Optimal. Leaf size=100 \[ \frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}} \]
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Rubi [A] time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}} \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^{10}} \, dx &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}-\frac {(6 b) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^9} \, dx}{13 a}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}+\frac {\left (24 b^2\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^8} \, dx}{143 a^2}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}-\frac {\left (16 b^3\right ) \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7} \, dx}{429 a^3}\\ &=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 0.58 \begin {gather*} \frac {2 (a+b x)^3 \sqrt {x (a+b x)} \left (-231 a^3+126 a^2 b x-56 a b^2 x^2+16 b^3 x^3\right )}{3003 a^4 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 86, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt {a x+b x^2} \left (-231 a^6-567 a^5 b x-371 a^4 b^2 x^2-5 a^3 b^3 x^3+6 a^2 b^4 x^4-8 a b^5 x^5+16 b^6 x^6\right )}{3003 a^4 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 82, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (16 \, b^{6} x^{6} - 8 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{4} b^{2} x^{2} - 567 \, a^{5} b x - 231 \, a^{6}\right )} \sqrt {b x^{2} + a x}}{3003 \, a^{4} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 281, normalized size = 2.81 \begin {gather*} \frac {2 \, {\left (6006 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9} b^{\frac {9}{2}} + 36036 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} a b^{4} + 99099 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a^{2} b^{\frac {7}{2}} + 161733 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{3} b^{3} + 171171 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{4} b^{\frac {5}{2}} + 121121 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{5} b^{2} + 57057 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{6} b^{\frac {3}{2}} + 17199 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{7} b + 3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{8} \sqrt {b} + 231 \, a^{9}\right )}}{3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 55, normalized size = 0.55 \begin {gather*} -\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+56 a \,b^{2} x^{2}-126 a^{2} b x +231 a^{3}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{3003 a^{4} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 178, normalized size = 1.78 \begin {gather*} \frac {32 \, \sqrt {b x^{2} + a x} b^{6}}{3003 \, a^{4} x} - \frac {16 \, \sqrt {b x^{2} + a x} b^{5}}{3003 \, a^{3} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} b^{4}}{1001 \, a^{2} x^{3}} - \frac {10 \, \sqrt {b x^{2} + a x} b^{3}}{3003 \, a x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} b^{2}}{1716 \, x^{5}} - \frac {3 \, \sqrt {b x^{2} + a x} a b}{1144 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} a^{2}}{104 \, x^{7}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{8 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{4 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 145, normalized size = 1.45 \begin {gather*} \frac {4\,b^4\,\sqrt {b\,x^2+a\,x}}{1001\,a^2\,x^3}-\frac {106\,b^2\,\sqrt {b\,x^2+a\,x}}{429\,x^5}-\frac {10\,b^3\,\sqrt {b\,x^2+a\,x}}{3003\,a\,x^4}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{13\,x^7}-\frac {16\,b^5\,\sqrt {b\,x^2+a\,x}}{3003\,a^3\,x^2}+\frac {32\,b^6\,\sqrt {b\,x^2+a\,x}}{3003\,a^4\,x}-\frac {54\,a\,b\,\sqrt {b\,x^2+a\,x}}{143\,x^6} \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^{10}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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