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

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*(10*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 = 0.495004, 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, Rules used = {897, 1153} \[ -\frac{2 e (f+g x)^{7/2} \left (e g (-a e g-3 b d g+4 b e f)-c \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )\right )}{7 g^6}+\frac{2 (f+g x)^{5/2} (e f-d g) \left (3 e g (-a e g-b d g+2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )}{5 g^6}-\frac{2 \sqrt{f+g x} (e f-d g)^3 \left (a g^2-b f g+c f^2\right )}{g^6}+\frac{2 (f+g x)^{3/2} (e f-d g)^2 (c f (5 e f-2 d g)-g (-3 a e g-b d g+4 b e f))}{3 g^6}-\frac{2 e^2 (f+g x)^{9/2} (-b e g-3 c d g+5 c e f)}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6} \]

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*(10*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)

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int \frac{(d+e x)^3 \left (a+b x+c x^2\right )}{\sqrt{f+g x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{-e f+d g}{g}+\frac{e x^2}{g}\right )^3 \left (\frac{c f^2-b f g+a g^2}{g^2}-\frac{(2 c f-b g) x^2}{g^2}+\frac{c x^4}{g^2}\right ) \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{(-e f+d g)^3 \left (c f^2-b f g+a g^2\right )}{g^5}+\frac{(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)) x^2}{g^5}+\frac{(e f-d g) \left (3 e g (2 b e f-b d g-a e g)-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) x^4}{g^5}+\frac{e \left (-e g (4 b e f-3 b d g-a e g)+c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) x^6}{g^5}+\frac{e^2 (-5 c e f+3 c d g+b e g) x^8}{g^5}+\frac{c e^3 x^{10}}{g^5}\right ) \, dx,x,\sqrt{f+g x}\right )}{g}\\ &=-\frac{2 (e f-d g)^3 \left (c f^2-b f g+a g^2\right ) \sqrt{f+g x}}{g^6}+\frac{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}+\frac{2 (e f-d g) \left (3 e g (2 b e f-b d g-a e g)-c \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^6}-\frac{2 e \left (e g (4 b e f-3 b d g-a e g)-c \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) (f+g x)^{7/2}}{7 g^6}-\frac{2 e^2 (5 c e f-3 c d g-b e g) (f+g x)^{9/2}}{9 g^6}+\frac{2 c e^3 (f+g x)^{11/2}}{11 g^6}\\ \end{align*}

Mathematica [A]  time = 0.433423, size = 249, normalized size = 0.87 \[ \frac{2 \sqrt{f+g x} \left (-495 e (f+g x)^3 \left (c \left (-3 d^2 g^2+12 d e f g-10 e^2 f^2\right )-e g (a e g+3 b d g-4 b e f)\right )+693 (f+g x)^2 (e f-d g) \left (-3 e g (a e g+b d g-2 b e f)-c \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )\right )-3465 (e f-d g)^3 \left (g (a g-b f)+c f^2\right )+1155 (f+g x) (e f-d g)^2 (g (3 a e g+b d g-4 b e f)+c f (5 e f-2 d g))-385 e^2 (f+g x)^4 (-b e g-3 c d g+5 c e f)+315 c e^3 (f+g x)^5\right )}{3465 g^6} \]

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]*(-3465*(e*f - d*g)^3*(c*f^2 + g*(-(b*f) + a*g)) + 1155*(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) + 693*(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)^2 - 495*e*(-(e*g*(-4*b*e*f + 3*b*d*g + a*e*g)) + c*(-10*e^2*f^2 + 12*d*e*f*g
- 3*d^2*g^2))*(f + g*x)^3 - 385*e^2*(5*c*e*f - 3*c*d*g - b*e*g)*(f + g*x)^4 + 315*c*e^3*(f + g*x)^5))/(3465*g
^6)

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Maple [B]  time = 0.053, size = 540, normalized size = 1.9 \begin{align*}{\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}} \end{align*}

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^4-350*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+1485*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-27
72*a*d*e^2*f*g^4*x+792*a*e^3*f^2*g^3*x+1155*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^5-693
0*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 = 0.997175, size = 579, normalized size = 2.02 \begin{align*} \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}} \end{align*}

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, 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 = 2.13294, size = 971, normalized size = 3.38 \begin{align*} \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}} \end{align*}

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, 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 - 176*(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 = 169.264, size = 1544, normalized size = 5.38 \begin{align*} \text{result too large to display} \end{align*}

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*sq
rt(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/s
qrt(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*s
qrt(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 [B]  time = 1.14787, size = 763, normalized size = 2.66 \begin{align*} \text{result too large to display} \end{align*}

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, algorithm="giac")

[Out]

2/3465*(3465*sqrt(g*x + f)*a*d^3 + 1155*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*b*d^3/g + 3465*((g*x + f)^(3/2)
- 3*sqrt(g*x + f)*f)*a*d^2*e/g + 231*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*c*d^3/g
^2 + 693*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*b*d^2*e/g^2 + 693*(3*(g*x + f)^(5/2
) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a*d*e^2/g^2 + 297*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f +
35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*c*d^2*e/g^3 + 297*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f +
35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*b*d*e^2/g^3 + 99*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35
*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*a*e^3/g^3 + 33*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*
(g*x + f)^(5/2)*f^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*c*d*e^2/g^4 + 11*(35*(g*x + f)^(9/2) -
180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*b*e^3/g^4 +
5*(63*(g*x + f)^(11/2) - 385*(g*x + f)^(9/2)*f + 990*(g*x + f)^(7/2)*f^2 - 1386*(g*x + f)^(5/2)*f^3 + 1155*(g
*x + f)^(3/2)*f^4 - 693*sqrt(g*x + f)*f^5)*c*e^3/g^5)/g