Optimal. Leaf size=189 \[ -\frac {99 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{8 a^{13/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.29, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2023, 2025, 2029, 206} \[ -\frac {99 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{8 a^{13/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2023
Rule 2025
Rule 2029
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (a x+b x^3\right )^{9/2}} \, dx &=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11 \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {99 \int \frac {1}{x^{5/2} \left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33 \int \frac {1}{x^{7/2} \left (a x+b x^3\right )^{3/2}} \, dx}{5 a^3}\\ &=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}+\frac {33 \int \frac {1}{x^{9/2} \sqrt {a x+b x^3}} \, dx}{a^4}\\ &=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}-\frac {(99 b) \int \frac {1}{x^{5/2} \sqrt {a x+b x^3}} \, dx}{4 a^5}\\ &=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}+\frac {\left (99 b^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a x+b x^3}} \, dx}{8 a^6}\\ &=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}-\frac {\left (99 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a x+b x^3}}\right )}{8 a^6}\\ &=\frac {1}{7 a \sqrt {x} \left (a x+b x^3\right )^{7/2}}+\frac {11}{35 a^2 x^{3/2} \left (a x+b x^3\right )^{5/2}}+\frac {33}{35 a^3 x^{5/2} \left (a x+b x^3\right )^{3/2}}+\frac {33}{5 a^4 x^{7/2} \sqrt {a x+b x^3}}-\frac {33 \sqrt {a x+b x^3}}{4 a^5 x^{9/2}}+\frac {99 b \sqrt {a x+b x^3}}{8 a^6 x^{5/2}}-\frac {99 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{8 a^{13/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 46, normalized size = 0.24 \[ \frac {b^2 x^{7/2} \, _2F_1\left (-\frac {7}{2},3;-\frac {5}{2};\frac {b x^2}{a}+1\right )}{7 a^3 \left (x \left (a+b x^2\right )\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 422, normalized size = 2.23 \[ \left [\frac {3465 \, {\left (b^{6} x^{13} + 4 \, a b^{5} x^{11} + 6 \, a^{2} b^{4} x^{9} + 4 \, a^{3} b^{3} x^{7} + a^{4} b^{2} x^{5}\right )} \sqrt {a} \log \left (\frac {b x^{3} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x} \sqrt {a} \sqrt {x}}{x^{3}}\right ) + 2 \, {\left (3465 \, a b^{5} x^{10} + 11550 \, a^{2} b^{4} x^{8} + 13398 \, a^{3} b^{3} x^{6} + 5808 \, a^{4} b^{2} x^{4} + 385 \, a^{5} b x^{2} - 70 \, a^{6}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{560 \, {\left (a^{7} b^{4} x^{13} + 4 \, a^{8} b^{3} x^{11} + 6 \, a^{9} b^{2} x^{9} + 4 \, a^{10} b x^{7} + a^{11} x^{5}\right )}}, \frac {3465 \, {\left (b^{6} x^{13} + 4 \, a b^{5} x^{11} + 6 \, a^{2} b^{4} x^{9} + 4 \, a^{3} b^{3} x^{7} + a^{4} b^{2} x^{5}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x} \sqrt {-a}}{a \sqrt {x}}\right ) + {\left (3465 \, a b^{5} x^{10} + 11550 \, a^{2} b^{4} x^{8} + 13398 \, a^{3} b^{3} x^{6} + 5808 \, a^{4} b^{2} x^{4} + 385 \, a^{5} b x^{2} - 70 \, a^{6}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{280 \, {\left (a^{7} b^{4} x^{13} + 4 \, a^{8} b^{3} x^{11} + 6 \, a^{9} b^{2} x^{9} + 4 \, a^{10} b x^{7} + a^{11} x^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 138, normalized size = 0.73 \[ \frac {99 \, b^{2} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{6}} + \frac {350 \, {\left (b x^{2} + a\right )}^{3} b^{2} + 70 \, {\left (b x^{2} + a\right )}^{2} a b^{2} + 21 \, {\left (b x^{2} + a\right )} a^{2} b^{2} + 5 \, a^{3} b^{2}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{6}} + \frac {19 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2} - 21 \, \sqrt {b x^{2} + a} a b^{2}}{8 \, a^{6} b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 247, normalized size = 1.31 \[ -\frac {\sqrt {\left (b \,x^{2}+a \right ) x}\, \left (3465 \sqrt {b \,x^{2}+a}\, b^{5} x^{10} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-3465 \sqrt {a}\, b^{5} x^{10}+10395 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-11550 a^{\frac {3}{2}} b^{4} x^{8}+10395 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-13398 a^{\frac {5}{2}} b^{3} x^{6}+3465 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-5808 a^{\frac {7}{2}} b^{2} x^{4}-385 a^{\frac {9}{2}} b \,x^{2}+70 a^{\frac {11}{2}}\right )}{280 \left (b \,x^{2}+a \right )^{4} a^{\frac {13}{2}} x^{\frac {9}{2}}} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}} \sqrt {x}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {x}\,{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {x} \left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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