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

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 = 0.708561, 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 \[ -\frac{2 (f+g x)^{3/2} \left (e g (-a e g-2 b d g+3 b e f)-c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac{2 (e f-d g)^2 \left (a g^2-b f g+c f^2\right )}{g^5 \sqrt{f+g x}}-\frac{2 \sqrt{f+g x} (e f-d g) (2 c f (2 e f-d g)-g (-2 a e g-b d g+3 b e f))}{g^5}-\frac{2 e (f+g x)^{5/2} (-b e g-2 c d g+4 c e f)}{5 g^5}+\frac{2 c e^2 (f+g x)^{7/2}}{7 g^5} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 c e^{2} \left (f + g x\right )^{\frac{7}{2}}}{7 g^{5}} + \frac{2 e \left (f + g x\right )^{\frac{5}{2}} \left (b e g + 2 c d g - 4 c e f\right )}{5 g^{5}} + \frac{2 \left (d g - e f\right ) \left (2 a e g^{2} + b d g^{2} - 3 b e f g - 2 c d f g + 4 c e f^{2}\right ) \int ^{\sqrt{f + g x}} \frac{1}{g^{4}}\, dx}{g} + \frac{2 \left (f + g x\right )^{\frac{3}{2}} \left (a e^{2} g^{2} + 2 b d e g^{2} - 3 b e^{2} f g + c d^{2} g^{2} - 6 c d e f g + 6 c e^{2} f^{2}\right )}{3 g^{5}} - \frac{2 \left (d g - e f\right )^{2} \left (a g^{2} - b f g + c f^{2}\right )}{g^{5} \sqrt{f + g x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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) + 2*e*(f + g*x)**(5/2)*(b*e*g + 2*c*d*g - 4*c
*e*f)/(5*g**5) + 2*(d*g - e*f)*(2*a*e*g**2 + b*d*g**2 - 3*b*e*f*g - 2*c*d*f*g +
4*c*e*f**2)*Integral(g**(-4), (x, sqrt(f + g*x)))/g + 2*(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**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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Mathematica [A]  time = 0.342102, size = 252, normalized size = 1.2 \[ \frac{2 \left (7 g \left (5 a g \left (-3 d^2 g^2+6 d e g (2 f+g x)+e^2 \left (-8 f^2-4 f g x+g^2 x^2\right )\right )+b \left (15 d^2 g^2 (2 f+g x)+10 d e g \left (-8 f^2-4 f g x+g^2 x^2\right )+3 e^2 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )\right )\right )+c \left (35 d^2 g^2 \left (-8 f^2-4 f g x+g^2 x^2\right )+42 d e g \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )-3 e^2 \left (128 f^4+64 f^3 g x-16 f^2 g^2 x^2+8 f g^3 x^3-5 g^4 x^4\right )\right )\right )}{105 g^5 \sqrt{f+g x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(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])

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Maple [A]  time = 0.011, size = 315, normalized size = 1.5 \[ -{\frac{-30\,{e}^{2}c{x}^{4}{g}^{4}-42\,b{e}^{2}{g}^{4}{x}^{3}-84\,cde{g}^{4}{x}^{3}+48\,c{e}^{2}f{g}^{3}{x}^{3}-70\,a{e}^{2}{g}^{4}{x}^{2}-140\,bde{g}^{4}{x}^{2}+84\,b{e}^{2}f{g}^{3}{x}^{2}-70\,c{d}^{2}{g}^{4}{x}^{2}+168\,cdef{g}^{3}{x}^{2}-96\,c{e}^{2}{f}^{2}{g}^{2}{x}^{2}-420\,ade{g}^{4}x+280\,a{e}^{2}f{g}^{3}x-210\,b{d}^{2}{g}^{4}x+560\,bdef{g}^{3}x-336\,b{e}^{2}{f}^{2}{g}^{2}x+280\,c{d}^{2}f{g}^{3}x-672\,cde{f}^{2}{g}^{2}x+384\,c{e}^{2}{f}^{3}gx+210\,{d}^{2}a{g}^{4}-840\,adef{g}^{3}+560\,a{e}^{2}{f}^{2}{g}^{2}-420\,b{d}^{2}f{g}^{3}+1120\,bde{f}^{2}{g}^{2}-672\,b{e}^{2}{f}^{3}g+560\,c{d}^{2}{f}^{2}{g}^{2}-1344\,cde{f}^{3}g+768\,c{e}^{2}{f}^{4}}{105\,{g}^{5}}{\frac{1}{\sqrt{gx+f}}}} \]

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+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+3
84*c*e^2*f^4)/g^5

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Maxima [A]  time = 0.698885, size = 363, normalized size = 1.73 \[ \frac{2 \,{\left (\frac{15 \,{\left (g x + f\right )}^{\frac{7}{2}} c e^{2} - 21 \,{\left (4 \, c e^{2} f -{\left (2 \, c d e + b e^{2}\right )} g\right )}{\left (g x + f\right )}^{\frac{5}{2}} + 35 \,{\left (6 \, c e^{2} f^{2} - 3 \,{\left (2 \, c d e + b e^{2}\right )} f g +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{3}{2}} - 105 \,{\left (4 \, c e^{2} f^{3} - 3 \,{\left (2 \, c d e + b e^{2}\right )} f^{2} g + 2 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{2} -{\left (b d^{2} + 2 \, a d e\right )} g^{3}\right )} \sqrt{g x + f}}{g^{4}} - \frac{105 \,{\left (c e^{2} f^{4} + a d^{2} g^{4} -{\left (2 \, c d e + b e^{2}\right )} f^{3} g +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} -{\left (b d^{2} + 2 \, a d e\right )} f g^{3}\right )}}{\sqrt{g x + f} g^{4}}\right )}}{105 \, g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^2/(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 = 0.27179, size = 350, normalized size = 1.67 \[ \frac{2 \,{\left (15 \, c e^{2} g^{4} x^{4} - 384 \, c e^{2} f^{4} - 105 \, a d^{2} g^{4} + 336 \,{\left (2 \, c d e + b e^{2}\right )} f^{3} g - 280 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f^{2} g^{2} + 210 \,{\left (b d^{2} + 2 \, a d e\right )} f g^{3} - 3 \,{\left (8 \, c e^{2} f g^{3} - 7 \,{\left (2 \, c d e + b e^{2}\right )} g^{4}\right )} x^{3} +{\left (48 \, c e^{2} f^{2} g^{2} - 42 \,{\left (2 \, c d e + b e^{2}\right )} f g^{3} + 35 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} g^{4}\right )} x^{2} -{\left (192 \, c e^{2} f^{3} g - 168 \,{\left (2 \, c d e + b e^{2}\right )} f^{2} g^{2} + 140 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} f g^{3} - 105 \,{\left (b d^{2} + 2 \, a d e\right )} g^{4}\right )} x\right )}}{105 \, \sqrt{g x + f} g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^2/(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^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )}{\left (f + g x\right )^{\frac{3}{2}}}\, dx \]

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]

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

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GIAC/XCAS [A]  time = 0.272074, size = 545, normalized size = 2.6 \[ -\frac{2 \,{\left (c d^{2} f^{2} g^{2} - b d^{2} f g^{3} + a d^{2} g^{4} - 2 \, c d f^{3} g e + 2 \, b d f^{2} g^{2} e - 2 \, a d f g^{3} e + c f^{4} e^{2} - b f^{3} g e^{2} + a f^{2} g^{2} e^{2}\right )}}{\sqrt{g x + f} g^{5}} + \frac{2 \,{\left (35 \,{\left (g x + f\right )}^{\frac{3}{2}} c d^{2} g^{32} - 210 \, \sqrt{g x + f} c d^{2} f g^{32} + 105 \, \sqrt{g x + f} b d^{2} g^{33} + 42 \,{\left (g x + f\right )}^{\frac{5}{2}} c d g^{31} e - 210 \,{\left (g x + f\right )}^{\frac{3}{2}} c d f g^{31} e + 630 \, \sqrt{g x + f} c d f^{2} g^{31} e + 70 \,{\left (g x + f\right )}^{\frac{3}{2}} b d g^{32} e - 420 \, \sqrt{g x + f} b d f g^{32} e + 210 \, \sqrt{g x + f} a d g^{33} e + 15 \,{\left (g x + f\right )}^{\frac{7}{2}} c g^{30} e^{2} - 84 \,{\left (g x + f\right )}^{\frac{5}{2}} c f g^{30} e^{2} + 210 \,{\left (g x + f\right )}^{\frac{3}{2}} c f^{2} g^{30} e^{2} - 420 \, \sqrt{g x + f} c f^{3} g^{30} e^{2} + 21 \,{\left (g x + f\right )}^{\frac{5}{2}} b g^{31} e^{2} - 105 \,{\left (g x + f\right )}^{\frac{3}{2}} b f g^{31} e^{2} + 315 \, \sqrt{g x + f} b f^{2} g^{31} e^{2} + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} a g^{32} e^{2} - 210 \, \sqrt{g x + f} a f g^{32} e^{2}\right )}}{105 \, g^{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^2/(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 + 630*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