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

Optimal. Leaf size=173 \[ \frac{2 (f+g x)^{3/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac{2 \left (a g^2+c f^2\right ) (e f-d g)^2}{g^5 \sqrt{f+g x}}-\frac{4 \sqrt{f+g x} (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )}{g^5}-\frac{4 c e (f+g x)^{5/2} (2 e f-d g)}{5 g^5}+\frac{2 c e^2 (f+g x)^{7/2}}{7 g^5} \]

[Out]

(-2*(e*f - d*g)^2*(c*f^2 + a*g^2))/(g^5*Sqrt[f + g*x]) - (4*(e*f - d*g)*(a*e*g^2
+ c*f*(2*e*f - d*g))*Sqrt[f + g*x])/g^5 + (2*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d*e*
f*g + d^2*g^2))*(f + g*x)^(3/2))/(3*g^5) - (4*c*e*(2*e*f - d*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 = 0.492887, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 (f+g x)^{3/2} \left (a e^2 g^2+c \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )\right )}{3 g^5}-\frac{2 \left (a g^2+c f^2\right ) (e f-d g)^2}{g^5 \sqrt{f+g x}}-\frac{4 \sqrt{f+g x} (e f-d g) \left (a e g^2+c f (2 e f-d g)\right )}{g^5}-\frac{4 c e (f+g x)^{5/2} (2 e f-d g)}{5 g^5}+\frac{2 c e^2 (f+g x)^{7/2}}{7 g^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(e*f - d*g)^2*(c*f^2 + a*g^2))/(g^5*Sqrt[f + g*x]) - (4*(e*f - d*g)*(a*e*g^2
+ c*f*(2*e*f - d*g))*Sqrt[f + g*x])/g^5 + (2*(a*e^2*g^2 + c*(6*e^2*f^2 - 6*d*e*
f*g + d^2*g^2))*(f + g*x)^(3/2))/(3*g^5) - (4*c*e*(2*e*f - d*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 [A]  time = 79.7398, size = 173, normalized size = 1. \[ \frac{2 c e^{2} \left (f + g x\right )^{\frac{7}{2}}}{7 g^{5}} + \frac{4 c e \left (f + g x\right )^{\frac{5}{2}} \left (d g - 2 e f\right )}{5 g^{5}} + \frac{2 \left (f + g x\right )^{\frac{3}{2}} \left (a e^{2} g^{2} + c d^{2} g^{2} - 6 c d e f g + 6 c e^{2} f^{2}\right )}{3 g^{5}} + \frac{4 \sqrt{f + g x} \left (d g - e f\right ) \left (a e g^{2} - c d f g + 2 c e f^{2}\right )}{g^{5}} - \frac{2 \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )^{2}}{g^{5} \sqrt{f + g x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.279657, size = 177, normalized size = 1.02 \[ \frac{2 c \left (35 d^2 g^2 \left (-8 f^2-4 f g x+g^2 x^2\right )+42 d e g \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )-3 e^2 \left (128 f^4+64 f^3 g x-16 f^2 g^2 x^2+8 f g^3 x^3-5 g^4 x^4\right )\right )-70 a g^2 \left (3 d^2 g^2-6 d e g (2 f+g x)+e^2 \left (8 f^2+4 f g x-g^2 x^2\right )\right )}{105 g^5 \sqrt{f+g x}} \]

Antiderivative was successfully verified.

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

[Out]

(-70*a*g^2*(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)) +
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)))/(105*g^5*Sqrt[f + g*x])

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Maple [A]  time = 0.01, size = 215, normalized size = 1.2 \[ -{\frac{-30\,{e}^{2}c{x}^{4}{g}^{4}-84\,cde{g}^{4}{x}^{3}+48\,c{e}^{2}f{g}^{3}{x}^{3}-70\,a{e}^{2}{g}^{4}{x}^{2}-70\,c{d}^{2}{g}^{4}{x}^{2}+168\,cdef{g}^{3}{x}^{2}-96\,c{e}^{2}{f}^{2}{g}^{2}{x}^{2}-420\,ade{g}^{4}x+280\,a{e}^{2}f{g}^{3}x+280\,c{d}^{2}f{g}^{3}x-672\,cde{f}^{2}{g}^{2}x+384\,c{e}^{2}{f}^{3}gx+210\,{d}^{2}a{g}^{4}-840\,adef{g}^{3}+560\,a{e}^{2}{f}^{2}{g}^{2}+560\,c{d}^{2}{f}^{2}{g}^{2}-1344\,cde{f}^{3}g+768\,c{e}^{2}{f}^{4}}{105\,{g}^{5}}{\frac{1}{\sqrt{gx+f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105/(g*x+f)^(1/2)*(-15*c*e^2*g^4*x^4-42*c*d*e*g^4*x^3+24*c*e^2*f*g^3*x^3-35*a
*e^2*g^4*x^2-35*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+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+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 = 0.700947, size = 277, normalized size = 1.6 \[ \frac{2 \,{\left (\frac{15 \,{\left (g x + f\right )}^{\frac{7}{2}} c e^{2} - 42 \,{\left (2 \, c e^{2} f - c d e g\right )}{\left (g x + f\right )}^{\frac{5}{2}} + 35 \,{\left (6 \, c e^{2} f^{2} - 6 \, c d e f g +{\left (c d^{2} + a e^{2}\right )} g^{2}\right )}{\left (g x + f\right )}^{\frac{3}{2}} - 210 \,{\left (2 \, c e^{2} f^{3} - 3 \, c d e f^{2} g - a d e g^{3} +{\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt{g x + f}}{g^{4}} - \frac{105 \,{\left (c e^{2} f^{4} - 2 \, c d e f^{3} g - 2 \, a d e f g^{3} + a d^{2} g^{4} +{\left (c d^{2} + a e^{2}\right )} f^{2} g^{2}\right )}}{\sqrt{g x + f} g^{4}}\right )}}{105 \, g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + 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 - 42*(2*c*e^2*f - c*d*e*g)*(g*x + f)^(5/2) + 35
*(6*c*e^2*f^2 - 6*c*d*e*f*g + (c*d^2 + a*e^2)*g^2)*(g*x + f)^(3/2) - 210*(2*c*e^
2*f^3 - 3*c*d*e*f^2*g - a*d*e*g^3 + (c*d^2 + a*e^2)*f*g^2)*sqrt(g*x + f))/g^4 -
105*(c*e^2*f^4 - 2*c*d*e*f^3*g - 2*a*d*e*f*g^3 + a*d^2*g^4 + (c*d^2 + a*e^2)*f^2
*g^2)/(sqrt(g*x + f)*g^4))/g

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Fricas [A]  time = 0.274904, size = 265, normalized size = 1.53 \[ \frac{2 \,{\left (15 \, c e^{2} g^{4} x^{4} - 384 \, c e^{2} f^{4} + 672 \, c d e f^{3} g + 420 \, a d e f g^{3} - 105 \, a d^{2} g^{4} - 280 \,{\left (c d^{2} + a e^{2}\right )} f^{2} g^{2} - 6 \,{\left (4 \, c e^{2} f g^{3} - 7 \, c d e g^{4}\right )} x^{3} +{\left (48 \, c e^{2} f^{2} g^{2} - 84 \, c d e f g^{3} + 35 \,{\left (c d^{2} + a e^{2}\right )} g^{4}\right )} x^{2} - 2 \,{\left (96 \, c e^{2} f^{3} g - 168 \, c d e f^{2} g^{2} - 105 \, a d e g^{4} + 70 \,{\left (c d^{2} + a e^{2}\right )} f g^{3}\right )} x\right )}}{105 \, \sqrt{g x + f} g^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + 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 + 672*c*d*e*f^3*g + 420*a*d*e*f*g^3 - 10
5*a*d^2*g^4 - 280*(c*d^2 + a*e^2)*f^2*g^2 - 6*(4*c*e^2*f*g^3 - 7*c*d*e*g^4)*x^3
+ (48*c*e^2*f^2*g^2 - 84*c*d*e*f*g^3 + 35*(c*d^2 + a*e^2)*g^4)*x^2 - 2*(96*c*e^2
*f^3*g - 168*c*d*e*f^2*g^2 - 105*a*d*e*g^4 + 70*(c*d^2 + a*e^2)*f*g^3)*x)/(sqrt(
g*x + f)*g^5)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282372, size = 371, normalized size = 2.14 \[ -\frac{2 \,{\left (c d^{2} f^{2} g^{2} + a d^{2} g^{4} - 2 \, c d f^{3} g e - 2 \, a d f g^{3} e + c f^{4} e^{2} + a f^{2} g^{2} e^{2}\right )}}{\sqrt{g x + f} g^{5}} + \frac{2 \,{\left (35 \,{\left (g x + f\right )}^{\frac{3}{2}} c d^{2} g^{32} - 210 \, \sqrt{g x + f} c d^{2} f g^{32} + 42 \,{\left (g x + f\right )}^{\frac{5}{2}} c d g^{31} e - 210 \,{\left (g x + f\right )}^{\frac{3}{2}} c d f g^{31} e + 630 \, \sqrt{g x + f} c d f^{2} g^{31} e + 210 \, \sqrt{g x + f} a d g^{33} e + 15 \,{\left (g x + f\right )}^{\frac{7}{2}} c g^{30} e^{2} - 84 \,{\left (g x + f\right )}^{\frac{5}{2}} c f g^{30} e^{2} + 210 \,{\left (g x + f\right )}^{\frac{3}{2}} c f^{2} g^{30} e^{2} - 420 \, \sqrt{g x + f} c f^{3} g^{30} e^{2} + 35 \,{\left (g x + f\right )}^{\frac{3}{2}} a g^{32} e^{2} - 210 \, \sqrt{g x + f} a f g^{32} e^{2}\right )}}{105 \, g^{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(c*d^2*f^2*g^2 + a*d^2*g^4 - 2*c*d*f^3*g*e - 2*a*d*f*g^3*e + c*f^4*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 + 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 + 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 + 35*(g*x + f)^(3/2)*a*g^3
2*e^2 - 210*sqrt(g*x + f)*a*f*g^32*e^2)/g^35