Optimal. Leaf size=108 \[ \frac {32 b^3 \sqrt {a x^3+b x^4}}{35 a^4 x^2}-\frac {16 b^2 \sqrt {a x^3+b x^4}}{35 a^3 x^3}+\frac {12 b \sqrt {a x^3+b x^4}}{35 a^2 x^4}-\frac {2 \sqrt {a x^3+b x^4}}{7 a x^5} \]
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Rubi [A] time = 0.13, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2000} \begin {gather*} \frac {32 b^3 \sqrt {a x^3+b x^4}}{35 a^4 x^2}-\frac {16 b^2 \sqrt {a x^3+b x^4}}{35 a^3 x^3}+\frac {12 b \sqrt {a x^3+b x^4}}{35 a^2 x^4}-\frac {2 \sqrt {a x^3+b x^4}}{7 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2000
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a x^3+b x^4}} \, dx &=-\frac {2 \sqrt {a x^3+b x^4}}{7 a x^5}-\frac {(6 b) \int \frac {1}{x^2 \sqrt {a x^3+b x^4}} \, dx}{7 a}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{7 a x^5}+\frac {12 b \sqrt {a x^3+b x^4}}{35 a^2 x^4}+\frac {\left (24 b^2\right ) \int \frac {1}{x \sqrt {a x^3+b x^4}} \, dx}{35 a^2}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{7 a x^5}+\frac {12 b \sqrt {a x^3+b x^4}}{35 a^2 x^4}-\frac {16 b^2 \sqrt {a x^3+b x^4}}{35 a^3 x^3}-\frac {\left (16 b^3\right ) \int \frac {1}{\sqrt {a x^3+b x^4}} \, dx}{35 a^3}\\ &=-\frac {2 \sqrt {a x^3+b x^4}}{7 a x^5}+\frac {12 b \sqrt {a x^3+b x^4}}{35 a^2 x^4}-\frac {16 b^2 \sqrt {a x^3+b x^4}}{35 a^3 x^3}+\frac {32 b^3 \sqrt {a x^3+b x^4}}{35 a^4 x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {x^3 (a+b x)} \left (-5 a^3+6 a^2 b x-8 a b^2 x^2+16 b^3 x^3\right )}{35 a^4 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 55, normalized size = 0.51 \begin {gather*} \frac {2 \left (-5 a^3+6 a^2 b x-8 a b^2 x^2+16 b^3 x^3\right ) \sqrt {a x^3+b x^4}}{35 a^4 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 51, normalized size = 0.47 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} x^{3} - 8 \, a b^{2} x^{2} + 6 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt {b x^{4} + a x^{3}}}{35 \, a^{4} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 57, normalized size = 0.53 \begin {gather*} -\frac {2 \, {\left (5 \, {\left (b + \frac {a}{x}\right )}^{\frac {7}{2}} - 21 \, {\left (b + \frac {a}{x}\right )}^{\frac {5}{2}} b + 35 \, {\left (b + \frac {a}{x}\right )}^{\frac {3}{2}} b^{2} - 35 \, \sqrt {b + \frac {a}{x}} b^{3}\right )}}{35 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 57, normalized size = 0.53 \begin {gather*} -\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+8 a \,b^{2} x^{2}-6 a^{2} b x +5 a^{3}\right )}{35 \sqrt {b \,x^{4}+a \,x^{3}}\, a^{4} x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {b x^{4} + a x^{3}} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.14, size = 92, normalized size = 0.85 \begin {gather*} \frac {12\,b\,\sqrt {b\,x^4+a\,x^3}}{35\,a^2\,x^4}-\frac {2\,\sqrt {b\,x^4+a\,x^3}}{7\,a\,x^5}-\frac {16\,b^2\,\sqrt {b\,x^4+a\,x^3}}{35\,a^3\,x^3}+\frac {32\,b^3\,\sqrt {b\,x^4+a\,x^3}}{35\,a^4\,x^2} \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 {1}{x^{3} \sqrt {x^{3} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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