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

Optimal. Leaf size=210 \[ -\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e^2 (2 c d-b e) (d+e x)^9}-\frac {2 (4 c e f+18 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{99 e^2 (2 c d-b e)^2 (d+e x)^8}-\frac {4 c (4 c e f+18 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{693 e^2 (2 c d-b e)^3 (d+e x)^7} \]

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Rubi [A]
time = 0.21, antiderivative size = 210, normalized size of antiderivative = 1.00, 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 = {806, 672, 664} \begin {gather*} -\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)} \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)^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*}

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Mathematica [A]
time = 0.25, size = 169, normalized size = 0.80 \begin {gather*} -\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (7 b^2 e^2 (9 e f+2 d g+11 e g x)-2 b c e \left (25 d^2 g+e^2 x (14 f+11 g x)+2 d e (70 f+81 g x)\right )+4 c^2 \left (9 d^3 g+2 e^3 f x^2+9 d e^2 x (2 f+g x)+d^2 e (79 f+81 g x)\right )\right )}{693 e^2 (-2 c d+b e)^3 (d+e x)^6} \end {gather*}

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method	result	size

gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (-22 b c \,e^{3} g \,x^{2}+36 c^{2} d \,e^{2} g \,x^{2}+8 c^{2} e^{3} f \,x^{2}+77 b^{2} e^{3} g x -324 b c d \,e^{2} g x -28 b c \,e^{3} f x +324 c^{2} d^{2} e g x +72 c^{2} d \,e^{2} f x +14 b^{2} d \,e^{2} g +63 b^{2} e^{3} f -50 b c \,d^{2} e g -280 b c d \,e^{2} f +36 c^{2} d^{3} g +316 c^{2} d^{2} e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{693 \left (e x +d \right )^{8} \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2}}\)	\(236\)
default	\(\frac {\left (-d g +e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{9}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{10}}+\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 c d e \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{e^{9}}\)	\(365\)
trager	\(-\frac {2 \left (-22 b \,c^{4} e^{6} g \,x^{5}+36 c^{5} d \,e^{5} g \,x^{5}+8 c^{5} e^{6} f \,x^{5}+11 b^{2} c^{3} e^{6} g \,x^{4}-150 b \,c^{4} d \,e^{5} g \,x^{4}-4 b \,c^{4} e^{6} f \,x^{4}+216 c^{5} d^{2} e^{4} g \,x^{4}+48 c^{5} d \,e^{5} f \,x^{4}+165 b^{3} c^{2} e^{6} g \,x^{3}-949 b^{2} c^{3} d \,e^{5} g \,x^{3}+3 b^{2} c^{3} e^{6} f \,x^{3}+1612 b \,c^{4} d^{2} e^{4} g \,x^{3}-28 b \,c^{4} d \,e^{5} f \,x^{3}-828 c^{5} d^{3} e^{3} g \,x^{3}+124 c^{5} d^{2} e^{4} f \,x^{3}+209 b^{4} c \,e^{6} g \,x^{2}-1290 b^{3} c^{2} d \,e^{5} g \,x^{2}+113 b^{3} c^{2} e^{6} f \,x^{2}+2781 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-669 b^{2} c^{3} d \,e^{5} f \,x^{2}-2528 b \,c^{4} d^{3} e^{3} g \,x^{2}+1296 b \,c^{4} d^{2} e^{4} f \,x^{2}+828 c^{5} d^{4} e^{2} g \,x^{2}-740 c^{5} d^{3} e^{3} f \,x^{2}+77 b^{5} e^{6} g x -513 b^{4} c d \,e^{5} g x +161 b^{4} c \,e^{6} f x +1293 b^{3} c^{2} d^{2} e^{4} g x -1062 b^{3} c^{2} d \,e^{5} f x -1571 b^{2} c^{3} d^{3} e^{3} g x +2517 b^{2} c^{3} d^{2} e^{4} f x +930 b \,c^{4} d^{4} e^{2} g x -2492 b \,c^{4} d^{3} e^{3} f x -216 c^{5} d^{5} e g x +876 c^{5} d^{4} e^{2} f x +14 b^{5} d \,e^{5} g +63 b^{5} e^{6} f -92 b^{4} c \,d^{2} e^{4} g -469 b^{4} c d \,e^{5} f +228 b^{3} c^{2} d^{3} e^{3} g +1345 b^{3} c^{2} d^{2} e^{4} f -272 b^{2} c^{3} d^{4} e^{2} g -1851 b^{2} c^{3} d^{3} e^{3} f +158 b \,c^{4} d^{5} e g +1228 b \,c^{4} d^{4} e^{2} f -36 c^{5} d^{6} g -316 c^{5} d^{5} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{693 \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2} \left (e x +d \right )^{6}}\)	\(728\)

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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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{9}}\, dx \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4740 vs. \(2 (203) = 406\).
time = 1.05, size = 4740, normalized size = 22.57 \begin {gather*} \text {Too large to display} \end {gather*}

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Mupad [B]
time = 53.40, size = 2500, normalized size = 11.90 \begin {gather*} \text {Too large to display} \end {gather*}

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3.23.6 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^9} \, dx\) [2206]