Optimal. Leaf size=200 \[ \frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{35 c^2 d^2 e \sqrt{d+e x}}-\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}} \]
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Rubi [A] time = 0.228383, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac{8 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)}{35 c^2 d^2 e \sqrt{d+e x}}-\frac{8 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right )}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac{2 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 870
Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(f+g x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx &=\frac{2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}+\frac{\left (4 \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{7 c d e^2}\\ &=\frac{8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 c^2 d^2 e \sqrt{d+e x}}+\frac{2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}-\frac{\left (4 (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right )\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}} \, dx}{35 c^2 d^2 e}\\ &=-\frac{8 (c d f-a e g) \left (2 a e^2 g-c d (5 e f-3 d g)\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 e (d+e x)^{3/2}}+\frac{8 g (c d f-a e g) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 c^2 d^2 e \sqrt{d+e x}}+\frac{2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 c d (d+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0859647, size = 90, normalized size = 0.45 \[ \frac{2 ((d+e x) (a e+c d x))^{3/2} \left (8 a^2 e^2 g^2-4 a c d e g (7 f+3 g x)+c^2 d^2 \left (35 f^2+42 f g x+15 g^2 x^2\right )\right )}{105 c^3 d^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 116, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 15\,{g}^{2}{x}^{2}{c}^{2}{d}^{2}-12\,acde{g}^{2}x+42\,{c}^{2}{d}^{2}fgx+8\,{a}^{2}{e}^{2}{g}^{2}-28\,acdefg+35\,{f}^{2}{c}^{2}{d}^{2} \right ) }{105\,{c}^{3}{d}^{3}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13287, size = 180, normalized size = 0.9 \begin{align*} \frac{2 \,{\left (c d x + a e\right )}^{\frac{3}{2}} f^{2}}{3 \, c d} + \frac{4 \,{\left (3 \, c^{2} d^{2} x^{2} + a c d e x - 2 \, a^{2} e^{2}\right )} \sqrt{c d x + a e} f g}{15 \, c^{2} d^{2}} + \frac{2 \,{\left (15 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} - 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} g^{2}}{105 \, c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62733, size = 369, normalized size = 1.84 \begin{align*} \frac{2 \,{\left (15 \, c^{3} d^{3} g^{2} x^{3} + 35 \, a c^{2} d^{2} e f^{2} - 28 \, a^{2} c d e^{2} f g + 8 \, a^{3} e^{3} g^{2} + 3 \,{\left (14 \, c^{3} d^{3} f g + a c^{2} d^{2} e g^{2}\right )} x^{2} +{\left (35 \, c^{3} d^{3} f^{2} + 14 \, a c^{2} d^{2} e f g - 4 \, a^{2} c d e^{2} g^{2}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{105 \,{\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \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{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{2}}{\sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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