Optimal. Leaf size=210 \[ -\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+18 c d g+4 c e f)}{693 e^2 (d+e x)^7 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+18 c d g+4 c e f)}{99 e^2 (d+e x)^8 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (d+e x)^9 (2 c d-b e)} \]
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Rubi [A] time = 0.327171, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {792, 658, 650} \[ -\frac{4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+18 c d g+4 c e f)}{693 e^2 (d+e x)^7 (2 c d-b e)^3}-\frac{2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+18 c d g+4 c e f)}{99 e^2 (d+e x)^8 (2 c d-b e)^2}-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (d+e x)^9 (2 c d-b e)} \]
Antiderivative was successfully verified.
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Rule 792
Rule 658
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)^9} \, 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}}{11 e^2 (2 c d-b e) (d+e x)^9}+\frac{(4 c e f+18 c d g-11 b e g) \int \frac{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx}{11 e (2 c d-b e)}\\ &=-\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (2 c d-b e) (d+e x)^9}-\frac{2 (4 c e f+18 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^8}+\frac{(2 c (4 c e f+18 c d g-11 b e g)) \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}{99 e (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}}{11 e^2 (2 c d-b e) (d+e x)^9}-\frac{2 (4 c e f+18 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^8}-\frac{4 c (4 c e f+18 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{693 e^2 (2 c d-b e)^3 (d+e x)^7}\\ \end{align*}
Mathematica [A] time = 0.178911, size = 169, normalized size = 0.8 \[ -\frac{2 (b e-c d+c e x)^3 \sqrt{(d+e x) (c (d-e x)-b e)} \left (7 b^2 e^2 (2 d g+9 e f+11 e g x)-2 b c e \left (25 d^2 g+2 d e (70 f+81 g x)+e^2 x (14 f+11 g x)\right )+4 c^2 \left (d^2 e (79 f+81 g x)+9 d^3 g+9 d e^2 x (2 f+g x)+2 e^3 f x^2\right )\right )}{693 e^2 (d+e x)^6 (b e-2 c d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 236, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -22\,bc{e}^{3}g{x}^{2}+36\,{c}^{2}d{e}^{2}g{x}^{2}+8\,{c}^{2}{e}^{3}f{x}^{2}+77\,{b}^{2}{e}^{3}gx-324\,bcd{e}^{2}gx-28\,bc{e}^{3}fx+324\,{c}^{2}{d}^{2}egx+72\,{c}^{2}d{e}^{2}fx+14\,{b}^{2}d{e}^{2}g+63\,{b}^{2}{e}^{3}f-50\,bc{d}^{2}eg-280\,bcd{e}^{2}f+36\,{c}^{2}{d}^{3}g+316\,{c}^{2}{d}^{2}ef \right ) }{693\, \left ( ex+d \right ) ^{8} \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+12\,b{c}^{2}{d}^{2}e-8\,{c}^{3}{d}^{3} \right ){e}^{2}} \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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