Optimal. Leaf size=80 \[ -\frac {16 b^2 \sqrt {a x^3+b x^4}}{15 a^3 x^2}+\frac {8 b \sqrt {a x^3+b x^4}}{15 a^2 x^3}-\frac {2 \sqrt {a x^3+b x^4}}{5 a x^4} \]
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Rubi [A] time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2000} \[ -\frac {16 b^2 \sqrt {a x^3+b x^4}}{15 a^3 x^2}+\frac {8 b \sqrt {a x^3+b x^4}}{15 a^2 x^3}-\frac {2 \sqrt {a x^3+b x^4}}{5 a x^4} \]
Antiderivative was successfully verified.
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Rule 2000
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a x^3+b x^4}} \, dx &=-\frac {2 \sqrt {a x^3+b x^4}}{5 a x^4}-\frac {(4 b) \int \frac {1}{x \sqrt {a x^3+b x^4}} \, dx}{5 a}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{5 a x^4}+\frac {8 b \sqrt {a x^3+b x^4}}{15 a^2 x^3}+\frac {\left (8 b^2\right ) \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx}{15 a^2}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{5 a x^4}+\frac {8 b \sqrt {a x^3+b x^4}}{15 a^2 x^3}-\frac {16 b^2 \sqrt {a x^3+b x^4}}{15 a^3 x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.52 \[ -\frac {2 \sqrt {x^3 (a+b x)} \left (3 a^2-4 a b x+8 b^2 x^2\right )}{15 a^3 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 40, normalized size = 0.50 \[ -\frac {2 \, \sqrt {b x^{4} + a x^{3}} {\left (8 \, b^{2} x^{2} - 4 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 43, normalized size = 0.54 \[ -\frac {2 \, {\left (3 \, {\left (b + \frac {a}{x}\right )}^{\frac {5}{2}} - 10 \, {\left (b + \frac {a}{x}\right )}^{\frac {3}{2}} b + 15 \, \sqrt {b + \frac {a}{x}} b^{2}\right )}}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 0.58 \[ -\frac {2 \left (b x +a \right ) \left (8 b^{2} x^{2}-4 a b x +3 a^{2}\right )}{15 \sqrt {b \,x^{4}+a \,x^{3}}\, a^{3} x} \]
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 \[ \int \frac {1}{\sqrt {b x^{4} + a x^{3}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.14, size = 40, normalized size = 0.50 \[ -\frac {2\,\sqrt {b\,x^4+a\,x^3}\,\left (3\,a^2-4\,a\,b\,x+8\,b^2\,x^2\right )}{15\,a^3\,x^4} \]
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 \[ \int \frac {1}{x^{2} \sqrt {x^{3} \left (a + b x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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