Optimal. Leaf size=32 \[ -\frac {a^2}{3 x^3}-\frac {4 a b}{5 x^{5/2}}-\frac {b^2}{2 x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {a^2}{3 x^3}-\frac {4 a b}{5 x^{5/2}}-\frac {b^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^2}{x^4} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^7} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {a^2}{x^7}+\frac {2 a b}{x^6}+\frac {b^2}{x^5}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^2}{3 x^3}-\frac {4 a b}{5 x^{5/2}}-\frac {b^2}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.88 \[ -\frac {10 a^2+24 a b \sqrt {x}+15 b^2 x}{30 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 24, normalized size = 0.75 \[ -\frac {15 \, b^{2} x + 24 \, a b \sqrt {x} + 10 \, a^{2}}{30 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 24, normalized size = 0.75 \[ -\frac {15 \, b^{2} x + 24 \, a b \sqrt {x} + 10 \, a^{2}}{30 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.78 \[ -\frac {b^{2}}{2 x^{2}}-\frac {4 a b}{5 x^{\frac {5}{2}}}-\frac {a^{2}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 24, normalized size = 0.75 \[ -\frac {15 \, b^{2} x + 24 \, a b \sqrt {x} + 10 \, a^{2}}{30 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 24, normalized size = 0.75 \[ -\frac {15\,b^2\,x+10\,a^2+24\,a\,b\,\sqrt {x}}{30\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 29, normalized size = 0.91 \[ - \frac {a^{2}}{3 x^{3}} - \frac {4 a b}{5 x^{\frac {5}{2}}} - \frac {b^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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