Optimal. Leaf size=336 \[ \frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{715 c^3 d^3 (d+e x)^{7/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3}{6435 c^4 d^4 e (d+e x)^{5/2}}-\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \]
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Rubi [A] time = 0.618059, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {870, 794, 648} \[ \frac{16 (f+g x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{32 (f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^2}{715 c^3 d^3 (d+e x)^{7/2}}+\frac{128 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3}{6435 c^4 d^4 e (d+e x)^{5/2}}-\frac{128 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2} (c d f-a e g)^3 \left (2 a e^2 g-c d (9 e f-7 d g)\right )}{45045 c^5 d^5 e (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 870
Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx &=\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac{(8 (c d f-a e g)) \int \frac{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{15 c d}\\ &=\frac{16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac{\left (16 (c d f-a e g)^2\right ) \int \frac{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{65 c^2 d^2}\\ &=\frac{32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac{16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac{\left (64 (c d f-a e g)^3\right ) \int \frac{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{715 c^3 d^3}\\ &=\frac{128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac{32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac{16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}+\frac{\left (64 (c d f-a e g)^3 \left (9 f-\frac{7 d g}{e}-\frac{2 a e g}{c d}\right )\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{6435 c^3 d^3}\\ &=\frac{128 (c d f-a e g)^3 \left (9 f-\frac{7 d g}{e}-\frac{2 a e g}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{45045 c^4 d^4 (d+e x)^{7/2}}+\frac{128 g (c d f-a e g)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{6435 c^4 d^4 e (d+e x)^{5/2}}+\frac{32 (c d f-a e g)^2 (f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{7/2}}+\frac{16 (c d f-a e g) (f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{195 c^2 d^2 (d+e x)^{7/2}}+\frac{2 (f+g x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d (d+e x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.22377, size = 205, normalized size = 0.61 \[ \frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (48 a^2 c^2 d^2 e^2 g^2 \left (65 f^2+70 f g x+21 g^2 x^2\right )-64 a^3 c d e^3 g^3 (15 f+7 g x)+128 a^4 e^4 g^4-8 a c^3 d^3 e g \left (1365 f^2 g x+715 f^3+945 f g^2 x^2+231 g^3 x^3\right )+c^4 d^4 \left (24570 f^2 g^2 x^2+20020 f^3 g x+6435 f^4+13860 f g^3 x^3+3003 g^4 x^4\right )\right )}{45045 c^5 d^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 283, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3003\,{g}^{4}{x}^{4}{c}^{4}{d}^{4}-1848\,a{c}^{3}{d}^{3}e{g}^{4}{x}^{3}+13860\,{c}^{4}{d}^{4}f{g}^{3}{x}^{3}+1008\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{g}^{4}{x}^{2}-7560\,a{c}^{3}{d}^{3}ef{g}^{3}{x}^{2}+24570\,{c}^{4}{d}^{4}{f}^{2}{g}^{2}{x}^{2}-448\,{a}^{3}cd{e}^{3}{g}^{4}x+3360\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}f{g}^{3}x-10920\,a{c}^{3}{d}^{3}e{f}^{2}{g}^{2}x+20020\,{c}^{4}{d}^{4}{f}^{3}gx+128\,{a}^{4}{e}^{4}{g}^{4}-960\,{a}^{3}cd{e}^{3}f{g}^{3}+3120\,{a}^{2}{c}^{2}{d}^{2}{e}^{2}{f}^{2}{g}^{2}-5720\,a{c}^{3}{d}^{3}e{f}^{3}g+6435\,{f}^{4}{c}^{4}{d}^{4} \right ) }{45045\,{c}^{5}{d}^{5}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.25476, size = 672, normalized size = 2. \begin{align*} \frac{2 \,{\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{2} e x^{2} + 3 \, a^{2} c d e^{2} x + a^{3} e^{3}\right )} \sqrt{c d x + a e} f^{4}}{7 \, c d} + \frac{8 \,{\left (7 \, c^{4} d^{4} x^{4} + 19 \, a c^{3} d^{3} e x^{3} + 15 \, a^{2} c^{2} d^{2} e^{2} x^{2} + a^{3} c d e^{3} x - 2 \, a^{4} e^{4}\right )} \sqrt{c d x + a e} f^{3} g}{63 \, c^{2} d^{2}} + \frac{4 \,{\left (63 \, c^{5} d^{5} x^{5} + 161 \, a c^{4} d^{4} e x^{4} + 113 \, a^{2} c^{3} d^{3} e^{2} x^{3} + 3 \, a^{3} c^{2} d^{2} e^{3} x^{2} - 4 \, a^{4} c d e^{4} x + 8 \, a^{5} e^{5}\right )} \sqrt{c d x + a e} f^{2} g^{2}}{231 \, c^{3} d^{3}} + \frac{8 \,{\left (231 \, c^{6} d^{6} x^{6} + 567 \, a c^{5} d^{5} e x^{5} + 371 \, a^{2} c^{4} d^{4} e^{2} x^{4} + 5 \, a^{3} c^{3} d^{3} e^{3} x^{3} - 6 \, a^{4} c^{2} d^{2} e^{4} x^{2} + 8 \, a^{5} c d e^{5} x - 16 \, a^{6} e^{6}\right )} \sqrt{c d x + a e} f g^{3}}{3003 \, c^{4} d^{4}} + \frac{2 \,{\left (3003 \, c^{7} d^{7} x^{7} + 7161 \, a c^{6} d^{6} e x^{6} + 4473 \, a^{2} c^{5} d^{5} e^{2} x^{5} + 35 \, a^{3} c^{4} d^{4} e^{3} x^{4} - 40 \, a^{4} c^{3} d^{3} e^{4} x^{3} + 48 \, a^{5} c^{2} d^{2} e^{5} x^{2} - 64 \, a^{6} c d e^{6} x + 128 \, a^{7} e^{7}\right )} \sqrt{c d x + a e} g^{4}}{45045 \, c^{5} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6954, size = 1214, normalized size = 3.61 \begin{align*} \frac{2 \,{\left (3003 \, c^{7} d^{7} g^{4} x^{7} + 6435 \, a^{3} c^{4} d^{4} e^{3} f^{4} - 5720 \, a^{4} c^{3} d^{3} e^{4} f^{3} g + 3120 \, a^{5} c^{2} d^{2} e^{5} f^{2} g^{2} - 960 \, a^{6} c d e^{6} f g^{3} + 128 \, a^{7} e^{7} g^{4} + 231 \,{\left (60 \, c^{7} d^{7} f g^{3} + 31 \, a c^{6} d^{6} e g^{4}\right )} x^{6} + 63 \,{\left (390 \, c^{7} d^{7} f^{2} g^{2} + 540 \, a c^{6} d^{6} e f g^{3} + 71 \, a^{2} c^{5} d^{5} e^{2} g^{4}\right )} x^{5} + 35 \,{\left (572 \, c^{7} d^{7} f^{3} g + 1794 \, a c^{6} d^{6} e f^{2} g^{2} + 636 \, a^{2} c^{5} d^{5} e^{2} f g^{3} + a^{3} c^{4} d^{4} e^{3} g^{4}\right )} x^{4} + 5 \,{\left (1287 \, c^{7} d^{7} f^{4} + 10868 \, a c^{6} d^{6} e f^{3} g + 8814 \, a^{2} c^{5} d^{5} e^{2} f^{2} g^{2} + 60 \, a^{3} c^{4} d^{4} e^{3} f g^{3} - 8 \, a^{4} c^{3} d^{3} e^{4} g^{4}\right )} x^{3} + 3 \,{\left (6435 \, a c^{6} d^{6} e f^{4} + 14300 \, a^{2} c^{5} d^{5} e^{2} f^{3} g + 390 \, a^{3} c^{4} d^{4} e^{3} f^{2} g^{2} - 120 \, a^{4} c^{3} d^{3} e^{4} f g^{3} + 16 \, a^{5} c^{2} d^{2} e^{5} g^{4}\right )} x^{2} +{\left (19305 \, a^{2} c^{5} d^{5} e^{2} f^{4} + 2860 \, a^{3} c^{4} d^{4} e^{3} f^{3} g - 1560 \, a^{4} c^{3} d^{3} e^{4} f^{2} g^{2} + 480 \, a^{5} c^{2} d^{2} e^{5} f g^{3} - 64 \, a^{6} c d e^{6} g^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{45045 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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