Optimal. Leaf size=23 \[ -\frac{2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \]
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Rubi [A] time = 0.0074363, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {650} \[ -\frac{2 \left (a x+b x^2\right )^{7/2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin{align*} \int \frac{\left (a x+b x^2\right )^{5/2}}{x^7} \, dx &=-\frac{2 \left (a x+b x^2\right )^{7/2}}{7 a x^7}\\ \end{align*}
Mathematica [A] time = 0.0154628, size = 21, normalized size = 0.91 \[ -\frac{2 (x (a+b x))^{7/2}}{7 a x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 25, normalized size = 1.1 \begin{align*} -{\frac{2\,bx+2\,a}{7\,{x}^{6}a} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9822, size = 100, normalized size = 4.35 \begin{align*} -\frac{2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x^{2} + a x}}{7 \, a x^{4}} \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{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{7}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22441, size = 259, normalized size = 11.26 \begin{align*} \frac{2 \,{\left (7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} b^{3} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a b^{\frac{5}{2}} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{2} b^{2} + 35 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{3} b^{\frac{3}{2}} + 21 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{4} b + 7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{5} \sqrt{b} + a^{6}\right )}}{7 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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