Optimal. Leaf size=146 \[ \frac{2 (d+e x)^2 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 (d+e x) (b e g-4 c d g+2 c e f)}{3 c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]
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Rubi [A] time = 0.171988, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {788, 636} \[ \frac{2 (d+e x)^2 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 (d+e x) (b e g-4 c d g+2 c e f)}{3 c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 788
Rule 636
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^2}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{(2 c e f-4 c d g+b e g) \int \frac{d+e x}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^2}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 (2 c e f-4 c d g+b e g) (d+e x)}{3 c e^2 (2 c d-b e)^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0647882, size = 100, normalized size = 0.68 \[ \frac{2 (d+e x) \left (b e (-2 d g+3 e f+e g x)+2 c \left (d^2 g-2 d e (f+g x)+e^2 f x\right )\right )}{3 e^2 (b e-2 c d)^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 128, normalized size = 0.9 \begin{align*} -{\frac{2\, \left ( ex+d \right ) ^{3} \left ( cex+be-cd \right ) \left ( -b{e}^{2}gx+4\,cdegx-2\,c{e}^{2}fx+2\,bdeg-3\,b{e}^{2}f-2\,c{d}^{2}g+4\,cdef \right ) }{3\,{e}^{2} \left ({b}^{2}{e}^{2}-4\,bcde+4\,{c}^{2}{d}^{2} \right ) } \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \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 [A] time = 17.5717, size = 454, normalized size = 3.11 \begin{align*} \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left ({\left (4 \, c d e - 3 \, b e^{2}\right )} f - 2 \,{\left (c d^{2} - b d e\right )} g -{\left (2 \, c e^{2} f -{\left (4 \, c d e - b e^{2}\right )} g\right )} x\right )}}{3 \,{\left (4 \, c^{4} d^{4} e^{2} - 12 \, b c^{3} d^{3} e^{3} + 13 \, b^{2} c^{2} d^{2} e^{4} - 6 \, b^{3} c d e^{5} + b^{4} e^{6} +{\left (4 \, c^{4} d^{2} e^{4} - 4 \, b c^{3} d e^{5} + b^{2} c^{2} e^{6}\right )} x^{2} - 2 \,{\left (4 \, c^{4} d^{3} e^{3} - 8 \, b c^{3} d^{2} e^{4} + 5 \, b^{2} c^{2} d e^{5} - b^{3} c e^{6}\right )} x\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{\left (d + e x\right )^{2} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22587, size = 792, normalized size = 5.42 \begin{align*} \frac{2 \, \sqrt{-c x^{2} e^{2} + c d^{2} - b x e^{2} - b d e}{\left ({\left ({\left (\frac{{\left (16 \, c^{3} d^{3} g e^{3} - 8 \, c^{3} d^{2} f e^{4} - 20 \, b c^{2} d^{2} g e^{4} + 8 \, b c^{2} d f e^{5} + 8 \, b^{2} c d g e^{5} - 2 \, b^{2} c f e^{6} - b^{3} g e^{6}\right )} x}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}} + \frac{3 \,{\left (8 \, c^{3} d^{4} g e^{2} - 8 \, b c^{2} d^{3} g e^{3} - 4 \, b c^{2} d^{2} f e^{4} + 2 \, b^{2} c d^{2} g e^{4} + 4 \, b^{2} c d f e^{5} - b^{3} f e^{6}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x + \frac{3 \,{\left (8 \, c^{3} d^{4} f e^{2} + 4 \, b c^{2} d^{4} g e^{2} - 16 \, b c^{2} d^{3} f e^{3} - 4 \, b^{2} c d^{3} g e^{3} + 10 \, b^{2} c d^{2} f e^{4} + b^{3} d^{2} g e^{4} - 2 \, b^{3} d f e^{5}\right )}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )} x - \frac{8 \, c^{3} d^{6} g - 16 \, c^{3} d^{5} f e - 16 \, b c^{2} d^{5} g e + 28 \, b c^{2} d^{4} f e^{2} + 10 \, b^{2} c d^{4} g e^{2} - 16 \, b^{2} c d^{3} f e^{3} - 2 \, b^{3} d^{3} g e^{3} + 3 \, b^{3} d^{2} f e^{4}}{16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}}\right )}}{3 \,{\left (c x^{2} e^{2} - c d^{2} + b x e^{2} + b d e\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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