8 \, x^{3} + 18 \, x^{2} - 27\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 71, normalized size = 0.51
method | result | size |
risch | \(\frac {15 x^{2}-70 x +43}{256 \left (-3+x \right ) \sqrt {\left (1+x \right ) \left (-3+x \right )^{2}}}+\frac {\left (-\frac {15 \ln \left (\sqrt {1+x}+2\right )}{1024}+\frac {15 \ln \left (\sqrt {1+x}-2\right )}{1024}\right ) \sqrt {1+x}\, \left (-3+x \right )}{\sqrt {\left (1+x \right ) \left (-3+x \right )^{2}}}\) | \(71\) |
trager | \(\frac {\left (15 x^{2}-70 x +43\right ) \sqrt {x^{3}-5 x^{2}+3 x +9}}{256 \left (-3+x \right )^{3} \left (1+x \right )}+\frac {15 \ln \left (\frac {-x^{2}+4 \sqrt {x^{3}-5 x^{2}+3 x +9}-2 x +15}{\left (-3+x \right )^{2}}\right )}{1024}\) | \(75\) |
default | \(\frac {\left (-3+x \right )^{3} \left (1+x \right ) \left (15 \left (1+x \right )^{\frac {5}{2}} \ln \left (\sqrt {1+x}-2\right )-15 \left (1+x \right )^{\frac {5}{2}} \ln \left (\sqrt {1+x}+2\right )-120 \left (1+x \right )^{\frac {3}{2}} \ln \left (\sqrt {1+x}-2\right )+120 \left (1+x \right )^{\frac {3}{2}} \ln \left (\sqrt {1+x}+2\right )+240 \ln \left (\sqrt {1+x}-2\right ) \sqrt {1+x}-240 \ln \left (\sqrt {1+x}+2\right ) \sqrt {1+x}+60 x^{2}-280 x +172\right )}{1024 \left (x^{3}-5 x^{2}+3 x +9\right )^{\frac {3}{2}} \left (\sqrt {1+x}-2\right )^{2} \left (\sqrt {1+x}+2\right )^{2}}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (x^3-5\,x^2+3\,x+9\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x^{3} - 5 x^{2} + 3 x + 9\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________