Optimal. Leaf size=138 \[ \frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-2 c (8 d g+e f))}{63 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (d+e x)^8 (2 c d-b e)} \]
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Rubi [A] time = 0.221061, antiderivative size = 138, 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 = {792, 650} \[ \frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-2 c (8 d g+e f))}{63 e^2 (d+e x)^7 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (d+e x)^8 (2 c d-b e)} \]
Antiderivative was successfully verified.
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Rule 792
Rule 650
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (2 c d-b e) (d+e x)^8}-\frac{\left (2 \left (\frac{7}{2} e \left (-2 c e^2 f+b e^2 g\right )-8 \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^7} \, dx}{9 e \left (-2 c d e^2+b e^3\right )}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (2 c d-b e) (d+e x)^8}+\frac{2 (9 b e g-2 c (e f+8 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 e^2 (2 c d-b e)^2 (d+e x)^7}\\ \end{align*}
Mathematica [A] time = 0.117391, size = 104, normalized size = 0.75 \[ \frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (2 c \left (d^2 g+8 d e (f+g x)+e^2 f x\right )-b e (2 d g+7 e f+9 e g x)\right )}{63 e^2 (d+e x)^5 (b e-2 c d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 128, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 9\,b{e}^{2}gx-16\,cdegx-2\,c{e}^{2}fx+2\,bdeg+7\,b{e}^{2}f-2\,c{d}^{2}g-16\,cdef \right ) }{63\, \left ( ex+d \right ) ^{7}{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 [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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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 [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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