Optimal. Leaf size=76 \[ \frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5} \]
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Rubi [A] time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {646, 45, 37} \begin {gather*} \frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 646
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x^6} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x^5} \, dx}{5 a b \left (a b+b^2 x\right )}\\ &=-\frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{5 a x^5}+\frac {b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{20 a^2 x^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (4 a^3+15 a^2 b x+20 a b^2 x^2+10 b^3 x^3\right )}{20 x^5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.97, size = 344, normalized size = 4.53 \begin {gather*} \frac {4 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-4 a^7 b-31 a^6 b^2 x-104 a^5 b^3 x^2-196 a^4 b^4 x^3-224 a^3 b^5 x^4-155 a^2 b^6 x^5-60 a b^7 x^6-10 b^8 x^7\right )+4 \sqrt {b^2} b^4 \left (4 a^8+35 a^7 b x+135 a^6 b^2 x^2+300 a^5 b^3 x^3+420 a^4 b^4 x^4+379 a^3 b^5 x^5+215 a^2 b^6 x^6+70 a b^7 x^7+10 b^8 x^8\right )}{5 \sqrt {b^2} x^5 \sqrt {a^2+2 a b x+b^2 x^2} \left (-16 a^4 b^4-64 a^3 b^5 x-96 a^2 b^6 x^2-64 a b^7 x^3-16 b^8 x^4\right )+5 x^5 \left (16 a^5 b^5+80 a^4 b^6 x+160 a^3 b^7 x^2+160 a^2 b^8 x^3+80 a b^9 x^4+16 b^{10} x^5\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 35, normalized size = 0.46 \begin {gather*} -\frac {10 \, b^{3} x^{3} + 20 \, a b^{2} x^{2} + 15 \, a^{2} b x + 4 \, a^{3}}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 74, normalized size = 0.97 \begin {gather*} \frac {b^{5} \mathrm {sgn}\left (b x + a\right )}{20 \, a^{2}} - \frac {10 \, b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 4 \, a^{3} \mathrm {sgn}\left (b x + a\right )}{20 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 0.68 \begin {gather*} -\frac {\left (10 b^{3} x^{3}+20 a \,b^{2} x^{2}+15 a^{2} b x +4 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 \left (b x +a \right )^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.57, size = 167, normalized size = 2.20 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{5}}{4 \, a^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}}{4 \, a^{4} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{4 \, a^{5} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{2}}{4 \, a^{4} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b}{4 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}{5 \, a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 135, normalized size = 1.78 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {a\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\left (a+b\,x\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )} \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 (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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