Optimal. Leaf size=108 \[ -\frac{32 b^3 \sqrt{b x+c x^2}}{35 c^4 \sqrt{x}}+\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c} \]
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Rubi [A] time = 0.040222, 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 = {656, 648} \[ -\frac{32 b^3 \sqrt{b x+c x^2}}{35 c^4 \sqrt{x}}+\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{7/2}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c}-\frac{(6 b) \int \frac{x^{5/2}}{\sqrt{b x+c x^2}} \, dx}{7 c}\\ &=-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{\left (24 b^2\right ) \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx}{35 c^2}\\ &=\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c}-\frac{\left (16 b^3\right ) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{35 c^3}\\ &=-\frac{32 b^3 \sqrt{b x+c x^2}}{35 c^4 \sqrt{x}}+\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c}\\ \end{align*}
Mathematica [A] time = 0.0323661, size = 53, normalized size = 0.49 \[ \frac{2 \sqrt{x (b+c x)} \left (8 b^2 c x-16 b^3-6 b c^2 x^2+5 c^3 x^3\right )}{35 c^4 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 55, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -5\,{x}^{3}{c}^{3}+6\,b{x}^{2}{c}^{2}-8\,{b}^{2}xc+16\,{b}^{3} \right ) }{35\,{c}^{4}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14649, size = 72, normalized size = 0.67 \begin{align*} \frac{2 \,{\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )}}{35 \, \sqrt{c x + b} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0068, size = 115, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (5 \, c^{3} x^{3} - 6 \, b c^{2} x^{2} + 8 \, b^{2} c x - 16 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{35 \, c^{4} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19238, size = 78, normalized size = 0.72 \begin{align*} \frac{32 \, b^{\frac{7}{2}}}{35 \, c^{4}} + \frac{2 \,{\left (5 \,{\left (c x + b\right )}^{\frac{7}{2}} - 21 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} - 35 \, \sqrt{c x + b} b^{3}\right )}}{35 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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