\(\int \frac {(d+e x)^2 (a+b x+c x^2)}{(f+g x)^{3/2}} \, dx\) [827]

Optimal result

Integrand size = 27, antiderivative size = 210 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=-\frac {2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )}{g^5 \sqrt {f+g x}}-\frac {2 (e f-d g) (2 c f (2 e f-d g)-g (3 b e f-b d g-2 a e g)) \sqrt {f+g x}}{g^5}-\frac {2 \left (e g (3 b e f-2 b d g-a e g)-c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^5}-\frac {2 e (4 c e f-2 c d g-b e g) (f+g x)^{5/2}}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5} \]

[Out]

-2/3*(e*g*(-a*e*g-2*b*d*g+3*b*e*f)-c*(d^2*g^2-6*d*e*f*g+6*e^2*f^2))*(g*x+f)^(3/2)/g^5-2/5*e*(-b*e*g-2*c*d*g+4*
c*e*f)*(g*x+f)^(5/2)/g^5+2/7*c*e^2*(g*x+f)^(7/2)/g^5-2*(-d*g+e*f)^2*(a*g^2-b*f*g+c*f^2)/g^5/(g*x+f)^(1/2)-2*(-
d*g+e*f)*(2*c*f*(-d*g+2*e*f)-g*(-2*a*e*g-b*d*g+3*b*e*f))*(g*x+f)^(1/2)/g^5

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {911, 1275} \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=-\frac {2 (f+g x)^{3/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac {2 (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5 \sqrt {f+g x}}-\frac {2 \sqrt {f+g x} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{g^5}-\frac {2 e (f+g x)^{5/2} (-b e g-2 c d g+4 c e f)}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5} \]

[In]

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

[Out]

(-2*(e*f - d*g)^2*(c*f^2 - b*f*g + a*g^2))/(g^5*Sqrt[f + g*x]) - (2*(e*f - d*g)*(2*c*f*(2*e*f - d*g) - g*(3*b*
e*f - b*d*g - 2*a*e*g))*Sqrt[f + g*x])/g^5 - (2*(e*g*(3*b*e*f - 2*b*d*g - a*e*g) - c*(6*e^2*f^2 - 6*d*e*f*g +
d^2*g^2))*(f + g*x)^(3/2))/(3*g^5) - (2*e*(4*c*e*f - 2*c*d*g - b*e*g)*(f + g*x)^(5/2))/(5*g^5) + (2*c*e^2*(f +
g*x)^(7/2))/(7*g^5)

Rule 911

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 1275

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2 \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 )}{x^2} \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {(e f-d g) (-2 c f (2 e f-d g)+g (3 b e f-b d g-2 a e g))}{g^4}+\frac {(-e f+d g)^2 \left (c f^2-b f g+a g^2\right )}{g^4 x^2}+\frac {\left (-e g (3 b e f-2 b d g-a e g)+c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) x^2}{g^4}+\frac {e (-4 c e f+2 c d g+b e g) x^4}{g^4}+\frac {c e^2 x^6}{g^4}\right ) \, dx,x,\sqrt {f+g x}\right )}{g} \\ & = -\frac {2 (e f-d g)^2 \left (c f^2-b f g+a g^2\right )}{g^5 \sqrt {f+g x}}-\frac {2 (e f-d g) (2 c f (2 e f-d g)-g (3 b e f-b d g-2 a e g)) \sqrt {f+g x}}{g^5}-\frac {2 \left (e g (3 b e f-2 b d g-a e g)-c \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^5}-\frac {2 e (4 c e f-2 c d g-b e g) (f+g x)^{5/2}}{5 g^5}+\frac {2 c e^2 (f+g x)^{7/2}}{7 g^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.20 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \left (c \left (35 d^2 g^2 \left (-8 f^2-4 f g x+g^2 x^2\right )+42 d e g \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )-3 e^2 \left (128 f^4+64 f^3 g x-16 f^2 g^2 x^2+8 f g^3 x^3-5 g^4 x^4\right )\right )+7 g \left (5 a g \left (-3 d^2 g^2+6 d e g (2 f+g x)+e^2 \left (-8 f^2-4 f g x+g^2 x^2\right )\right )+b \left (15 d^2 g^2 (2 f+g x)+10 d e g \left (-8 f^2-4 f g x+g^2 x^2\right )+3 e^2 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )\right )\right )\right )}{105 g^5 \sqrt {f+g x}} \]

[In]

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

[Out]

(2*(c*(35*d^2*g^2*(-8*f^2 - 4*f*g*x + g^2*x^2) + 42*d*e*g*(16*f^3 + 8*f^2*g*x - 2*f*g^2*x^2 + g^3*x^3) - 3*e^2
*(128*f^4 + 64*f^3*g*x - 16*f^2*g^2*x^2 + 8*f*g^3*x^3 - 5*g^4*x^4)) + 7*g*(5*a*g*(-3*d^2*g^2 + 6*d*e*g*(2*f +
g*x) + e^2*(-8*f^2 - 4*f*g*x + g^2*x^2)) + b*(15*d^2*g^2*(2*f + g*x) + 10*d*e*g*(-8*f^2 - 4*f*g*x + g^2*x^2) +
3*e^2*(16*f^3 + 8*f^2*g*x - 2*f*g^2*x^2 + g^3*x^3)))))/(105*g^5*Sqrt[f + g*x])

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(\frac {\left (\left (30 c \,x^{4}+42 b \,x^{3}+70 a \,x^{2}\right ) e^{2}+420 d \left (\frac {1}{5} c \,x^{2}+\frac {1}{3} b x +a \right ) x e -210 d^{2} \left (a -\frac {1}{3} c \,x^{2}-b x \right )\right ) g^{4}+840 f \left (-\frac {\left (\frac {6}{35} c \,x^{2}+\frac {3}{10} b x +a \right ) x \,e^{2}}{3}+d \left (-\frac {1}{5} c \,x^{2}-\frac {2}{3} b x +a \right ) e +\frac {d^{2} \left (-\frac {2 c x}{3}+b \right )}{2}\right ) g^{3}-560 \left (\left (-\frac {6}{35} c \,x^{2}-\frac {3}{5} b x +a \right ) e^{2}+2 d \left (-\frac {3 c x}{5}+b \right ) e +c \,d^{2}\right ) f^{2} g^{2}+672 e \left (\left (-\frac {4 c x}{7}+b \right ) e +2 c d \right ) f^{3} g -768 c \,e^{2} f^{4}}{105 \sqrt {g x +f}\, g^{5}}\)	\(196\)
risch	\(\frac {2 \left (15 c \,e^{2} x^{3} g^{3}+21 b \,e^{2} g^{3} x^{2}+42 c d e \,g^{3} x^{2}-39 c \,e^{2} f \,g^{2} x^{2}+35 a \,e^{2} g^{3} x +70 b d e \,g^{3} x -63 b \,e^{2} f \,g^{2} x +35 c \,d^{2} g^{3} x -126 c d e f \,g^{2} x +87 c \,e^{2} f^{2} g x +210 a d e \,g^{3}-175 a \,e^{2} f \,g^{2}+105 b \,d^{2} g^{3}-350 b d e f \,g^{2}+231 b \,e^{2} f^{2} g -175 c \,d^{2} f \,g^{2}+462 c d e \,f^{2} g -279 c \,e^{2} f^{3}\right ) \sqrt {g x +f}}{105 g^{5}}-\frac {2 \left (a \,d^{2} g^{4}-2 a d e f \,g^{3}+a \,e^{2} f^{2} g^{2}-b \,d^{2} f \,g^{3}+2 b d e \,f^{2} g^{2}-b \,e^{2} f^{3} g +c \,d^{2} f^{2} g^{2}-2 c d e \,f^{3} g +c \,e^{2} f^{4}\right )}{g^{5} \sqrt {g x +f}}\)	\(298\)
gosper	\(-\frac {2 \left (-15 c \,e^{2} x^{4} g^{4}-21 b \,e^{2} g^{4} x^{3}-42 c d e \,g^{4} x^{3}+24 c \,e^{2} f \,g^{3} x^{3}-35 a \,e^{2} g^{4} x^{2}-70 b d e \,g^{4} x^{2}+42 b \,e^{2} f \,g^{3} x^{2}-35 c \,d^{2} g^{4} x^{2}+84 c d e f \,g^{3} x^{2}-48 c \,e^{2} f^{2} g^{2} x^{2}-210 a d e \,g^{4} x +140 a \,e^{2} f \,g^{3} x -105 b \,d^{2} g^{4} x +280 b d e f \,g^{3} x -168 b \,e^{2} f^{2} g^{2} x +140 c \,d^{2} f \,g^{3} x -336 c d e \,f^{2} g^{2} x +192 c \,e^{2} f^{3} g x +105 a \,d^{2} g^{4}-420 a d e f \,g^{3}+280 a \,e^{2} f^{2} g^{2}-210 b \,d^{2} f \,g^{3}+560 b d e \,f^{2} g^{2}-336 b \,e^{2} f^{3} g +280 c \,d^{2} f^{2} g^{2}-672 c d e \,f^{3} g +384 c \,e^{2} f^{4}\right )}{105 \sqrt {g x +f}\, g^{5}}\)	\(315\)
trager	\(-\frac {2 \left (-15 c \,e^{2} x^{4} g^{4}-21 b \,e^{2} g^{4} x^{3}-42 c d e \,g^{4} x^{3}+24 c \,e^{2} f \,g^{3} x^{3}-35 a \,e^{2} g^{4} x^{2}-70 b d e \,g^{4} x^{2}+42 b \,e^{2} f \,g^{3} x^{2}-35 c \,d^{2} g^{4} x^{2}+84 c d e f \,g^{3} x^{2}-48 c \,e^{2} f^{2} g^{2} x^{2}-210 a d e \,g^{4} x +140 a \,e^{2} f \,g^{3} x -105 b \,d^{2} g^{4} x +280 b d e f \,g^{3} x -168 b \,e^{2} f^{2} g^{2} x +140 c \,d^{2} f \,g^{3} x -336 c d e \,f^{2} g^{2} x +192 c \,e^{2} f^{3} g x +105 a \,d^{2} g^{4}-420 a d e f \,g^{3}+280 a \,e^{2} f^{2} g^{2}-210 b \,d^{2} f \,g^{3}+560 b d e \,f^{2} g^{2}-336 b \,e^{2} f^{3} g +280 c \,d^{2} f^{2} g^{2}-672 c d e \,f^{3} g +384 c \,e^{2} f^{4}\right )}{105 \sqrt {g x +f}\, g^{5}}\)	\(315\)
derivativedivides	\(\frac {\frac {2 c \,e^{2} \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 b \,e^{2} g \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {4 c d e g \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {8 c \,e^{2} f \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 a \,e^{2} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}+\frac {4 b d e \,g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}-2 b \,e^{2} f g \left (g x +f \right )^{\frac {3}{2}}+\frac {2 c \,d^{2} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}-4 c d e f g \left (g x +f \right )^{\frac {3}{2}}+4 c \,e^{2} f^{2} \left (g x +f \right )^{\frac {3}{2}}+4 a d e \,g^{3} \sqrt {g x +f}-4 a \,e^{2} f \,g^{2} \sqrt {g x +f}+2 b \,d^{2} g^{3} \sqrt {g x +f}-8 b d e f \,g^{2} \sqrt {g x +f}+6 b \,e^{2} f^{2} g \sqrt {g x +f}-4 c \,d^{2} f \,g^{2} \sqrt {g x +f}+12 c d e \,f^{2} g \sqrt {g x +f}-8 c \,e^{2} f^{3} \sqrt {g x +f}-\frac {2 \left (a \,d^{2} g^{4}-2 a d e f \,g^{3}+a \,e^{2} f^{2} g^{2}-b \,d^{2} f \,g^{3}+2 b d e \,f^{2} g^{2}-b \,e^{2} f^{3} g +c \,d^{2} f^{2} g^{2}-2 c d e \,f^{3} g +c \,e^{2} f^{4}\right )}{\sqrt {g x +f}}}{g^{5}}\)	\(379\)
default	\(\frac {\frac {2 c \,e^{2} \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 b \,e^{2} g \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {4 c d e g \left (g x +f \right )^{\frac {5}{2}}}{5}-\frac {8 c \,e^{2} f \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 a \,e^{2} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}+\frac {4 b d e \,g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}-2 b \,e^{2} f g \left (g x +f \right )^{\frac {3}{2}}+\frac {2 c \,d^{2} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}-4 c d e f g \left (g x +f \right )^{\frac {3}{2}}+4 c \,e^{2} f^{2} \left (g x +f \right )^{\frac {3}{2}}+4 a d e \,g^{3} \sqrt {g x +f}-4 a \,e^{2} f \,g^{2} \sqrt {g x +f}+2 b \,d^{2} g^{3} \sqrt {g x +f}-8 b d e f \,g^{2} \sqrt {g x +f}+6 b \,e^{2} f^{2} g \sqrt {g x +f}-4 c \,d^{2} f \,g^{2} \sqrt {g x +f}+12 c d e \,f^{2} g \sqrt {g x +f}-8 c \,e^{2} f^{3} \sqrt {g x +f}-\frac {2 \left (a \,d^{2} g^{4}-2 a d e f \,g^{3}+a \,e^{2} f^{2} g^{2}-b \,d^{2} f \,g^{3}+2 b d e \,f^{2} g^{2}-b \,e^{2} f^{3} g +c \,d^{2} f^{2} g^{2}-2 c d e \,f^{3} g +c \,e^{2} f^{4}\right )}{\sqrt {g x +f}}}{g^{5}}\)	\(379\)

[In]

int((e*x+d)^2*(c*x^2+b*x+a)/(g*x+f)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/105*(((30*c*x^4+42*b*x^3+70*a*x^2)*e^2+420*d*(1/5*c*x^2+1/3*b*x+a)*x*e-210*d^2*(a-1/3*c*x^2-b*x))*g^4+840*f*
(-1/3*(6/35*c*x^2+3/10*b*x+a)*x*e^2+d*(-1/5*c*x^2-2/3*b*x+a)*e+1/2*d^2*(-2/3*c*x+b))*g^3-560*((-6/35*c*x^2-3/5
*b*x+a)*e^2+2*d*(-3/5*c*x+b)*e+c*d^2)*f^2*g^2+672*e*((-4/7*c*x+b)*e+2*c*d)*f^3*g-768*c*e^2*f^4)/(g*x+f)^(1/2)/
g^5

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (15 \, c e^{2} g^{4} x^{4} - 384 \, c e^{2} f^{4} - 105 \, a d^{2} g^{4} + 336 \, {\left (2 \, c d e + b e^{2}\right )} f^{3} g - 280 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} + 210 \, {\left (b d^{2} + 2 \, a d e\right )} f g^{3} - 3 \, {\left (8 \, c e^{2} f g^{3} - 7 \, {\left (2 \, c d e + b e^{2}\right )} g^{4}\right )} x^{3} + {\left (48 \, c e^{2} f^{2} g^{2} - 42 \, {\left (2 \, c d e + b e^{2}\right )} f g^{3} + 35 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{4}\right )} x^{2} - {\left (192 \, c e^{2} f^{3} g - 168 \, {\left (2 \, c d e + b e^{2}\right )} f^{2} g^{2} + 140 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{3} - 105 \, {\left (b d^{2} + 2 \, a d e\right )} g^{4}\right )} x\right )} \sqrt {g x + f}}{105 \, {\left (g^{6} x + f g^{5}\right )}} \]

[In]

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

[Out]

2/105*(15*c*e^2*g^4*x^4 - 384*c*e^2*f^4 - 105*a*d^2*g^4 + 336*(2*c*d*e + b*e^2)*f^3*g - 280*(c*d^2 + 2*b*d*e +
a*e^2)*f^2*g^2 + 210*(b*d^2 + 2*a*d*e)*f*g^3 - 3*(8*c*e^2*f*g^3 - 7*(2*c*d*e + b*e^2)*g^4)*x^3 + (48*c*e^2*f^
2*g^2 - 42*(2*c*d*e + b*e^2)*f*g^3 + 35*(c*d^2 + 2*b*d*e + a*e^2)*g^4)*x^2 - (192*c*e^2*f^3*g - 168*(2*c*d*e +
b*e^2)*f^2*g^2 + 140*(c*d^2 + 2*b*d*e + a*e^2)*f*g^3 - 105*(b*d^2 + 2*a*d*e)*g^4)*x)*sqrt(g*x + f)/(g^6*x + f
*g^5)

Sympy [A] (verification not implemented)

Time = 11.82 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.67 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c e^{2} \left (f + g x\right )^{\frac {7}{2}}}{7 g^{4}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (b e^{2} g + 2 c d e g - 4 c e^{2} f\right )}{5 g^{4}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \left (a e^{2} g^{2} + 2 b d e g^{2} - 3 b e^{2} f g + c d^{2} g^{2} - 6 c d e f g + 6 c e^{2} f^{2}\right )}{3 g^{4}} + \frac {\sqrt {f + g x} \left (2 a d e g^{3} - 2 a e^{2} f g^{2} + b d^{2} g^{3} - 4 b d e f g^{2} + 3 b e^{2} f^{2} g - 2 c d^{2} f g^{2} + 6 c d e f^{2} g - 4 c e^{2} f^{3}\right )}{g^{4}} - \frac {\left (d g - e f\right )^{2} \left (a g^{2} - b f g + c f^{2}\right )}{g^{4} \sqrt {f + g x}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a d^{2} x + \frac {c e^{2} x^{5}}{5} + \frac {x^{4} \left (b e^{2} + 2 c d e\right )}{4} + \frac {x^{3} \left (a e^{2} + 2 b d e + c d^{2}\right )}{3} + \frac {x^{2} \cdot \left (2 a d e + b d^{2}\right )}{2}}{f^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (g x + f\right )}^{\frac {7}{2}} c e^{2} - 21 \, {\left (4 \, c e^{2} f - {\left (2 \, c d e + b e^{2}\right )} g\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, c e^{2} f^{2} - 3 \, {\left (2 \, c d e + b e^{2}\right )} f g + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {3}{2}} - 105 \, {\left (4 \, c e^{2} f^{3} - 3 \, {\left (2 \, c d e + b e^{2}\right )} f^{2} g + 2 \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{2} - {\left (b d^{2} + 2 \, a d e\right )} g^{3}\right )} \sqrt {g x + f}}{g^{4}} - \frac {105 \, {\left (c e^{2} f^{4} + a d^{2} g^{4} - {\left (2 \, c d e + b e^{2}\right )} f^{3} g + {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} - {\left (b d^{2} + 2 \, a d e\right )} f g^{3}\right )}}{\sqrt {g x + f} g^{4}}\right )}}{105 \, g} \]

[In]

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

[Out]

2/105*((15*(g*x + f)^(7/2)*c*e^2 - 21*(4*c*e^2*f - (2*c*d*e + b*e^2)*g)*(g*x + f)^(5/2) + 35*(6*c*e^2*f^2 - 3*
(2*c*d*e + b*e^2)*f*g + (c*d^2 + 2*b*d*e + a*e^2)*g^2)*(g*x + f)^(3/2) - 105*(4*c*e^2*f^3 - 3*(2*c*d*e + b*e^2
)*f^2*g + 2*(c*d^2 + 2*b*d*e + a*e^2)*f*g^2 - (b*d^2 + 2*a*d*e)*g^3)*sqrt(g*x + f))/g^4 - 105*(c*e^2*f^4 + a*d
^2*g^4 - (2*c*d*e + b*e^2)*f^3*g + (c*d^2 + 2*b*d*e + a*e^2)*f^2*g^2 - (b*d^2 + 2*a*d*e)*f*g^3)/(sqrt(g*x + f)
*g^4))/g

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (194) = 388\).

Time = 0.30 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.94 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=-\frac {2 \, {\left (c e^{2} f^{4} - 2 \, c d e f^{3} g - b e^{2} f^{3} g + c d^{2} f^{2} g^{2} + 2 \, b d e f^{2} g^{2} + a e^{2} f^{2} g^{2} - b d^{2} f g^{3} - 2 \, a d e f g^{3} + a d^{2} g^{4}\right )}}{\sqrt {g x + f} g^{5}} + \frac {2 \, {\left (15 \, {\left (g x + f\right )}^{\frac {7}{2}} c e^{2} g^{30} - 84 \, {\left (g x + f\right )}^{\frac {5}{2}} c e^{2} f g^{30} + 210 \, {\left (g x + f\right )}^{\frac {3}{2}} c e^{2} f^{2} g^{30} - 420 \, \sqrt {g x + f} c e^{2} f^{3} g^{30} + 42 \, {\left (g x + f\right )}^{\frac {5}{2}} c d e g^{31} + 21 \, {\left (g x + f\right )}^{\frac {5}{2}} b e^{2} g^{31} - 210 \, {\left (g x + f\right )}^{\frac {3}{2}} c d e f g^{31} - 105 \, {\left (g x + f\right )}^{\frac {3}{2}} b e^{2} f g^{31} + 630 \, \sqrt {g x + f} c d e f^{2} g^{31} + 315 \, \sqrt {g x + f} b e^{2} f^{2} g^{31} + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} g^{32} + 70 \, {\left (g x + f\right )}^{\frac {3}{2}} b d e g^{32} + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} a e^{2} g^{32} - 210 \, \sqrt {g x + f} c d^{2} f g^{32} - 420 \, \sqrt {g x + f} b d e f g^{32} - 210 \, \sqrt {g x + f} a e^{2} f g^{32} + 105 \, \sqrt {g x + f} b d^{2} g^{33} + 210 \, \sqrt {g x + f} a d e g^{33}\right )}}{105 \, g^{35}} \]

[In]

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

[Out]

-2*(c*e^2*f^4 - 2*c*d*e*f^3*g - b*e^2*f^3*g + c*d^2*f^2*g^2 + 2*b*d*e*f^2*g^2 + a*e^2*f^2*g^2 - b*d^2*f*g^3 -
2*a*d*e*f*g^3 + a*d^2*g^4)/(sqrt(g*x + f)*g^5) + 2/105*(15*(g*x + f)^(7/2)*c*e^2*g^30 - 84*(g*x + f)^(5/2)*c*e
^2*f*g^30 + 210*(g*x + f)^(3/2)*c*e^2*f^2*g^30 - 420*sqrt(g*x + f)*c*e^2*f^3*g^30 + 42*(g*x + f)^(5/2)*c*d*e*g
^31 + 21*(g*x + f)^(5/2)*b*e^2*g^31 - 210*(g*x + f)^(3/2)*c*d*e*f*g^31 - 105*(g*x + f)^(3/2)*b*e^2*f*g^31 + 63
0*sqrt(g*x + f)*c*d*e*f^2*g^31 + 315*sqrt(g*x + f)*b*e^2*f^2*g^31 + 35*(g*x + f)^(3/2)*c*d^2*g^32 + 70*(g*x +
f)^(3/2)*b*d*e*g^32 + 35*(g*x + f)^(3/2)*a*e^2*g^32 - 210*sqrt(g*x + f)*c*d^2*f*g^32 - 420*sqrt(g*x + f)*b*d*e
*f*g^32 - 210*sqrt(g*x + f)*a*e^2*f*g^32 + 105*sqrt(g*x + f)*b*d^2*g^33 + 210*sqrt(g*x + f)*a*d*e*g^33)/g^35

Mupad [B] (verification not implemented)

Time = 11.86 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {{\left (f+g\,x\right )}^{5/2}\,\left (2\,b\,e^2\,g-8\,c\,e^2\,f+4\,c\,d\,e\,g\right )}{5\,g^5}-\frac {2\,c\,d^2\,f^2\,g^2-2\,b\,d^2\,f\,g^3+2\,a\,d^2\,g^4-4\,c\,d\,e\,f^3\,g+4\,b\,d\,e\,f^2\,g^2-4\,a\,d\,e\,f\,g^3+2\,c\,e^2\,f^4-2\,b\,e^2\,f^3\,g+2\,a\,e^2\,f^2\,g^2}{g^5\,\sqrt {f+g\,x}}+\frac {{\left (f+g\,x\right )}^{3/2}\,\left (2\,c\,d^2\,g^2-12\,c\,d\,e\,f\,g+4\,b\,d\,e\,g^2+12\,c\,e^2\,f^2-6\,b\,e^2\,f\,g+2\,a\,e^2\,g^2\right )}{3\,g^5}+\frac {2\,\sqrt {f+g\,x}\,\left (d\,g-e\,f\right )\,\left (2\,a\,e\,g^2+b\,d\,g^2+4\,c\,e\,f^2-3\,b\,e\,f\,g-2\,c\,d\,f\,g\right )}{g^5}+\frac {2\,c\,e^2\,{\left (f+g\,x\right )}^{7/2}}{7\,g^5} \]

[In]

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

[Out]

((f + g*x)^(5/2)*(2*b*e^2*g - 8*c*e^2*f + 4*c*d*e*g))/(5*g^5) - (2*a*d^2*g^4 + 2*c*e^2*f^4 + 2*a*e^2*f^2*g^2 +
2*c*d^2*f^2*g^2 - 2*b*d^2*f*g^3 - 2*b*e^2*f^3*g + 4*b*d*e*f^2*g^2 - 4*a*d*e*f*g^3 - 4*c*d*e*f^3*g)/(g^5*(f +
g*x)^(1/2)) + ((f + g*x)^(3/2)*(2*a*e^2*g^2 + 2*c*d^2*g^2 + 12*c*e^2*f^2 + 4*b*d*e*g^2 - 6*b*e^2*f*g - 12*c*d*
e*f*g))/(3*g^5) + (2*(f + g*x)^(1/2)*(d*g - e*f)*(2*a*e*g^2 + b*d*g^2 + 4*c*e*f^2 - 3*b*e*f*g - 2*c*d*f*g))/g^
5 + (2*c*e^2*(f + g*x)^(7/2))/(7*g^5)