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

Optimal. Leaf size=287 \[ -\frac{2 e (f+g x)^{7/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}+\frac{2 (f+g x)^{5/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 \sqrt{f+g x} (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{3 g^6}-\frac{2 e^2 (f+g x)^{9/2} (-b e g-3 c d g+5 c e f)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6} \]

[Out]

(-2*(e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^6 + (2*(e*f - d*g)^2*
(c*f*(5*e*f - 2*d*g) - g*(4*b*e*f - b*d*g - 3*a*e*g))*(f + g*x)^(3/2))/(3*g^6) +
(2*(e*f - d*g)*(3*e*g*(2*b*e*f - b*d*g - a*e*g) - c*(10*e^2*f^2 - 8*d*e*f*g + d
^2*g^2))*(f + g*x)^(5/2))/(5*g^6) - (2*e*(e*g*(4*b*e*f - 3*b*d*g - a*e*g) - c*(1
0*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(7/2))/(7*g^6) - (2*e^2*(5*c*e*f
- 3*c*d*g - b*e*g)*(f + g*x)^(9/2))/(9*g^6) + (2*c*e^3*(f + g*x)^(11/2))/(11*g^6
)

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Rubi [A]  time = 1.02091, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{2 e (f+g x)^{7/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}+\frac{2 (f+g x)^{5/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 \sqrt{f+g x} (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{3 g^6}-\frac{2 e^2 (f+g x)^{9/2} (-b e g-3 c d g+5 c e f)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(e*f - d*g)^3*(c*f^2 - b*f*g + a*g^2)*Sqrt[f + g*x])/g^6 + (2*(e*f - d*g)^2*
(c*f*(5*e*f - 2*d*g) - g*(4*b*e*f - b*d*g - 3*a*e*g))*(f + g*x)^(3/2))/(3*g^6) +
(2*(e*f - d*g)*(3*e*g*(2*b*e*f - b*d*g - a*e*g) - c*(10*e^2*f^2 - 8*d*e*f*g + d
^2*g^2))*(f + g*x)^(5/2))/(5*g^6) - (2*e*(e*g*(4*b*e*f - 3*b*d*g - a*e*g) - c*(1
0*e^2*f^2 - 12*d*e*f*g + 3*d^2*g^2))*(f + g*x)^(7/2))/(7*g^6) - (2*e^2*(5*c*e*f
- 3*c*d*g - b*e*g)*(f + g*x)^(9/2))/(9*g^6) + (2*c*e^3*(f + g*x)^(11/2))/(11*g^6
)

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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*(c*x**2+b*x+a)/(g*x+f)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.04055, size = 412, normalized size = 1.44 \[ \frac{2 \sqrt{f+g x} \left (11 g \left (9 a g \left (35 d^3 g^3+35 d^2 e g^2 (g x-2 f)+7 d e^2 g \left (8 f^2-4 f g x+3 g^2 x^2\right )+e^3 \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )\right )+b \left (105 d^3 g^3 (g x-2 f)+63 d^2 e g^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )+27 d e^2 g \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )+e^3 \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )\right )\right )+c \left (231 d^3 g^3 \left (8 f^2-4 f g x+3 g^2 x^2\right )+297 d^2 e g^2 \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )+33 d e^2 g \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )-5 e^3 \left (256 f^5-128 f^4 g x+96 f^3 g^2 x^2-80 f^2 g^3 x^3+70 f g^4 x^4-63 g^5 x^5\right )\right )\right )}{3465 g^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[f + g*x]*(c*(231*d^3*g^3*(8*f^2 - 4*f*g*x + 3*g^2*x^2) + 297*d^2*e*g^2*(
-16*f^3 + 8*f^2*g*x - 6*f*g^2*x^2 + 5*g^3*x^3) + 33*d*e^2*g*(128*f^4 - 64*f^3*g*
x + 48*f^2*g^2*x^2 - 40*f*g^3*x^3 + 35*g^4*x^4) - 5*e^3*(256*f^5 - 128*f^4*g*x +
96*f^3*g^2*x^2 - 80*f^2*g^3*x^3 + 70*f*g^4*x^4 - 63*g^5*x^5)) + 11*g*(9*a*g*(35
*d^3*g^3 + 35*d^2*e*g^2*(-2*f + g*x) + 7*d*e^2*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) +
e^3*(-16*f^3 + 8*f^2*g*x - 6*f*g^2*x^2 + 5*g^3*x^3)) + b*(105*d^3*g^3*(-2*f + g
*x) + 63*d^2*e*g^2*(8*f^2 - 4*f*g*x + 3*g^2*x^2) + 27*d*e^2*g*(-16*f^3 + 8*f^2*g
*x - 6*f*g^2*x^2 + 5*g^3*x^3) + e^3*(128*f^4 - 64*f^3*g*x + 48*f^2*g^2*x^2 - 40*
f*g^3*x^3 + 35*g^4*x^4)))))/(3465*g^6)

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Maple [B]  time = 0.011, size = 540, normalized size = 1.9 \[{\frac{630\,{e}^{3}c{x}^{5}{g}^{5}+770\,b{e}^{3}{g}^{5}{x}^{4}+2310\,cd{e}^{2}{g}^{5}{x}^{4}-700\,c{e}^{3}f{g}^{4}{x}^{4}+990\,a{e}^{3}{g}^{5}{x}^{3}+2970\,bd{e}^{2}{g}^{5}{x}^{3}-880\,b{e}^{3}f{g}^{4}{x}^{3}+2970\,c{d}^{2}e{g}^{5}{x}^{3}-2640\,cd{e}^{2}f{g}^{4}{x}^{3}+800\,c{e}^{3}{f}^{2}{g}^{3}{x}^{3}+4158\,ad{e}^{2}{g}^{5}{x}^{2}-1188\,a{e}^{3}f{g}^{4}{x}^{2}+4158\,b{d}^{2}e{g}^{5}{x}^{2}-3564\,bd{e}^{2}f{g}^{4}{x}^{2}+1056\,b{e}^{3}{f}^{2}{g}^{3}{x}^{2}+1386\,c{d}^{3}{g}^{5}{x}^{2}-3564\,c{d}^{2}ef{g}^{4}{x}^{2}+3168\,cd{e}^{2}{f}^{2}{g}^{3}{x}^{2}-960\,c{e}^{3}{f}^{3}{g}^{2}{x}^{2}+6930\,a{d}^{2}e{g}^{5}x-5544\,ad{e}^{2}f{g}^{4}x+1584\,a{e}^{3}{f}^{2}{g}^{3}x+2310\,b{d}^{3}{g}^{5}x-5544\,b{d}^{2}ef{g}^{4}x+4752\,bd{e}^{2}{f}^{2}{g}^{3}x-1408\,b{e}^{3}{f}^{3}{g}^{2}x-1848\,c{d}^{3}f{g}^{4}x+4752\,c{d}^{2}e{f}^{2}{g}^{3}x-4224\,cd{e}^{2}{f}^{3}{g}^{2}x+1280\,c{e}^{3}{f}^{4}gx+6930\,{d}^{3}a{g}^{5}-13860\,a{d}^{2}ef{g}^{4}+11088\,ad{e}^{2}{f}^{2}{g}^{3}-3168\,a{e}^{3}{f}^{3}{g}^{2}-4620\,b{d}^{3}f{g}^{4}+11088\,b{d}^{2}e{f}^{2}{g}^{3}-9504\,bd{e}^{2}{f}^{3}{g}^{2}+2816\,b{e}^{3}{f}^{4}g+3696\,c{d}^{3}{f}^{2}{g}^{3}-9504\,c{d}^{2}e{f}^{3}{g}^{2}+8448\,cd{e}^{2}{f}^{4}g-2560\,c{e}^{3}{f}^{5}}{3465\,{g}^{6}}\sqrt{gx+f}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(g*x+f)^(1/2)*(315*c*e^3*g^5*x^5+385*b*e^3*g^5*x^4+1155*c*d*e^2*g^5*x^4-3
50*c*e^3*f*g^4*x^4+495*a*e^3*g^5*x^3+1485*b*d*e^2*g^5*x^3-440*b*e^3*f*g^4*x^3+14
85*c*d^2*e*g^5*x^3-1320*c*d*e^2*f*g^4*x^3+400*c*e^3*f^2*g^3*x^3+2079*a*d*e^2*g^5
*x^2-594*a*e^3*f*g^4*x^2+2079*b*d^2*e*g^5*x^2-1782*b*d*e^2*f*g^4*x^2+528*b*e^3*f
^2*g^3*x^2+693*c*d^3*g^5*x^2-1782*c*d^2*e*f*g^4*x^2+1584*c*d*e^2*f^2*g^3*x^2-480
*c*e^3*f^3*g^2*x^2+3465*a*d^2*e*g^5*x-2772*a*d*e^2*f*g^4*x+792*a*e^3*f^2*g^3*x+1
155*b*d^3*g^5*x-2772*b*d^2*e*f*g^4*x+2376*b*d*e^2*f^2*g^3*x-704*b*e^3*f^3*g^2*x-
924*c*d^3*f*g^4*x+2376*c*d^2*e*f^2*g^3*x-2112*c*d*e^2*f^3*g^2*x+640*c*e^3*f^4*g*
x+3465*a*d^3*g^5-6930*a*d^2*e*f*g^4+5544*a*d*e^2*f^2*g^3-1584*a*e^3*f^3*g^2-2310
*b*d^3*f*g^4+5544*b*d^2*e*f^2*g^3-4752*b*d*e^2*f^3*g^2+1408*b*e^3*f^4*g+1848*c*d
^3*f^2*g^3-4752*c*d^2*e*f^3*g^2+4224*c*d*e^2*f^4*g-1280*c*e^3*f^5)/g^6

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Maxima [A]  time = 0.698721, size = 579, normalized size = 2.02 \[ \frac{2 \,{\left (315 \,{\left (g x + f\right )}^{\frac{11}{2}} c e^{3} - 385 \,{\left (5 \, c e^{3} f -{\left (3 \, c d e^{2} + b e^{3}\right )} g\right )}{\left (g x + f\right )}^{\frac{9}{2}} + 495 \,{\left (10 \, c e^{3} f^{2} - 4 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f g +{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{7}{2}} - 693 \,{\left (10 \, c e^{3} f^{3} - 6 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g + 3 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{2} -{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{3}\right )}{\left (g x + f\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, c e^{3} f^{4} - 4 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g + 3 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{2} - 2 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{3} +{\left (b d^{3} + 3 \, a d^{2} e\right )} g^{4}\right )}{\left (g x + f\right )}^{\frac{3}{2}} - 3465 \,{\left (c e^{3} f^{5} - a d^{3} g^{5} -{\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g +{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} -{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} +{\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4}\right )} \sqrt{g x + f}\right )}}{3465 \, g^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*(g*x + f)^(11/2)*c*e^3 - 385*(5*c*e^3*f - (3*c*d*e^2 + b*e^3)*g)*(g*
x + f)^(9/2) + 495*(10*c*e^3*f^2 - 4*(3*c*d*e^2 + b*e^3)*f*g + (3*c*d^2*e + 3*b*
d*e^2 + a*e^3)*g^2)*(g*x + f)^(7/2) - 693*(10*c*e^3*f^3 - 6*(3*c*d*e^2 + b*e^3)*
f^2*g + 3*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f*g^2 - (c*d^3 + 3*b*d^2*e + 3*a*d*e^2
)*g^3)*(g*x + f)^(5/2) + 1155*(5*c*e^3*f^4 - 4*(3*c*d*e^2 + b*e^3)*f^3*g + 3*(3*
c*d^2*e + 3*b*d*e^2 + a*e^3)*f^2*g^2 - 2*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*f*g^3 +
(b*d^3 + 3*a*d^2*e)*g^4)*(g*x + f)^(3/2) - 3465*(c*e^3*f^5 - a*d^3*g^5 - (3*c*d
*e^2 + b*e^3)*f^4*g + (3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^3*g^2 - (c*d^3 + 3*b*d^2
*e + 3*a*d*e^2)*f^2*g^3 + (b*d^3 + 3*a*d^2*e)*f*g^4)*sqrt(g*x + f))/g^6

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Fricas [A]  time = 0.292644, size = 579, normalized size = 2.02 \[ \frac{2 \,{\left (315 \, c e^{3} g^{5} x^{5} - 1280 \, c e^{3} f^{5} + 3465 \, a d^{3} g^{5} + 1408 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{4} g - 1584 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{3} g^{2} + 1848 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f^{2} g^{3} - 2310 \,{\left (b d^{3} + 3 \, a d^{2} e\right )} f g^{4} - 35 \,{\left (10 \, c e^{3} f g^{4} - 11 \,{\left (3 \, c d e^{2} + b e^{3}\right )} g^{5}\right )} x^{4} + 5 \,{\left (80 \, c e^{3} f^{2} g^{3} - 88 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f g^{4} + 99 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} g^{5}\right )} x^{3} - 3 \,{\left (160 \, c e^{3} f^{3} g^{2} - 176 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{2} g^{3} + 198 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f g^{4} - 231 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} g^{5}\right )} x^{2} +{\left (640 \, c e^{3} f^{4} g - 704 \,{\left (3 \, c d e^{2} + b e^{3}\right )} f^{3} g^{2} + 792 \,{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} f^{2} g^{3} - 924 \,{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} f g^{4} + 1155 \,{\left (b d^{3} + 3 \, a d^{2} e\right )} g^{5}\right )} x\right )} \sqrt{g x + f}}{3465 \, g^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*(315*c*e^3*g^5*x^5 - 1280*c*e^3*f^5 + 3465*a*d^3*g^5 + 1408*(3*c*d*e^2 +
b*e^3)*f^4*g - 1584*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^3*g^2 + 1848*(c*d^3 + 3*b*
d^2*e + 3*a*d*e^2)*f^2*g^3 - 2310*(b*d^3 + 3*a*d^2*e)*f*g^4 - 35*(10*c*e^3*f*g^4
- 11*(3*c*d*e^2 + b*e^3)*g^5)*x^4 + 5*(80*c*e^3*f^2*g^3 - 88*(3*c*d*e^2 + b*e^3
)*f*g^4 + 99*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*g^5)*x^3 - 3*(160*c*e^3*f^3*g^2 - 1
76*(3*c*d*e^2 + b*e^3)*f^2*g^3 + 198*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f*g^4 - 231
*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*g^5)*x^2 + (640*c*e^3*f^4*g - 704*(3*c*d*e^2 +
b*e^3)*f^3*g^2 + 792*(3*c*d^2*e + 3*b*d*e^2 + a*e^3)*f^2*g^3 - 924*(c*d^3 + 3*b*
d^2*e + 3*a*d*e^2)*f*g^4 + 1155*(b*d^3 + 3*a*d^2*e)*g^5)*x)*sqrt(g*x + f)/g^6

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Sympy [A]  time = 110.025, size = 1544, normalized size = 5.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-(2*a*d**3*f/sqrt(f + g*x) + 2*a*d**3*(-f/sqrt(f + g*x) - sqrt(f + g*
x)) + 6*a*d**2*e*f*(-f/sqrt(f + g*x) - sqrt(f + g*x))/g + 6*a*d**2*e*(f**2/sqrt(
f + g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g + 6*a*d*e**2*f*(f**2/sqrt(f
+ g*x) + 2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 6*a*d*e**2*(-f**3/sqrt(
f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2
+ 2*a*e**3*f*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) -
(f + g*x)**(5/2)/5)/g**3 + 2*a*e**3*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) -
2*f**2*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 + 2
*b*d**3*f*(-f/sqrt(f + g*x) - sqrt(f + g*x))/g + 2*b*d**3*(f**2/sqrt(f + g*x) +
2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g + 6*b*d**2*e*f*(f**2/sqrt(f + g*x) + 2
*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 6*b*d**2*e*(-f**3/sqrt(f + g*x) -
3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 + 6*b*d*e**
2*f*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)
**(5/2)/5)/g**3 + 6*b*d*e**2*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2
*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 + 2*b*e**3
*f*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*(f
+ g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**4 + 2*b*e**3*(-f**5/sqrt(f + g*x) - 5*
f**4*sqrt(f + g*x) + 10*f**3*(f + g*x)**(3/2)/3 - 2*f**2*(f + g*x)**(5/2) + 5*f*
(f + g*x)**(7/2)/7 - (f + g*x)**(9/2)/9)/g**4 + 2*c*d**3*f*(f**2/sqrt(f + g*x) +
2*f*sqrt(f + g*x) - (f + g*x)**(3/2)/3)/g**2 + 2*c*d**3*(-f**3/sqrt(f + g*x) -
3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)**(5/2)/5)/g**2 + 6*c*d**2*
e*f*(-f**3/sqrt(f + g*x) - 3*f**2*sqrt(f + g*x) + f*(f + g*x)**(3/2) - (f + g*x)
**(5/2)/5)/g**3 + 6*c*d**2*e*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2
*(f + g*x)**(3/2) + 4*f*(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**3 + 6*c*d*e*
*2*f*(f**4/sqrt(f + g*x) + 4*f**3*sqrt(f + g*x) - 2*f**2*(f + g*x)**(3/2) + 4*f*
(f + g*x)**(5/2)/5 - (f + g*x)**(7/2)/7)/g**4 + 6*c*d*e**2*(-f**5/sqrt(f + g*x)
- 5*f**4*sqrt(f + g*x) + 10*f**3*(f + g*x)**(3/2)/3 - 2*f**2*(f + g*x)**(5/2) +
5*f*(f + g*x)**(7/2)/7 - (f + g*x)**(9/2)/9)/g**4 + 2*c*e**3*f*(-f**5/sqrt(f + g
*x) - 5*f**4*sqrt(f + g*x) + 10*f**3*(f + g*x)**(3/2)/3 - 2*f**2*(f + g*x)**(5/2
) + 5*f*(f + g*x)**(7/2)/7 - (f + g*x)**(9/2)/9)/g**5 + 2*c*e**3*(f**6/sqrt(f +
g*x) + 6*f**5*sqrt(f + g*x) - 5*f**4*(f + g*x)**(3/2) + 4*f**3*(f + g*x)**(5/2)
- 15*f**2*(f + g*x)**(7/2)/7 + 2*f*(f + g*x)**(9/2)/3 - (f + g*x)**(11/2)/11)/g*
*5)/g, Ne(g, 0)), ((a*d**3*x + c*e**3*x**6/6 + x**5*(b*e**3 + 3*c*d*e**2)/5 + x*
*4*(a*e**3 + 3*b*d*e**2 + 3*c*d**2*e)/4 + x**3*(3*a*d*e**2 + 3*b*d**2*e + c*d**3
)/3 + x**2*(3*a*d**2*e + b*d**3)/2)/sqrt(f), True))

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GIAC/XCAS [A]  time = 0.267718, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done