Optimal. Leaf size=159 \[ -\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {3 x^{7/2}}{b^4 \sqrt {a x+b x^3}}+\frac {9 \sqrt {x} \sqrt {a x+b x^3}}{2 b^5}-\frac {9 a \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x+b x^3}}\right )}{2 b^{11/2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2047, 2049,
2054, 212} \begin {gather*} -\frac {9 a \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x+b x^3}}\right )}{2 b^{11/2}}+\frac {9 \sqrt {x} \sqrt {a x+b x^3}}{2 b^5}-\frac {3 x^{7/2}}{b^4 \sqrt {a x+b x^3}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2047
Rule 2049
Rule 2054
Rubi steps
\begin {align*} \int \frac {x^{29/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}+\frac {9 \int \frac {x^{23/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 b}\\ &=-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}+\frac {9 \int \frac {x^{17/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{5 b^2}\\ &=-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}+\frac {3 \int \frac {x^{11/2}}{\left (a x+b x^3\right )^{3/2}} \, dx}{b^3}\\ &=-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {3 x^{7/2}}{b^4 \sqrt {a x+b x^3}}+\frac {9 \int \frac {x^{5/2}}{\sqrt {a x+b x^3}} \, dx}{b^4}\\ &=-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {3 x^{7/2}}{b^4 \sqrt {a x+b x^3}}+\frac {9 \sqrt {x} \sqrt {a x+b x^3}}{2 b^5}-\frac {(9 a) \int \frac {\sqrt {x}}{\sqrt {a x+b x^3}} \, dx}{2 b^5}\\ &=-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {3 x^{7/2}}{b^4 \sqrt {a x+b x^3}}+\frac {9 \sqrt {x} \sqrt {a x+b x^3}}{2 b^5}-\frac {(9 a) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {a x+b x^3}}\right )}{2 b^5}\\ &=-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {3 x^{7/2}}{b^4 \sqrt {a x+b x^3}}+\frac {9 \sqrt {x} \sqrt {a x+b x^3}}{2 b^5}-\frac {9 a \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x+b x^3}}\right )}{2 b^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 122, normalized size = 0.77 \begin {gather*} \frac {\sqrt {x} \left (\sqrt {b} x \left (315 a^4+1050 a^3 b x^2+1218 a^2 b^2 x^4+528 a b^3 x^6+35 b^4 x^8\right )+315 a \left (a+b x^2\right )^{7/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{70 b^{11/2} \left (a+b x^2\right )^3 \sqrt {x \left (a+b x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 212, normalized size = 1.33
method | result | size |
default | \(-\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (-35 x^{9} b^{\frac {9}{2}}+315 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a \,b^{3} x^{6} \sqrt {b \,x^{2}+a}-528 b^{\frac {7}{2}} a \,x^{7}+945 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a^{2} b^{2} x^{4} \sqrt {b \,x^{2}+a}-1218 b^{\frac {5}{2}} a^{2} x^{5}+945 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a^{3} b \,x^{2} \sqrt {b \,x^{2}+a}-1050 b^{\frac {3}{2}} a^{3} x^{3}+315 \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right ) a^{4} \sqrt {b \,x^{2}+a}-315 \sqrt {b}\, a^{4} x \right )}{70 b^{\frac {11}{2}} \sqrt {x}\, \left (b \,x^{2}+a \right )^{4}}\) | \(212\) |
risch | \(\frac {x^{\frac {3}{2}} \left (b \,x^{2}+a \right )}{2 b^{5} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {\left (-\frac {9 a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {11}{2}}}+\frac {a^{3} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{112 b^{7} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{4}}-\frac {53 a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{560 b^{7} \left (x -\frac {\sqrt {-a b}}{b}\right )^{3}}-\frac {571 a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 b^{6} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {97 a \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{35 b^{6} \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {a^{3} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{112 b^{7} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{4}}-\frac {53 a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{560 b^{7} \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}+\frac {571 a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 b^{6} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}+\frac {97 a \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{35 b^{6} \left (x +\frac {\sqrt {-a b}}{b}\right )}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x}}{\sqrt {x \left (b \,x^{2}+a \right )}}\) | \(595\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.84, size = 376, normalized size = 2.36 \begin {gather*} \left [\frac {315 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {b} \log \left (2 \, b x^{2} - 2 \, \sqrt {b x^{3} + a x} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (35 \, b^{5} x^{8} + 528 \, a b^{4} x^{6} + 1218 \, a^{2} b^{3} x^{4} + 1050 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{140 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}}, \frac {315 \, {\left (a b^{4} x^{8} + 4 \, a^{2} b^{3} x^{6} + 6 \, a^{3} b^{2} x^{4} + 4 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{3} + a x} \sqrt {-b}}{b x^{\frac {3}{2}}}\right ) + {\left (35 \, b^{5} x^{8} + 528 \, a b^{4} x^{6} + 1218 \, a^{2} b^{3} x^{4} + 1050 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{70 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.51, size = 100, normalized size = 0.63 \begin {gather*} \frac {{\left ({\left ({\left (x^{2} {\left (\frac {35 \, x^{2}}{b} + \frac {528 \, a}{b^{2}}\right )} + \frac {1218 \, a^{2}}{b^{3}}\right )} x^{2} + \frac {1050 \, a^{3}}{b^{4}}\right )} x^{2} + \frac {315 \, a^{4}}{b^{5}}\right )} x}{70 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {9 \, a \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, b^{\frac {11}{2}}} - \frac {9 \, a \log \left ({\left | a \right |}\right )}{4 \, b^{\frac {11}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {x^{29/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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