Optimal. Leaf size=74 \[ -\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac {8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9} \]
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Rubi [A] time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {658, 650} \begin {gather*} -\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7}+\frac {8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 658
Rubi steps
\begin {align*} \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}-\frac {(4 b) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{11 a}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}+\frac {8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}+\frac {\left (8 b^2\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{99 a^2}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}+\frac {8 b \left (a x+b x^2\right )^{7/2}}{99 a^2 x^8}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{693 a^3 x^7}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 47, normalized size = 0.64 \begin {gather*} -\frac {2 (a+b x)^3 \sqrt {x (a+b x)} \left (63 a^2-28 a b x+8 b^2 x^2\right )}{693 a^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 75, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt {a x+b x^2} \left (63 a^5+161 a^4 b x+113 a^3 b^2 x^2+3 a^2 b^3 x^3-4 a b^4 x^4+8 b^5 x^5\right )}{693 a^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 71, normalized size = 0.96 \begin {gather*} -\frac {2 \, {\left (8 \, b^{5} x^{5} - 4 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{3} + 113 \, a^{3} b^{2} x^{2} + 161 \, a^{4} b x + 63 \, a^{5}\right )} \sqrt {b x^{2} + a x}}{693 \, a^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 252, normalized size = 3.41 \begin {gather*} \frac {2 \, {\left (924 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} b^{4} + 4851 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a b^{\frac {7}{2}} + 11781 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{2} b^{3} + 16863 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{3} b^{\frac {5}{2}} + 15345 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{4} b^{2} + 9009 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{5} b^{\frac {3}{2}} + 3311 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{6} b + 693 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{7} \sqrt {b} + 63 \, a^{8}\right )}}{693 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 44, normalized size = 0.59 \begin {gather*} -\frac {2 \left (b x +a \right ) \left (8 b^{2} x^{2}-28 a b x +63 a^{2}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{693 a^{3} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.45, size = 156, normalized size = 2.11 \begin {gather*} -\frac {16 \, \sqrt {b x^{2} + a x} b^{5}}{693 \, a^{3} x} + \frac {8 \, \sqrt {b x^{2} + a x} b^{4}}{693 \, a^{2} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} b^{3}}{231 \, a x^{3}} + \frac {5 \, \sqrt {b x^{2} + a x} b^{2}}{693 \, x^{4}} - \frac {5 \, \sqrt {b x^{2} + a x} a b}{792 \, x^{5}} - \frac {5 \, \sqrt {b x^{2} + a x} a^{2}}{88 \, x^{6}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{24 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{3 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 123, normalized size = 1.66 \begin {gather*} \frac {8\,b^4\,\sqrt {b\,x^2+a\,x}}{693\,a^2\,x^2}-\frac {226\,b^2\,\sqrt {b\,x^2+a\,x}}{693\,x^4}-\frac {2\,b^3\,\sqrt {b\,x^2+a\,x}}{231\,a\,x^3}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{11\,x^6}-\frac {16\,b^5\,\sqrt {b\,x^2+a\,x}}{693\,a^3\,x}-\frac {46\,a\,b\,\sqrt {b\,x^2+a\,x}}{99\,x^5} \end {gather*}
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 \begin {gather*} \int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{9}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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