3.783 \(\int \frac{(d+e x)^{3/2} (f+g x)^4}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=501 \[ \frac{128 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{3465 c^6 d^6 e g \sqrt{d+e x}}-\frac{128 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )}{3465 c^5 d^5 e}-\frac{32 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )}{1155 c^4 d^4 g \sqrt{d+e x}}-\frac{16 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )}{693 c^3 d^3 g \sqrt{d+e x}}-\frac{2 (f+g x)^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (10 a e^2 g+c d (e f-11 d g)\right )}{99 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt{d+e x}} \]

[Out]

(128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(2*a*e^2*g - c*d*(3*e*f
- d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3465*c^6*d^6*e*g*Sqrt[d +
e*x]) - (128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*d*g))*Sqrt[d + e*x]*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3465*c^5*d^5*e) - (32*(c*d*f - a*e
*g)^2*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(1155*c^4*d^4*g*Sqrt[d + e*x]) - (16*(c*d*f - a*e*g)*(10*a*e^2*
g + c*d*(e*f - 11*d*g))*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
/(693*c^3*d^3*g*Sqrt[d + e*x]) - (2*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^
4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(99*c^2*d^2*g*Sqrt[d + e*x]) + (2
*e*(f + g*x)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(11*c*d*g*Sqrt[d + e
*x])

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Rubi [A]  time = 2.27652, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{128 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right ) \left (2 a e^2 g-c d (3 e f-d g)\right )}{3465 c^6 d^6 e g \sqrt{d+e x}}-\frac{128 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3 \left (10 a e^2 g+c d (e f-11 d g)\right )}{3465 c^5 d^5 e}-\frac{32 (f+g x)^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (10 a e^2 g+c d (e f-11 d g)\right )}{1155 c^4 d^4 g \sqrt{d+e x}}-\frac{16 (f+g x)^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g) \left (10 a e^2 g+c d (e f-11 d g)\right )}{693 c^3 d^3 g \sqrt{d+e x}}-\frac{2 (f+g x)^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (10 a e^2 g+c d (e f-11 d g)\right )}{99 c^2 d^2 g \sqrt{d+e x}}+\frac{2 e (f+g x)^5 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{11 c d g \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(3/2)*(f + g*x)^4)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(2*a*e^2*g - c*d*(3*e*f
- d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3465*c^6*d^6*e*g*Sqrt[d +
e*x]) - (128*(c*d*f - a*e*g)^3*(10*a*e^2*g + c*d*(e*f - 11*d*g))*Sqrt[d + e*x]*
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3465*c^5*d^5*e) - (32*(c*d*f - a*e
*g)^2*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)
*x + c*d*e*x^2])/(1155*c^4*d^4*g*Sqrt[d + e*x]) - (16*(c*d*f - a*e*g)*(10*a*e^2*
g + c*d*(e*f - 11*d*g))*(f + g*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
/(693*c^3*d^3*g*Sqrt[d + e*x]) - (2*(10*a*e^2*g + c*d*(e*f - 11*d*g))*(f + g*x)^
4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(99*c^2*d^2*g*Sqrt[d + e*x]) + (2
*e*(f + g*x)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(11*c*d*g*Sqrt[d + e
*x])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(g*x+f)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.58769, size = 380, normalized size = 0.76 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (-1280 a^5 e^6 g^4+128 a^4 c d e^4 g^3 (11 d g+44 e f+5 e g x)-32 a^3 c^2 d^2 e^3 g^2 \left (22 d g (9 f+g x)+e \left (297 f^2+88 f g x+15 g^2 x^2\right )\right )+16 a^2 c^3 d^3 e^2 g \left (33 d g \left (21 f^2+6 f g x+g^2 x^2\right )+e \left (462 f^3+297 f^2 g x+132 f g^2 x^2+25 g^3 x^3\right )\right )-2 a c^4 d^4 e \left (44 d g \left (105 f^3+63 f^2 g x+27 f g^2 x^2+5 g^3 x^3\right )+e \left (1155 f^4+1848 f^3 g x+1782 f^2 g^2 x^2+880 f g^3 x^3+175 g^4 x^4\right )\right )+c^5 d^5 \left (11 d \left (315 f^4+420 f^3 g x+378 f^2 g^2 x^2+180 f g^3 x^3+35 g^4 x^4\right )+e x \left (1155 f^4+2772 f^3 g x+2970 f^2 g^2 x^2+1540 f g^3 x^3+315 g^4 x^4\right )\right )\right )}{3465 c^6 d^6 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^(3/2)*(f + g*x)^4)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-1280*a^5*e^6*g^4 + 128*a^4*c*d*e^4*g^3*(44*e*
f + 11*d*g + 5*e*g*x) - 32*a^3*c^2*d^2*e^3*g^2*(22*d*g*(9*f + g*x) + e*(297*f^2
+ 88*f*g*x + 15*g^2*x^2)) + 16*a^2*c^3*d^3*e^2*g*(33*d*g*(21*f^2 + 6*f*g*x + g^2
*x^2) + e*(462*f^3 + 297*f^2*g*x + 132*f*g^2*x^2 + 25*g^3*x^3)) - 2*a*c^4*d^4*e*
(44*d*g*(105*f^3 + 63*f^2*g*x + 27*f*g^2*x^2 + 5*g^3*x^3) + e*(1155*f^4 + 1848*f
^3*g*x + 1782*f^2*g^2*x^2 + 880*f*g^3*x^3 + 175*g^4*x^4)) + c^5*d^5*(11*d*(315*f
^4 + 420*f^3*g*x + 378*f^2*g^2*x^2 + 180*f*g^3*x^3 + 35*g^4*x^4) + e*x*(1155*f^4
+ 2772*f^3*g*x + 2970*f^2*g^2*x^2 + 1540*f*g^3*x^3 + 315*g^4*x^4))))/(3465*c^6*
d^6*Sqrt[d + e*x])

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Maple [A]  time = 0.014, size = 641, normalized size = 1.3 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -315\,e{g}^{4}{x}^{5}{c}^{5}{d}^{5}+350\,a{c}^{4}{d}^{4}{e}^{2}{g}^{4}{x}^{4}-385\,{c}^{5}{d}^{6}{g}^{4}{x}^{4}-1540\,{c}^{5}{d}^{5}ef{g}^{3}{x}^{4}-400\,{a}^{2}{c}^{3}{d}^{3}{e}^{3}{g}^{4}{x}^{3}+440\,a{c}^{4}{d}^{5}e{g}^{4}{x}^{3}+1760\,a{c}^{4}{d}^{4}{e}^{2}f{g}^{3}{x}^{3}-1980\,{c}^{5}{d}^{6}f{g}^{3}{x}^{3}-2970\,{c}^{5}{d}^{5}e{f}^{2}{g}^{2}{x}^{3}+480\,{a}^{3}{c}^{2}{d}^{2}{e}^{4}{g}^{4}{x}^{2}-528\,{a}^{2}{c}^{3}{d}^{4}{e}^{2}{g}^{4}{x}^{2}-2112\,{a}^{2}{c}^{3}{d}^{3}{e}^{3}f{g}^{3}{x}^{2}+2376\,a{c}^{4}{d}^{5}ef{g}^{3}{x}^{2}+3564\,a{c}^{4}{d}^{4}{e}^{2}{f}^{2}{g}^{2}{x}^{2}-4158\,{c}^{5}{d}^{6}{f}^{2}{g}^{2}{x}^{2}-2772\,{c}^{5}{d}^{5}e{f}^{3}g{x}^{2}-640\,{a}^{4}cd{e}^{5}{g}^{4}x+704\,{a}^{3}{c}^{2}{d}^{3}{e}^{3}{g}^{4}x+2816\,{a}^{3}{c}^{2}{d}^{2}{e}^{4}f{g}^{3}x-3168\,{a}^{2}{c}^{3}{d}^{4}{e}^{2}f{g}^{3}x-4752\,{a}^{2}{c}^{3}{d}^{3}{e}^{3}{f}^{2}{g}^{2}x+5544\,a{c}^{4}{d}^{5}e{f}^{2}{g}^{2}x+3696\,a{c}^{4}{d}^{4}{e}^{2}{f}^{3}gx-4620\,{c}^{5}{d}^{6}{f}^{3}gx-1155\,{c}^{5}{d}^{5}e{f}^{4}x+1280\,{a}^{5}{e}^{6}{g}^{4}-1408\,{a}^{4}c{d}^{2}{e}^{4}{g}^{4}-5632\,{a}^{4}cd{e}^{5}f{g}^{3}+6336\,{a}^{3}{c}^{2}{d}^{3}{e}^{3}f{g}^{3}+9504\,{a}^{3}{c}^{2}{d}^{2}{e}^{4}{f}^{2}{g}^{2}-11088\,{a}^{2}{c}^{3}{d}^{4}{e}^{2}{f}^{2}{g}^{2}-7392\,{a}^{2}{c}^{3}{d}^{3}{e}^{3}{f}^{3}g+9240\,a{c}^{4}{d}^{5}e{f}^{3}g+2310\,a{c}^{4}{d}^{4}{e}^{2}{f}^{4}-3465\,{d}^{6}{f}^{4}{c}^{5} \right ) }{3465\,{c}^{6}{d}^{6}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(g*x+f)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-2/3465*(c*d*x+a*e)*(-315*c^5*d^5*e*g^4*x^5+350*a*c^4*d^4*e^2*g^4*x^4-385*c^5*d^
6*g^4*x^4-1540*c^5*d^5*e*f*g^3*x^4-400*a^2*c^3*d^3*e^3*g^4*x^3+440*a*c^4*d^5*e*g
^4*x^3+1760*a*c^4*d^4*e^2*f*g^3*x^3-1980*c^5*d^6*f*g^3*x^3-2970*c^5*d^5*e*f^2*g^
2*x^3+480*a^3*c^2*d^2*e^4*g^4*x^2-528*a^2*c^3*d^4*e^2*g^4*x^2-2112*a^2*c^3*d^3*e
^3*f*g^3*x^2+2376*a*c^4*d^5*e*f*g^3*x^2+3564*a*c^4*d^4*e^2*f^2*g^2*x^2-4158*c^5*
d^6*f^2*g^2*x^2-2772*c^5*d^5*e*f^3*g*x^2-640*a^4*c*d*e^5*g^4*x+704*a^3*c^2*d^3*e
^3*g^4*x+2816*a^3*c^2*d^2*e^4*f*g^3*x-3168*a^2*c^3*d^4*e^2*f*g^3*x-4752*a^2*c^3*
d^3*e^3*f^2*g^2*x+5544*a*c^4*d^5*e*f^2*g^2*x+3696*a*c^4*d^4*e^2*f^3*g*x-4620*c^5
*d^6*f^3*g*x-1155*c^5*d^5*e*f^4*x+1280*a^5*e^6*g^4-1408*a^4*c*d^2*e^4*g^4-5632*a
^4*c*d*e^5*f*g^3+6336*a^3*c^2*d^3*e^3*f*g^3+9504*a^3*c^2*d^2*e^4*f^2*g^2-11088*a
^2*c^3*d^4*e^2*f^2*g^2-7392*a^2*c^3*d^3*e^3*f^3*g+9240*a*c^4*d^5*e*f^3*g+2310*a*
c^4*d^4*e^2*f^4-3465*c^5*d^6*f^4)*(e*x+d)^(1/2)/c^6/d^6/(c*d*e*x^2+a*e^2*x+c*d^2
*x+a*d*e)^(1/2)

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Maxima [A]  time = 0.773457, size = 936, normalized size = 1.87 \[ \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )} f^{4}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} + \frac{8 \,{\left (3 \, c^{3} d^{3} e x^{3} - 10 \, a^{2} c d^{2} e^{2} + 8 \, a^{3} e^{4} +{\left (5 \, c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} x^{2} -{\left (5 \, a c^{2} d^{3} e - 4 \, a^{2} c d e^{3}\right )} x\right )} f^{3} g}{15 \, \sqrt{c d x + a e} c^{3} d^{3}} + \frac{4 \,{\left (15 \, c^{4} d^{4} e x^{4} + 56 \, a^{3} c d^{2} e^{3} - 48 \, a^{4} e^{5} + 3 \,{\left (7 \, c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x^{3} -{\left (7 \, a c^{3} d^{4} e - 6 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (7 \, a^{2} c^{2} d^{3} e^{2} - 6 \, a^{3} c d e^{4}\right )} x\right )} f^{2} g^{2}}{35 \, \sqrt{c d x + a e} c^{4} d^{4}} + \frac{8 \,{\left (35 \, c^{5} d^{5} e x^{5} - 144 \, a^{4} c d^{2} e^{4} + 128 \, a^{5} e^{6} + 5 \,{\left (9 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} -{\left (9 \, a c^{4} d^{5} e - 8 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \,{\left (9 \, a^{2} c^{3} d^{4} e^{2} - 8 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} - 8 \,{\left (9 \, a^{3} c^{2} d^{3} e^{3} - 8 \, a^{4} c d e^{5}\right )} x\right )} f g^{3}}{315 \, \sqrt{c d x + a e} c^{5} d^{5}} + \frac{2 \,{\left (315 \, c^{6} d^{6} e x^{6} + 1408 \, a^{5} c d^{2} e^{5} - 1280 \, a^{6} e^{7} + 35 \,{\left (11 \, c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{5} - 5 \,{\left (11 \, a c^{5} d^{6} e - 10 \, a^{2} c^{4} d^{4} e^{3}\right )} x^{4} + 8 \,{\left (11 \, a^{2} c^{4} d^{5} e^{2} - 10 \, a^{3} c^{3} d^{3} e^{4}\right )} x^{3} - 16 \,{\left (11 \, a^{3} c^{3} d^{4} e^{3} - 10 \, a^{4} c^{2} d^{2} e^{5}\right )} x^{2} + 64 \,{\left (11 \, a^{4} c^{2} d^{3} e^{4} - 10 \, a^{5} c d e^{6}\right )} x\right )} g^{4}}{3465 \, \sqrt{c d x + a e} c^{6} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)^4/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

2/3*(c^2*d^2*e*x^2 + 3*a*c*d^2*e - 2*a^2*e^3 + (3*c^2*d^3 - a*c*d*e^2)*x)*f^4/(s
qrt(c*d*x + a*e)*c^2*d^2) + 8/15*(3*c^3*d^3*e*x^3 - 10*a^2*c*d^2*e^2 + 8*a^3*e^4
+ (5*c^3*d^4 - a*c^2*d^2*e^2)*x^2 - (5*a*c^2*d^3*e - 4*a^2*c*d*e^3)*x)*f^3*g/(s
qrt(c*d*x + a*e)*c^3*d^3) + 4/35*(15*c^4*d^4*e*x^4 + 56*a^3*c*d^2*e^3 - 48*a^4*e
^5 + 3*(7*c^4*d^5 - a*c^3*d^3*e^2)*x^3 - (7*a*c^3*d^4*e - 6*a^2*c^2*d^2*e^3)*x^2
+ 4*(7*a^2*c^2*d^3*e^2 - 6*a^3*c*d*e^4)*x)*f^2*g^2/(sqrt(c*d*x + a*e)*c^4*d^4)
+ 8/315*(35*c^5*d^5*e*x^5 - 144*a^4*c*d^2*e^4 + 128*a^5*e^6 + 5*(9*c^5*d^6 - a*c
^4*d^4*e^2)*x^4 - (9*a*c^4*d^5*e - 8*a^2*c^3*d^3*e^3)*x^3 + 2*(9*a^2*c^3*d^4*e^2
- 8*a^3*c^2*d^2*e^4)*x^2 - 8*(9*a^3*c^2*d^3*e^3 - 8*a^4*c*d*e^5)*x)*f*g^3/(sqrt
(c*d*x + a*e)*c^5*d^5) + 2/3465*(315*c^6*d^6*e*x^6 + 1408*a^5*c*d^2*e^5 - 1280*a
^6*e^7 + 35*(11*c^6*d^7 - a*c^5*d^5*e^2)*x^5 - 5*(11*a*c^5*d^6*e - 10*a^2*c^4*d^
4*e^3)*x^4 + 8*(11*a^2*c^4*d^5*e^2 - 10*a^3*c^3*d^3*e^4)*x^3 - 16*(11*a^3*c^3*d^
4*e^3 - 10*a^4*c^2*d^2*e^5)*x^2 + 64*(11*a^4*c^2*d^3*e^4 - 10*a^5*c*d*e^6)*x)*g^
4/(sqrt(c*d*x + a*e)*c^6*d^6)

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Fricas [A]  time = 0.284789, size = 1504, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)^4/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

2/3465*(315*c^6*d^6*e^2*g^4*x^7 + 35*(44*c^6*d^6*e^2*f*g^3 + (20*c^6*d^7*e - a*c
^5*d^5*e^3)*g^4)*x^6 + 5*(594*c^6*d^6*e^2*f^2*g^2 + 44*(16*c^6*d^7*e - a*c^5*d^5
*e^3)*f*g^3 + (77*c^6*d^8 - 18*a*c^5*d^6*e^2 + 10*a^2*c^4*d^4*e^4)*g^4)*x^5 + 11
55*(3*a*c^5*d^7*e - 2*a^2*c^4*d^5*e^3)*f^4 - 1848*(5*a^2*c^4*d^6*e^2 - 4*a^3*c^3
*d^4*e^4)*f^3*g + 1584*(7*a^3*c^3*d^5*e^3 - 6*a^4*c^2*d^3*e^5)*f^2*g^2 - 704*(9*
a^4*c^2*d^4*e^4 - 8*a^5*c*d^2*e^6)*f*g^3 + 128*(11*a^5*c*d^3*e^5 - 10*a^6*d*e^7)
*g^4 + (2772*c^6*d^6*e^2*f^3*g + 594*(12*c^6*d^7*e - a*c^5*d^5*e^3)*f^2*g^2 + 44
*(45*c^6*d^8 - 14*a*c^5*d^6*e^2 + 8*a^2*c^4*d^4*e^4)*f*g^3 - (55*a*c^5*d^7*e - 1
38*a^2*c^4*d^5*e^3 + 80*a^3*c^3*d^3*e^5)*g^4)*x^4 + (1155*c^6*d^6*e^2*f^4 + 924*
(8*c^6*d^7*e - a*c^5*d^5*e^3)*f^3*g + 198*(21*c^6*d^8 - 10*a*c^5*d^6*e^2 + 6*a^2
*c^4*d^4*e^4)*f^2*g^2 - 44*(9*a*c^5*d^7*e - 26*a^2*c^4*d^5*e^3 + 16*a^3*c^3*d^3*
e^5)*f*g^3 + 8*(11*a^2*c^4*d^6*e^2 - 32*a^3*c^3*d^4*e^4 + 20*a^4*c^2*d^2*e^6)*g^
4)*x^3 + (1155*(4*c^6*d^7*e - a*c^5*d^5*e^3)*f^4 + 924*(5*c^6*d^8 - 6*a*c^5*d^6*
e^2 + 4*a^2*c^4*d^4*e^4)*f^3*g - 198*(7*a*c^5*d^7*e - 34*a^2*c^4*d^5*e^3 + 24*a^
3*c^3*d^3*e^5)*f^2*g^2 + 88*(9*a^2*c^4*d^6*e^2 - 44*a^3*c^3*d^4*e^4 + 32*a^4*c^2
*d^2*e^6)*f*g^3 - 16*(11*a^3*c^3*d^5*e^3 - 54*a^4*c^2*d^3*e^5 + 40*a^5*c*d*e^7)*
g^4)*x^2 + (1155*(3*c^6*d^8 + 2*a*c^5*d^6*e^2 - 2*a^2*c^4*d^4*e^4)*f^4 - 924*(5*
a*c^5*d^7*e + 6*a^2*c^4*d^5*e^3 - 8*a^3*c^3*d^3*e^5)*f^3*g + 792*(7*a^2*c^4*d^6*
e^2 + 8*a^3*c^3*d^4*e^4 - 12*a^4*c^2*d^2*e^6)*f^2*g^2 - 352*(9*a^3*c^3*d^5*e^3 +
10*a^4*c^2*d^3*e^5 - 16*a^5*c*d*e^7)*f*g^3 + 64*(11*a^4*c^2*d^4*e^4 + 12*a^5*c*
d^2*e^6 - 20*a^6*e^8)*g^4)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(
e*x + d)*c^6*d^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)*(g*x+f)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.414998, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)^4/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

Done