∫ (f+gx)(cd2−bde−be2x−ce2x2)5∕2 __________________________________________________

Optimal. Leaf size=138 \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e^2 (2 c d-b e) (d+e x)^8}+\frac {2 (9 b e g-2 c (e f+8 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 e^2 (2 c d-b e)^2 (d+e x)^7} \]

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Rubi [A]
time = 0.13, antiderivative size = 138, normalized size of antiderivative = 1.00, 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 = {806, 664} \begin {gather*} \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)} \end {gather*}

\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*}

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Mathematica [A]
time = 0.22, size = 104, normalized size = 0.75 \begin {gather*} \frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-b e (7 e f+2 d g+9 e g x)+2 c \left (d^2 g+e^2 f x+8 d e (f+g x)\right )\right )}{63 e^2 (-2 c d+b e)^2 (d+e x)^5} \end {gather*}

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method	result	size

gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (9 b \,e^{2} g x -16 c d e g x -2 c \,e^{2} f x +2 b d e g +7 b \,e^{2} f -2 c \,d^{2} g -16 c d e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{7} e^{2} \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right )}\)	\(128\)
default	\(-\frac {2 g \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7 e^{8} \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{e^{9}}\)	\(212\)
trager	\(-\frac {2 \left (9 b \,c^{3} e^{5} g \,x^{4}-16 c^{4} d \,e^{4} g \,x^{4}-2 c^{4} e^{5} f \,x^{4}+27 b^{2} c^{2} e^{5} g \,x^{3}-73 b \,c^{3} d \,e^{4} g \,x^{3}+b \,c^{3} e^{5} f \,x^{3}+46 c^{4} d^{2} e^{3} g \,x^{3}-10 c^{4} d \,e^{4} f \,x^{3}+27 b^{3} c \,e^{5} g \,x^{2}-96 b^{2} c^{2} d \,e^{4} g \,x^{2}+15 b^{2} c^{2} e^{5} f \,x^{2}+111 b \,c^{3} d^{2} e^{3} g \,x^{2}-57 b \,c^{3} d \,e^{4} f \,x^{2}-42 c^{4} d^{3} e^{2} g \,x^{2}+42 c^{4} d^{2} e^{3} f \,x^{2}+9 b^{4} e^{5} g x -37 b^{3} c d \,e^{4} g x +19 b^{3} c \,e^{5} f x +57 b^{2} c^{2} d^{2} e^{3} g x -84 b^{2} c^{2} d \,e^{4} f x -39 b \,c^{3} d^{3} e^{2} g x +111 b \,c^{3} d^{2} e^{3} f x +10 c^{4} d^{4} e g x -46 c^{4} d^{3} e^{2} f x +2 b^{4} d \,e^{4} g +7 b^{4} e^{5} f -8 b^{3} c \,d^{2} e^{3} g -37 b^{3} c d \,e^{4} f +12 b^{2} c^{2} d^{3} e^{2} g +69 b^{2} c^{2} d^{2} e^{3} f -8 b \,c^{3} d^{4} e g -55 b \,c^{3} d^{3} e^{2} f +2 c^{4} d^{5} g +16 c^{4} d^{4} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{63 \left (b^{2} e^{2}-4 b c d e +4 c^{2} d^{2}\right ) e^{2} \left (e x +d \right )^{5}}\)	\(509\)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3739 vs. \(2 (133) = 266\).
time = 1.04, size = 3739, normalized size = 27.09 \begin {gather*} \text {Too large to display} \end {gather*}

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Mupad [B]
time = 26.34, size = 2500, normalized size = 18.12 \begin {gather*} \text {Too large to display} \end {gather*}

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3.23.5 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^8} \, dx\) [2205]