Optimal. Leaf size=270 \[ -\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac {8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{143 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}} \]
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Rubi [A] time = 0.44, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {794, 656, 648} \begin {gather*} -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{143 c^2 e^2 (d+e x)^{3/2}}-\frac {8 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac {16 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-6 b e g-c d g+13 c e f)}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
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}}{\sqrt {d+e x}} \, dx &=-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}-\frac {\left (2 \left (\frac {7}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {1}{2} \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}}{\sqrt {d+e x}} \, dx}{13 c e^3}\\ &=-\frac {2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}+\frac {(4 (2 c d-b e) (13 c e f-c d g-6 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)^{3/2}} \, dx}{143 c^2 e}\\ &=-\frac {8 (2 c d-b e) (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac {2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}+\frac {\left (8 (2 c d-b e)^2 (13 c e f-c d g-6 b e g)\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)^{5/2}} \, dx}{1287 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9009 c^4 e^2 (d+e x)^{7/2}}-\frac {8 (2 c d-b e) (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{1287 c^3 e^2 (d+e x)^{5/2}}-\frac {2 (13 c e f-c d g-6 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 c^2 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 c e^2 \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 183, normalized size = 0.68 \begin {gather*} \frac {2 (b e-c d+c e x)^3 \sqrt {(d+e x) (c (d-e x)-b e)} \left (-48 b^3 e^3 g+8 b^2 c e^2 (44 d g+13 e f+21 e g x)-2 b c^2 e \left (423 d^2 g+d e (390 f+532 g x)+7 e^2 x (26 f+27 g x)\right )+c^3 \left (542 d^3 g+d^2 e (1963 f+1897 g x)+14 d e^2 x (169 f+144 g x)+63 e^3 x^2 (13 f+11 g x)\right )\right )}{9009 c^4 e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.64, size = 248, normalized size = 0.92 \begin {gather*} -\frac {2 \left ((d+e x) (2 c d-b e)-c (d+e x)^2\right )^{7/2} \left (-48 b^3 e^3 g+168 b^2 c e^2 g (d+e x)+184 b^2 c d e^2 g+104 b^2 c e^3 f-160 b c^2 d^2 e g-364 b c^2 e^2 f (d+e x)-416 b c^2 d e^2 f-378 b c^2 e g (d+e x)^2-308 b c^2 d e g (d+e x)-32 c^3 d^3 g+416 c^3 d^2 e f-56 c^3 d^2 g (d+e x)+819 c^3 e f (d+e x)^2+728 c^3 d e f (d+e x)+693 c^3 g (d+e x)^3-63 c^3 d g (d+e x)^2\right )}{9009 c^4 e^2 (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 675, normalized size = 2.50 \begin {gather*} \frac {2 \, {\left (693 \, c^{6} e^{6} g x^{6} + 63 \, {\left (13 \, c^{6} e^{6} f - {\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} g\right )} x^{5} - 7 \, {\left (13 \, {\left (c^{6} d e^{5} - 23 \, b c^{5} e^{6}\right )} f + {\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} g\right )} x^{4} - {\left (13 \, {\left (206 \, c^{6} d^{2} e^{4} - 192 \, b c^{5} d e^{5} - 113 \, b^{2} c^{4} e^{6}\right )} f - {\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} g\right )} x^{3} + 3 \, {\left (13 \, {\left (10 \, c^{6} d^{3} e^{3} - 118 \, b c^{5} d^{2} e^{4} + 107 \, b^{2} c^{4} d e^{5} + b^{3} c^{3} e^{6}\right )} f + {\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} g\right )} x^{2} - 13 \, {\left (151 \, c^{6} d^{5} e - 513 \, b c^{5} d^{4} e^{2} + 641 \, b^{2} c^{4} d^{3} e^{3} - 355 \, b^{3} c^{3} d^{2} e^{4} + 84 \, b^{4} c^{2} d e^{5} - 8 \, b^{5} c e^{6}\right )} f - 2 \, {\left (271 \, c^{6} d^{6} - 1236 \, b c^{5} d^{5} e + 2258 \, b^{2} c^{4} d^{4} e^{2} - 2092 \, b^{3} c^{3} d^{3} e^{3} + 1023 \, b^{4} c^{2} d^{2} e^{4} - 248 \, b^{5} c d e^{5} + 24 \, b^{6} e^{6}\right )} g + {\left (13 \, {\left (271 \, c^{6} d^{4} e^{2} - 512 \, b c^{5} d^{3} e^{3} + 207 \, b^{2} c^{4} d^{2} e^{4} + 38 \, b^{3} c^{3} d e^{5} - 4 \, b^{4} c^{2} e^{6}\right )} f - {\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{9009 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{\sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 235, normalized size = 0.87 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-693 g \,e^{3} x^{3} c^{3}+378 b \,c^{2} e^{3} g \,x^{2}-2016 c^{3} d \,e^{2} g \,x^{2}-819 c^{3} e^{3} f \,x^{2}-168 b^{2} c \,e^{3} g x +1064 b \,c^{2} d \,e^{2} g x +364 b \,c^{2} e^{3} f x -1897 c^{3} d^{2} e g x -2366 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -352 b^{2} c d \,e^{2} g -104 b^{2} c \,e^{3} f +846 b \,c^{2} d^{2} e g +780 b \,c^{2} d \,e^{2} f -542 c^{3} d^{3} g -1963 f \,d^{2} c^{3} e \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}{9009 \left (e x +d \right )^{\frac {5}{2}} c^{4} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.75, size = 638, normalized size = 2.36 \begin {gather*} \frac {2 \, {\left (63 \, c^{5} e^{5} x^{5} - 151 \, c^{5} d^{5} + 513 \, b c^{4} d^{4} e - 641 \, b^{2} c^{3} d^{3} e^{2} + 355 \, b^{3} c^{2} d^{2} e^{3} - 84 \, b^{4} c d e^{4} + 8 \, b^{5} e^{5} - 7 \, {\left (c^{5} d e^{4} - 23 \, b c^{4} e^{5}\right )} x^{4} - {\left (206 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} - 113 \, b^{2} c^{3} e^{5}\right )} x^{3} + 3 \, {\left (10 \, c^{5} d^{3} e^{2} - 118 \, b c^{4} d^{2} e^{3} + 107 \, b^{2} c^{3} d e^{4} + b^{3} c^{2} e^{5}\right )} x^{2} + {\left (271 \, c^{5} d^{4} e - 512 \, b c^{4} d^{3} e^{2} + 207 \, b^{2} c^{3} d^{2} e^{3} + 38 \, b^{3} c^{2} d e^{4} - 4 \, b^{4} c e^{5}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{693 \, c^{3} e} + \frac {2 \, {\left (693 \, c^{6} e^{6} x^{6} - 542 \, c^{6} d^{6} + 2472 \, b c^{5} d^{5} e - 4516 \, b^{2} c^{4} d^{4} e^{2} + 4184 \, b^{3} c^{3} d^{3} e^{3} - 2046 \, b^{4} c^{2} d^{2} e^{4} + 496 \, b^{5} c d e^{5} - 48 \, b^{6} e^{6} - 63 \, {\left (c^{6} d e^{5} - 27 \, b c^{5} e^{6}\right )} x^{5} - 7 \, {\left (296 \, c^{6} d^{2} e^{4} - 280 \, b c^{5} d e^{5} - 159 \, b^{2} c^{4} e^{6}\right )} x^{4} + {\left (206 \, c^{6} d^{3} e^{3} - 3114 \, b c^{5} d^{2} e^{4} + 2893 \, b^{2} c^{4} d e^{5} + 15 \, b^{3} c^{3} e^{6}\right )} x^{3} + 3 \, {\left (683 \, c^{6} d^{4} e^{2} - 1328 \, b c^{5} d^{3} e^{3} + 601 \, b^{2} c^{4} d^{2} e^{4} + 50 \, b^{3} c^{3} d e^{5} - 6 \, b^{4} c^{2} e^{6}\right )} x^{2} - {\left (271 \, c^{6} d^{5} e - 965 \, b c^{5} d^{4} e^{2} + 1293 \, b^{2} c^{4} d^{3} e^{3} - 799 \, b^{3} c^{3} d^{2} e^{4} + 224 \, b^{4} c^{2} d e^{5} - 24 \, b^{5} c e^{6}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{9009 \, c^{4} e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.72, size = 491, normalized size = 1.82 \begin {gather*} \frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,e^2\,x^4\,\left (159\,g\,b^2\,e^2+280\,g\,b\,c\,d\,e+299\,f\,b\,c\,e^2-296\,g\,c^2\,d^2-13\,f\,c^2\,d\,e\right )}{1287}+\frac {2\,c^2\,e^4\,g\,x^6}{13}+\frac {2\,x^2\,\left (b\,e-c\,d\right )\,\left (-6\,g\,b^3\,e^3+44\,g\,b^2\,c\,d\,e^2+13\,f\,b^2\,c\,e^3+645\,g\,b\,c^2\,d^2\,e+1404\,f\,b\,c^2\,d\,e^2-683\,g\,c^3\,d^3-130\,f\,c^3\,d^2\,e\right )}{3003\,c^2}+\frac {x^3\,\left (30\,g\,b^3\,c^3\,e^6+5786\,g\,b^2\,c^4\,d\,e^5+2938\,f\,b^2\,c^4\,e^6-6228\,g\,b\,c^5\,d^2\,e^4+4992\,f\,b\,c^5\,d\,e^5+412\,g\,c^6\,d^3\,e^3-5356\,f\,c^6\,d^2\,e^4\right )}{9009\,c^4\,e^2}+\frac {2\,c\,e^3\,x^5\,\left (27\,b\,e\,g-c\,d\,g+13\,c\,e\,f\right )}{143}+\frac {2\,{\left (b\,e-c\,d\right )}^3\,\left (-48\,g\,b^3\,e^3+352\,g\,b^2\,c\,d\,e^2+104\,f\,b^2\,c\,e^3-846\,g\,b\,c^2\,d^2\,e-780\,f\,b\,c^2\,d\,e^2+542\,g\,c^3\,d^3+1963\,f\,c^3\,d^2\,e\right )}{9009\,c^4\,e^2}+\frac {2\,x\,{\left (b\,e-c\,d\right )}^2\,\left (24\,g\,b^3\,e^3-176\,g\,b^2\,c\,d\,e^2-52\,f\,b^2\,c\,e^3+423\,g\,b\,c^2\,d^2\,e+390\,f\,b\,c^2\,d\,e^2-271\,g\,c^3\,d^3+3523\,f\,c^3\,d^2\,e\right )}{9009\,c^3\,e}\right )}{\sqrt {d+e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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