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 = 0.134198, antiderivative size = 108, normalized size of antiderivative = 1., 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} \[ \frac{32 b^3 \sqrt{a x^3+b x^4}}{35 a^4 x^2}-\frac{16 b^2 \sqrt{a x^3+b x^4}}{35 a^3 x^3}+\frac{12 b \sqrt{a x^3+b x^4}}{35 a^2 x^4}-\frac{2 \sqrt{a x^3+b x^4}}{7 a x^5} \]
Antiderivative was successfully verified.
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Rule 2016
Rule 2000
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*}
Mathematica [A] time = 0.0170488, size = 53, normalized size = 0.49 \[ \frac{2 \sqrt{x^3 (a+b x)} \left (6 a^2 b x-5 a^3-8 a b^2 x^2+16 b^3 x^3\right )}{35 a^4 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -16\,{b}^{3}{x}^{3}+8\,a{b}^{2}{x}^{2}-6\,bx{a}^{2}+5\,{a}^{3} \right ) }{35\,{x}^{2}{a}^{4}}{\frac{1}{\sqrt{b{x}^{4}+a{x}^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{4} + a x^{3}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.790084, size = 112, normalized size = 1.04 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \sqrt{x^{3} \left (a + b x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2341, size = 77, normalized size = 0.71 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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