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

Optimal. Leaf size=210 \[ -\frac{2 (f+g x)^{3/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac{2 (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5 \sqrt{f+g x}}-\frac{2 \sqrt{f+g x} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{g^5}-\frac{2 e (f+g x)^{5/2} (-b e g-2 c d g+4 c e f)}{5 g^5}+\frac{2 c e^2 (f+g x)^{7/2}}{7 g^5} \]

[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)

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Rubi [A]  time = 0.288764, antiderivative size = 210, 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, 1261} \[ -\frac{2 (f+g x)^{3/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac{2 (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5 \sqrt{f+g x}}-\frac{2 \sqrt{f+g x} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{g^5}-\frac{2 e (f+g x)^{5/2} (-b e g-2 c d g+4 c e f)}{5 g^5}+\frac{2 c e^2 (f+g x)^{7/2}}{7 g^5} \]

Antiderivative was successfully verified.

[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 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 1261

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*} \int \frac{(d+e x)^2 \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx &=\frac{2 \operatorname{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 \operatorname{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]  time = 0.376082, size = 184, normalized size = 0.88 \[ \frac{2 \left (-35 (f+g x)^2 \left (-e g (a e g+2 b d g-3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )-105 (e f-d g)^2 \left (g (a g-b f)+c f^2\right )-105 (f+g x) (e f-d g) (g (2 a e g+b d g-3 b e f)+2 c f (2 e f-d g))-21 e (f+g x)^3 (-b e g-2 c d g+4 c e f)+15 c e^2 (f+g x)^4\right )}{105 g^5 \sqrt{f+g x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-105*(e*f - d*g)^2*(c*f^2 + g*(-(b*f) + a*g)) - 105*(e*f - d*g)*(2*c*f*(2*e*f - d*g) + g*(-3*b*e*f + b*d*g
+ 2*a*e*g))*(f + g*x) - 35*(-(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)^2 - 21*e*(4*c*e*f - 2*c*d*g - b*e*g)*(f + g*x)^3 + 15*c*e^2*(f + g*x)^4))/(105*g^5*Sqrt[f + g*x])

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Maple [A]  time = 0.051, size = 315, normalized size = 1.5 \begin{align*} -{\frac{-30\,c{e}^{2}{x}^{4}{g}^{4}-42\,b{e}^{2}{g}^{4}{x}^{3}-84\,cde{g}^{4}{x}^{3}+48\,c{e}^{2}f{g}^{3}{x}^{3}-70\,a{e}^{2}{g}^{4}{x}^{2}-140\,bde{g}^{4}{x}^{2}+84\,b{e}^{2}f{g}^{3}{x}^{2}-70\,c{d}^{2}{g}^{4}{x}^{2}+168\,cdef{g}^{3}{x}^{2}-96\,c{e}^{2}{f}^{2}{g}^{2}{x}^{2}-420\,ade{g}^{4}x+280\,a{e}^{2}f{g}^{3}x-210\,b{d}^{2}{g}^{4}x+560\,bdef{g}^{3}x-336\,b{e}^{2}{f}^{2}{g}^{2}x+280\,c{d}^{2}f{g}^{3}x-672\,cde{f}^{2}{g}^{2}x+384\,c{e}^{2}{f}^{3}gx+210\,a{d}^{2}{g}^{4}-840\,adef{g}^{3}+560\,a{e}^{2}{f}^{2}{g}^{2}-420\,b{d}^{2}f{g}^{3}+1120\,bde{f}^{2}{g}^{2}-672\,b{e}^{2}{f}^{3}g+560\,c{d}^{2}{f}^{2}{g}^{2}-1344\,cde{f}^{3}g+768\,c{e}^{2}{f}^{4}}{105\,{g}^{5}}{\frac{1}{\sqrt{gx+f}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105/(g*x+f)^(1/2)*(-15*c*e^2*g^4*x^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+1
40*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+1
92*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)/g^5

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Maxima [A]  time = 0.96979, size = 363, normalized size = 1.73 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.48136, size = 603, normalized size = 2.87 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 62.753, size = 272, normalized size = 1.3 \begin{align*} \frac{2 c e^{2} \left (f + g x\right )^{\frac{7}{2}}}{7 g^{5}} + \frac{\left (f + g x\right )^{\frac{5}{2}} \left (2 b e^{2} g + 4 c d e g - 8 c e^{2} f\right )}{5 g^{5}} + \frac{\left (f + g x\right )^{\frac{3}{2}} \left (2 a e^{2} g^{2} + 4 b d e g^{2} - 6 b e^{2} f g + 2 c d^{2} g^{2} - 12 c d e f g + 12 c e^{2} f^{2}\right )}{3 g^{5}} + \frac{\sqrt{f + g x} \left (4 a d e g^{3} - 4 a e^{2} f g^{2} + 2 b d^{2} g^{3} - 8 b d e f g^{2} + 6 b e^{2} f^{2} g - 4 c d^{2} f g^{2} + 12 c d e f^{2} g - 8 c e^{2} f^{3}\right )}{g^{5}} - \frac{2 \left (d g - e f\right )^{2} \left (a g^{2} - b f g + c f^{2}\right )}{g^{5} \sqrt{f + g x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.12825, size = 545, normalized size = 2.6 \begin{align*} -\frac{2 \,{\left (c d^{2} f^{2} g^{2} - b d^{2} f g^{3} + a d^{2} g^{4} - 2 \, c d f^{3} g e + 2 \, b d f^{2} g^{2} e - 2 \, a d f g^{3} e + c f^{4} e^{2} - b f^{3} g e^{2} + a f^{2} g^{2} e^{2}\right )}}{\sqrt{g x + f} g^{5}} + \frac{2 \,{\left (35 \,{\left (g x + f\right )}^{\frac{3}{2}} c d^{2} g^{32} - 210 \, \sqrt{g x + f} c d^{2} f g^{32} + 105 \, \sqrt{g x + f} b d^{2} g^{33} + 42 \,{\left (g x + f\right )}^{\frac{5}{2}} c d g^{31} e - 210 \,{\left (g x + f\right )}^{\frac{3}{2}} c d f g^{31} e + 630 \, \sqrt{g x + f} c d f^{2} g^{31} e + 70 \,{\left (g x + f\right )}^{\frac{3}{2}} b d g^{32} e - 420 \, \sqrt{g x + f} b d f g^{32} e + 210 \, \sqrt{g x + f} a d g^{33} e + 15 \,{\left (g x + f\right )}^{\frac{7}{2}} c g^{30} e^{2} - 84 \,{\left (g x + f\right )}^{\frac{5}{2}} c f g^{30} e^{2} + 210 \,{\left (g x + f\right )}^{\frac{3}{2}} c f^{2} g^{30} e^{2} - 420 \, \sqrt{g x + f} c f^{3} g^{30} e^{2} + 21 \,{\left (g x + f\right )}^{\frac{5}{2}} b g^{31} e^{2} - 105 \,{\left (g x + f\right )}^{\frac{3}{2}} b f g^{31} e^{2} + 315 \, \sqrt{g x + f} b f^{2} g^{31} e^{2} + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} a g^{32} e^{2} - 210 \, \sqrt{g x + f} a f g^{32} e^{2}\right )}}{105 \, g^{35}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(c*d^2*f^2*g^2 - b*d^2*f*g^3 + a*d^2*g^4 - 2*c*d*f^3*g*e + 2*b*d*f^2*g^2*e - 2*a*d*f*g^3*e + c*f^4*e^2 - b*
f^3*g*e^2 + a*f^2*g^2*e^2)/(sqrt(g*x + f)*g^5) + 2/105*(35*(g*x + f)^(3/2)*c*d^2*g^32 - 210*sqrt(g*x + f)*c*d^
2*f*g^32 + 105*sqrt(g*x + f)*b*d^2*g^33 + 42*(g*x + f)^(5/2)*c*d*g^31*e - 210*(g*x + f)^(3/2)*c*d*f*g^31*e + 6
30*sqrt(g*x + f)*c*d*f^2*g^31*e + 70*(g*x + f)^(3/2)*b*d*g^32*e - 420*sqrt(g*x + f)*b*d*f*g^32*e + 210*sqrt(g*
x + f)*a*d*g^33*e + 15*(g*x + f)^(7/2)*c*g^30*e^2 - 84*(g*x + f)^(5/2)*c*f*g^30*e^2 + 210*(g*x + f)^(3/2)*c*f^
2*g^30*e^2 - 420*sqrt(g*x + f)*c*f^3*g^30*e^2 + 21*(g*x + f)^(5/2)*b*g^31*e^2 - 105*(g*x + f)^(3/2)*b*f*g^31*e
^2 + 315*sqrt(g*x + f)*b*f^2*g^31*e^2 + 35*(g*x + f)^(3/2)*a*g^32*e^2 - 210*sqrt(g*x + f)*a*f*g^32*e^2)/g^35