3.828 \(\int \frac{(d+e x) \left (a+b x+c x^2\right )}{(f+g x)^{3/2}} \, dx\)
Optimal. Leaf size=135 \[ \frac{2 (e f-d g) \left (a g^2-b f g+c f^2\right )}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (c f (3 e f-2 d g)-g (-a e g-b d g+2 b e f))}{g^4}-\frac{2 (f+g x)^{3/2} (-b e g-c d g+3 c e f)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4} \]
[Out]
(2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2))/(g^4*Sqrt[f + g*x]) + (2*(c*f*(3*e*f - 2
*d*g) - g*(2*b*e*f - b*d*g - a*e*g))*Sqrt[f + g*x])/g^4 - (2*(3*c*e*f - c*d*g -
b*e*g)*(f + g*x)^(3/2))/(3*g^4) + (2*c*e*(f + g*x)^(5/2))/(5*g^4)
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Rubi [A] time = 0.238885, antiderivative size = 135, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04
\[ \frac{2 (e f-d g) \left (a g^2-b f g+c f^2\right )}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} (c f (3 e f-2 d g)-g (-a e g-b d g+2 b e f))}{g^4}-\frac{2 (f+g x)^{3/2} (-b e g-c d g+3 c e f)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)*(a + b*x + c*x^2))/(f + g*x)^(3/2),x]
[Out]
(2*(e*f - d*g)*(c*f^2 - b*f*g + a*g^2))/(g^4*Sqrt[f + g*x]) + (2*(c*f*(3*e*f - 2
*d*g) - g*(2*b*e*f - b*d*g - a*e*g))*Sqrt[f + g*x])/g^4 - (2*(3*c*e*f - c*d*g -
b*e*g)*(f + g*x)^(3/2))/(3*g^4) + (2*c*e*(f + g*x)^(5/2))/(5*g^4)
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Rubi in Sympy [A] time = 38.0806, size = 138, normalized size = 1.02 \[ \frac{2 c e \left (f + g x\right )^{\frac{5}{2}}}{5 g^{4}} + \frac{2 \left (f + g x\right )^{\frac{3}{2}} \left (b e g + c d g - 3 c e f\right )}{3 g^{4}} + \frac{2 \sqrt{f + g x} \left (a e g^{2} + b d g^{2} - 2 b e f g - 2 c d f g + 3 c e f^{2}\right )}{g^{4}} - \frac{2 \left (d g - e f\right ) \left (a g^{2} - b f g + c f^{2}\right )}{g^{4} \sqrt{f + g x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+b*x+a)/(g*x+f)**(3/2),x)
[Out]
2*c*e*(f + g*x)**(5/2)/(5*g**4) + 2*(f + g*x)**(3/2)*(b*e*g + c*d*g - 3*c*e*f)/(
3*g**4) + 2*sqrt(f + g*x)*(a*e*g**2 + b*d*g**2 - 2*b*e*f*g - 2*c*d*f*g + 3*c*e*f
**2)/g**4 - 2*(d*g - e*f)*(a*g**2 - b*f*g + c*f**2)/(g**4*sqrt(f + g*x))
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Mathematica [A] time = 0.159507, size = 128, normalized size = 0.95 \[ \frac{2 \left (5 g \left (3 a g (-d g+2 e f+e g x)+3 b d g (2 f+g x)+b e \left (-8 f^2-4 f g x+g^2 x^2\right )\right )+c \left (5 d g \left (-8 f^2-4 f g x+g^2 x^2\right )+3 e \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )\right )\right )}{15 g^4 \sqrt{f+g x}} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)*(a + b*x + c*x^2))/(f + g*x)^(3/2),x]
[Out]
(2*(5*g*(3*b*d*g*(2*f + g*x) + 3*a*g*(2*e*f - d*g + e*g*x) + b*e*(-8*f^2 - 4*f*g
*x + g^2*x^2)) + c*(5*d*g*(-8*f^2 - 4*f*g*x + g^2*x^2) + 3*e*(16*f^3 + 8*f^2*g*x
- 2*f*g^2*x^2 + g^3*x^3))))/(15*g^4*Sqrt[f + g*x])
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Maple [A] time = 0.008, size = 144, normalized size = 1.1 \[ -{\frac{-6\,ce{x}^{3}{g}^{3}-10\,be{g}^{3}{x}^{2}-10\,cd{g}^{3}{x}^{2}+12\,cef{g}^{2}{x}^{2}-30\,ae{g}^{3}x-30\,bd{g}^{3}x+40\,bef{g}^{2}x+40\,cdf{g}^{2}x-48\,ce{f}^{2}gx+30\,ad{g}^{3}-60\,aef{g}^{2}-60\,bdf{g}^{2}+80\,be{f}^{2}g+80\,cd{f}^{2}g-96\,ce{f}^{3}}{15\,{g}^{4}}{\frac{1}{\sqrt{gx+f}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+b*x+a)/(g*x+f)^(3/2),x)
[Out]
-2/15/(g*x+f)^(1/2)*(-3*c*e*g^3*x^3-5*b*e*g^3*x^2-5*c*d*g^3*x^2+6*c*e*f*g^2*x^2-
15*a*e*g^3*x-15*b*d*g^3*x+20*b*e*f*g^2*x+20*c*d*f*g^2*x-24*c*e*f^2*g*x+15*a*d*g^
3-30*a*e*f*g^2-30*b*d*f*g^2+40*b*e*f^2*g+40*c*d*f^2*g-48*c*e*f^3)/g^4
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Maxima [A] time = 0.695653, size = 185, normalized size = 1.37 \[ \frac{2 \,{\left (\frac{3 \,{\left (g x + f\right )}^{\frac{5}{2}} c e - 5 \,{\left (3 \, c e f -{\left (c d + b e\right )} g\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, c e f^{2} - 2 \,{\left (c d + b e\right )} f g +{\left (b d + a e\right )} g^{2}\right )} \sqrt{g x + f}}{g^{3}} + \frac{15 \,{\left (c e f^{3} - a d g^{3} -{\left (c d + b e\right )} f^{2} g +{\left (b d + a e\right )} f g^{2}\right )}}{\sqrt{g x + f} g^{3}}\right )}}{15 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)/(g*x + f)^(3/2),x, algorithm="maxima")
[Out]
2/15*((3*(g*x + f)^(5/2)*c*e - 5*(3*c*e*f - (c*d + b*e)*g)*(g*x + f)^(3/2) + 15*
(3*c*e*f^2 - 2*(c*d + b*e)*f*g + (b*d + a*e)*g^2)*sqrt(g*x + f))/g^3 + 15*(c*e*f
^3 - a*d*g^3 - (c*d + b*e)*f^2*g + (b*d + a*e)*f*g^2)/(sqrt(g*x + f)*g^3))/g
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Fricas [A] time = 0.269217, size = 169, normalized size = 1.25 \[ \frac{2 \,{\left (3 \, c e g^{3} x^{3} + 48 \, c e f^{3} - 15 \, a d g^{3} - 40 \,{\left (c d + b e\right )} f^{2} g + 30 \,{\left (b d + a e\right )} f g^{2} -{\left (6 \, c e f g^{2} - 5 \,{\left (c d + b e\right )} g^{3}\right )} x^{2} +{\left (24 \, c e f^{2} g - 20 \,{\left (c d + b e\right )} f g^{2} + 15 \,{\left (b d + a e\right )} g^{3}\right )} x\right )}}{15 \, \sqrt{g x + f} g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)/(g*x + f)^(3/2),x, algorithm="fricas")
[Out]
2/15*(3*c*e*g^3*x^3 + 48*c*e*f^3 - 15*a*d*g^3 - 40*(c*d + b*e)*f^2*g + 30*(b*d +
a*e)*f*g^2 - (6*c*e*f*g^2 - 5*(c*d + b*e)*g^3)*x^2 + (24*c*e*f^2*g - 20*(c*d +
b*e)*f*g^2 + 15*(b*d + a*e)*g^3)*x)/(sqrt(g*x + f)*g^4)
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Sympy [A] time = 26.1302, size = 2720, normalized size = 20.15 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+b*x+a)/(g*x+f)**(3/2),x)
[Out]
-2*a*d/(g*sqrt(f + g*x)) + a*e*Piecewise((4*f/(g**2*sqrt(f + g*x)) + 2*x/(g*sqrt
(f + g*x)), Ne(g, 0)), (x**2/(2*f**(3/2)), True)) + b*d*Piecewise((4*f/(g**2*sqr
t(f + g*x)) + 2*x/(g*sqrt(f + g*x)), Ne(g, 0)), (x**2/(2*f**(3/2)), True)) + b*e
*(-16*f**(19/2)*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2
+ 3*f**5*g**6*x**3) + 16*f**(19/2)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x*
*2 + 3*f**5*g**6*x**3) - 40*f**(17/2)*g*x*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*
g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) + 48*f**(17/2)*g*x/(3*f**8*g**3 +
9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) - 30*f**(15/2)*g**2*x**2*sq
rt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3
) + 48*f**(15/2)*g**2*x**2/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f
**5*g**6*x**3) - 4*f**(13/2)*g**3*x**3*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**
4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) + 16*f**(13/2)*g**3*x**3/(3*f**8*g**3
+ 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) + 2*f**(11/2)*g**4*x**4*
sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x*
*3)) + c*d*(-16*f**(19/2)*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*
g**5*x**2 + 3*f**5*g**6*x**3) + 16*f**(19/2)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f*
*6*g**5*x**2 + 3*f**5*g**6*x**3) - 40*f**(17/2)*g*x*sqrt(1 + g*x/f)/(3*f**8*g**3
+ 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) + 48*f**(17/2)*g*x/(3*f*
*8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) - 30*f**(15/2)*g*
*2*x**2*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5
*g**6*x**3) + 48*f**(15/2)*g**2*x**2/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*
x**2 + 3*f**5*g**6*x**3) - 4*f**(13/2)*g**3*x**3*sqrt(1 + g*x/f)/(3*f**8*g**3 +
9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) + 16*f**(13/2)*g**3*x**3/(3
*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f**5*g**6*x**3) + 2*f**(11/2)*
g**4*x**4*sqrt(1 + g*x/f)/(3*f**8*g**3 + 9*f**7*g**4*x + 9*f**6*g**5*x**2 + 3*f*
*5*g**6*x**3)) + c*e*(32*f**(45/2)*sqrt(1 + g*x/f)/(5*f**20*g**4 + 30*f**19*g**5
*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f**15*g*
*9*x**5 + 5*f**14*g**10*x**6) - 32*f**(45/2)/(5*f**20*g**4 + 30*f**19*g**5*x + 7
5*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f**15*g**9*x**
5 + 5*f**14*g**10*x**6) + 176*f**(43/2)*g*x*sqrt(1 + g*x/f)/(5*f**20*g**4 + 30*f
**19*g**5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30
*f**15*g**9*x**5 + 5*f**14*g**10*x**6) - 192*f**(43/2)*g*x/(5*f**20*g**4 + 30*f*
*19*g**5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*
f**15*g**9*x**5 + 5*f**14*g**10*x**6) + 396*f**(41/2)*g**2*x**2*sqrt(1 + g*x/f)/
(5*f**20*g**4 + 30*f**19*g**5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*
f**16*g**8*x**4 + 30*f**15*g**9*x**5 + 5*f**14*g**10*x**6) - 480*f**(41/2)*g**2*
x**2/(5*f**20*g**4 + 30*f**19*g**5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3
+ 75*f**16*g**8*x**4 + 30*f**15*g**9*x**5 + 5*f**14*g**10*x**6) + 462*f**(39/2)*
g**3*x**3*sqrt(1 + g*x/f)/(5*f**20*g**4 + 30*f**19*g**5*x + 75*f**18*g**6*x**2 +
100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f**15*g**9*x**5 + 5*f**14*g**10*x
**6) - 640*f**(39/2)*g**3*x**3/(5*f**20*g**4 + 30*f**19*g**5*x + 75*f**18*g**6*x
**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f**15*g**9*x**5 + 5*f**14*g*
*10*x**6) + 290*f**(37/2)*g**4*x**4*sqrt(1 + g*x/f)/(5*f**20*g**4 + 30*f**19*g**
5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f**15*g
**9*x**5 + 5*f**14*g**10*x**6) - 480*f**(37/2)*g**4*x**4/(5*f**20*g**4 + 30*f**1
9*g**5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f*
*15*g**9*x**5 + 5*f**14*g**10*x**6) + 92*f**(35/2)*g**5*x**5*sqrt(1 + g*x/f)/(5*
f**20*g**4 + 30*f**19*g**5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**
16*g**8*x**4 + 30*f**15*g**9*x**5 + 5*f**14*g**10*x**6) - 192*f**(35/2)*g**5*x**
5/(5*f**20*g**4 + 30*f**19*g**5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 7
5*f**16*g**8*x**4 + 30*f**15*g**9*x**5 + 5*f**14*g**10*x**6) + 16*f**(33/2)*g**6
*x**6*sqrt(1 + g*x/f)/(5*f**20*g**4 + 30*f**19*g**5*x + 75*f**18*g**6*x**2 + 100
*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f**15*g**9*x**5 + 5*f**14*g**10*x**6)
- 32*f**(33/2)*g**6*x**6/(5*f**20*g**4 + 30*f**19*g**5*x + 75*f**18*g**6*x**2 +
100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f**15*g**9*x**5 + 5*f**14*g**10*x
**6) + 6*f**(31/2)*g**7*x**7*sqrt(1 + g*x/f)/(5*f**20*g**4 + 30*f**19*g**5*x + 7
5*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4 + 30*f**15*g**9*x**
5 + 5*f**14*g**10*x**6) + 2*f**(29/2)*g**8*x**8*sqrt(1 + g*x/f)/(5*f**20*g**4 +
30*f**19*g**5*x + 75*f**18*g**6*x**2 + 100*f**17*g**7*x**3 + 75*f**16*g**8*x**4
+ 30*f**15*g**9*x**5 + 5*f**14*g**10*x**6))
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GIAC/XCAS [A] time = 0.270537, size = 275, normalized size = 2.04 \[ -\frac{2 \,{\left (c d f^{2} g - b d f g^{2} + a d g^{3} - c f^{3} e + b f^{2} g e - a f g^{2} e\right )}}{\sqrt{g x + f} g^{4}} + \frac{2 \,{\left (5 \,{\left (g x + f\right )}^{\frac{3}{2}} c d g^{17} - 30 \, \sqrt{g x + f} c d f g^{17} + 15 \, \sqrt{g x + f} b d g^{18} + 3 \,{\left (g x + f\right )}^{\frac{5}{2}} c g^{16} e - 15 \,{\left (g x + f\right )}^{\frac{3}{2}} c f g^{16} e + 45 \, \sqrt{g x + f} c f^{2} g^{16} e + 5 \,{\left (g x + f\right )}^{\frac{3}{2}} b g^{17} e - 30 \, \sqrt{g x + f} b f g^{17} e + 15 \, \sqrt{g x + f} a g^{18} e\right )}}{15 \, g^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(e*x + d)/(g*x + f)^(3/2),x, algorithm="giac")
[Out]
-2*(c*d*f^2*g - b*d*f*g^2 + a*d*g^3 - c*f^3*e + b*f^2*g*e - a*f*g^2*e)/(sqrt(g*x
+ f)*g^4) + 2/15*(5*(g*x + f)^(3/2)*c*d*g^17 - 30*sqrt(g*x + f)*c*d*f*g^17 + 15
*sqrt(g*x + f)*b*d*g^18 + 3*(g*x + f)^(5/2)*c*g^16*e - 15*(g*x + f)^(3/2)*c*f*g^
16*e + 45*sqrt(g*x + f)*c*f^2*g^16*e + 5*(g*x + f)^(3/2)*b*g^17*e - 30*sqrt(g*x
+ f)*b*f*g^17*e + 15*sqrt(g*x + f)*a*g^18*e)/g^20