Optimal. Leaf size=130 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{a^{9/2}}+\frac {\sqrt {x}}{a^4 \sqrt {a x+b x^3}}+\frac {x^{3/2}}{3 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2023, 2029, 206} \[ \frac {x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{3/2}}{3 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {\sqrt {x}}{a^4 \sqrt {a x+b x^3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{a^{9/2}}+\frac {x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2023
Rule 2029
Rubi steps
\begin {align*} \int \frac {x^{7/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac {x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {\int \frac {x^{5/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{a}\\ &=\frac {x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {\int \frac {x^{3/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{a^2}\\ &=\frac {x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{3/2}}{3 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {\int \frac {\sqrt {x}}{\left (a x+b x^3\right )^{3/2}} \, dx}{a^3}\\ &=\frac {x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{3/2}}{3 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {\sqrt {x}}{a^4 \sqrt {a x+b x^3}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {a x+b x^3}} \, dx}{a^4}\\ &=\frac {x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{3/2}}{3 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {\sqrt {x}}{a^4 \sqrt {a x+b x^3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a x+b x^3}}\right )}{a^4}\\ &=\frac {x^{7/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {x^{5/2}}{5 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {x^{3/2}}{3 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {\sqrt {x}}{a^4 \sqrt {a x+b x^3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{a^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 43, normalized size = 0.33 \[ \frac {x^{7/2} \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};\frac {b x^2}{a}+1\right )}{7 a \left (x \left (a+b x^2\right )\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 360, normalized size = 2.77 \[ \left [\frac {105 \, {\left (b^{4} x^{9} + 4 \, a b^{3} x^{7} + 6 \, a^{2} b^{2} x^{5} + 4 \, a^{3} b x^{3} + a^{4} x\right )} \sqrt {a} \log \left (\frac {b x^{3} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x} \sqrt {a} \sqrt {x}}{x^{3}}\right ) + 2 \, {\left (105 \, a b^{3} x^{6} + 350 \, a^{2} b^{2} x^{4} + 406 \, a^{3} b x^{2} + 176 \, a^{4}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{210 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}, \frac {105 \, {\left (b^{4} x^{9} + 4 \, a b^{3} x^{7} + 6 \, a^{2} b^{2} x^{5} + 4 \, a^{3} b x^{3} + a^{4} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a x} \sqrt {-a}}{a \sqrt {x}}\right ) + {\left (105 \, a b^{3} x^{6} + 350 \, a^{2} b^{2} x^{4} + 406 \, a^{3} b x^{2} + 176 \, a^{4}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{105 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 114, normalized size = 0.88 \[ \frac {\arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} - \frac {105 \, \sqrt {a} \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 176 \, \sqrt {-a}}{105 \, \sqrt {-a} a^{\frac {9}{2}}} + \frac {105 \, {\left (b x^{2} + a\right )}^{3} + 35 \, {\left (b x^{2} + a\right )}^{2} a + 21 \, {\left (b x^{2} + a\right )} a^{2} + 15 \, a^{3}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 217, normalized size = 1.67 \[ -\frac {\sqrt {\left (b \,x^{2}+a \right ) x}\, \left (105 \sqrt {b \,x^{2}+a}\, b^{3} x^{6} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-105 \sqrt {a}\, b^{3} x^{6}+315 \sqrt {b \,x^{2}+a}\, a \,b^{2} x^{4} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-350 a^{\frac {3}{2}} b^{2} x^{4}+315 \sqrt {b \,x^{2}+a}\, a^{2} b \,x^{2} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-406 a^{\frac {5}{2}} b \,x^{2}+105 \sqrt {b \,x^{2}+a}\, a^{3} \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-176 a^{\frac {7}{2}}\right )}{105 \left (b \,x^{2}+a \right )^{4} a^{\frac {9}{2}} \sqrt {x}} \]
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 \[ \int \frac {x^{\frac {7}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {x^{7/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
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 \[ \int \frac {x^{\frac {7}{2}}}{\left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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