Optimal. Leaf size=159 \[ -\frac {9 a \tanh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a x+b x^3}}\right )}{2 b^{11/2}}+\frac {9 \sqrt {x} \sqrt {a x+b x^3}}{2 b^5}-\frac {3 x^{7/2}}{b^4 \sqrt {a x+b x^3}}-\frac {3 x^{13/2}}{5 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {9 x^{19/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {x^{25/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.25, 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 = {2022, 2024, 2029, 206} \begin {gather*} -\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}}-\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 206
Rule 2022
Rule 2024
Rule 2029
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) \operatorname {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 = 0.21, size = 130, normalized size = 0.82 \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 )-\frac {315 \sqrt {a} \left (a+b x^2\right )^4 \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}\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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IntegrateAlgebraic [A] time = 0.13, size = 124, normalized size = 0.78 \begin {gather*} \frac {x^{9/2} \left (a+b x^2\right )^{9/2} \left (\frac {315 a^4 x+1050 a^3 b x^3+1218 a^2 b^2 x^5+528 a b^3 x^7+35 b^4 x^9}{70 b^5 \left (a+b x^2\right )^{7/2}}+\frac {9 a \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{2 b^{11/2}}\right )}{\left (x \left (a+b x^2\right )\right )^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, 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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giac [A] time = 0.33, 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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maple [A] time = 0.09, size = 212, normalized size = 1.33 \begin {gather*} -\frac {\sqrt {\left (b \,x^{2}+a \right ) x}\, \left (-35 b^{\frac {9}{2}} x^{9}-528 a \,b^{\frac {7}{2}} x^{7}+315 \sqrt {b \,x^{2}+a}\, a \,b^{3} x^{6} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )-1218 a^{2} b^{\frac {5}{2}} x^{5}+945 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} x^{4} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )-1050 a^{3} b^{\frac {3}{2}} x^{3}+945 \sqrt {b \,x^{2}+a}\, a^{3} b \,x^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )-315 a^{4} \sqrt {b}\, x +315 \sqrt {b \,x^{2}+a}\, a^{4} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )\right )}{70 \left (b \,x^{2}+a \right )^{4} b^{\frac {11}{2}} \sqrt {x}} \end {gather*}
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*} \int \frac {x^{\frac {29}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \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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sympy [F(-1)] 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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