3.598 \(\int \frac{(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx\)
Optimal. Leaf size=111 \[ \frac{2 \left (a g^2+c f^2\right ) (e f-d g)}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} \left (a e g^2+c f (3 e f-2 d g)\right )}{g^4}-\frac{2 c (f+g x)^{3/2} (3 e f-d g)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4} \]
[Out]
(2*(e*f - d*g)*(c*f^2 + a*g^2))/(g^4*Sqrt[f + g*x]) + (2*(a*e*g^2 + c*f*(3*e*f -
2*d*g))*Sqrt[f + g*x])/g^4 - (2*c*(3*e*f - d*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 = 0.171267, antiderivative size = 111, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045
\[ \frac{2 \left (a g^2+c f^2\right ) (e f-d g)}{g^4 \sqrt{f+g x}}+\frac{2 \sqrt{f+g x} \left (a e g^2+c f (3 e f-2 d g)\right )}{g^4}-\frac{2 c (f+g x)^{3/2} (3 e f-d g)}{3 g^4}+\frac{2 c e (f+g x)^{5/2}}{5 g^4} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)*(a + c*x^2))/(f + g*x)^(3/2),x]
[Out]
(2*(e*f - d*g)*(c*f^2 + a*g^2))/(g^4*Sqrt[f + g*x]) + (2*(a*e*g^2 + c*f*(3*e*f -
2*d*g))*Sqrt[f + g*x])/g^4 - (2*c*(3*e*f - d*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 = 23.2014, size = 110, normalized size = 0.99 \[ \frac{2 c e \left (f + g x\right )^{\frac{5}{2}}}{5 g^{4}} + \frac{2 c \left (f + g x\right )^{\frac{3}{2}} \left (d g - 3 e f\right )}{3 g^{4}} + \frac{2 \sqrt{f + g x} \left (a e g^{2} - 2 c d f g + 3 c e f^{2}\right )}{g^{4}} - \frac{2 \left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )}{g^{4} \sqrt{f + g x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(c*x**2+a)/(g*x+f)**(3/2),x)
[Out]
2*c*e*(f + g*x)**(5/2)/(5*g**4) + 2*c*(f + g*x)**(3/2)*(d*g - 3*e*f)/(3*g**4) +
2*sqrt(f + g*x)*(a*e*g**2 - 2*c*d*f*g + 3*c*e*f**2)/g**4 - 2*(a*g**2 + c*f**2)*(
d*g - e*f)/(g**4*sqrt(f + g*x))
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Mathematica [A] time = 0.125521, size = 92, normalized size = 0.83 \[ \frac{30 a g^2 (-d g+2 e f+e g x)+10 c d g \left (-8 f^2-4 f g x+g^2 x^2\right )+6 c e \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )}{15 g^4 \sqrt{f+g x}} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)*(a + c*x^2))/(f + g*x)^(3/2),x]
[Out]
(30*a*g^2*(2*e*f - d*g + e*g*x) + 10*c*d*g*(-8*f^2 - 4*f*g*x + g^2*x^2) + 6*c*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 = 0.006, size = 101, normalized size = 0.9 \[ -{\frac{-6\,ce{x}^{3}{g}^{3}-10\,cd{g}^{3}{x}^{2}+12\,cef{g}^{2}{x}^{2}-30\,ae{g}^{3}x+40\,cdf{g}^{2}x-48\,ce{f}^{2}gx+30\,ad{g}^{3}-60\,aef{g}^{2}+80\,cd{f}^{2}g-96\,ce{f}^{3}}{15\,{g}^{4}}{\frac{1}{\sqrt{gx+f}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(c*x^2+a)/(g*x+f)^(3/2),x)
[Out]
-2/15/(g*x+f)^(1/2)*(-3*c*e*g^3*x^3-5*c*d*g^3*x^2+6*c*e*f*g^2*x^2-15*a*e*g^3*x+2
0*c*d*f*g^2*x-24*c*e*f^2*g*x+15*a*d*g^3-30*a*e*f*g^2+40*c*d*f^2*g-48*c*e*f^3)/g^
4
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Maxima [A] time = 0.701284, size = 151, normalized size = 1.36 \[ \frac{2 \,{\left (\frac{3 \,{\left (g x + f\right )}^{\frac{5}{2}} c e - 5 \,{\left (3 \, c e f - c d g\right )}{\left (g x + f\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, c e f^{2} - 2 \, c d f g + a e g^{2}\right )} \sqrt{g x + f}}{g^{3}} + \frac{15 \,{\left (c e f^{3} - c d f^{2} g + a e f g^{2} - a d g^{3}\right )}}{\sqrt{g x + f} g^{3}}\right )}}{15 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + 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*g)*(g*x + f)^(3/2) + 15*(3*c*e*f
^2 - 2*c*d*f*g + a*e*g^2)*sqrt(g*x + f))/g^3 + 15*(c*e*f^3 - c*d*f^2*g + a*e*f*g
^2 - a*d*g^3)/(sqrt(g*x + f)*g^3))/g
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Fricas [A] time = 0.273172, size = 135, normalized size = 1.22 \[ \frac{2 \,{\left (3 \, c e g^{3} x^{3} + 48 \, c e f^{3} - 40 \, c d f^{2} g + 30 \, a e f g^{2} - 15 \, a d g^{3} -{\left (6 \, c e f g^{2} - 5 \, c d g^{3}\right )} x^{2} +{\left (24 \, c e f^{2} g - 20 \, c d f g^{2} + 15 \, a e g^{3}\right )} x\right )}}{15 \, \sqrt{g x + f} g^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + 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 - 40*c*d*f^2*g + 30*a*e*f*g^2 - 15*a*d*g^3 - (6
*c*e*f*g^2 - 5*c*d*g^3)*x^2 + (24*c*e*f^2*g - 20*c*d*f*g^2 + 15*a*e*g^3)*x)/(sqr
t(g*x + f)*g^4)
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Sympy [A] time = 20.2027, size = 2139, normalized size = 19.27 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(c*x**2+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)) + 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 + 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) + 176*f**(4
3/2)*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 + 1
00*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 + 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) - 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**1
6*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**1
7*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) - 48
0*f**(37/2)*g**4*x**4/(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)
+ 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 + 75*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 + 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) + 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**1
0*x**6))
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GIAC/XCAS [A] time = 0.270497, size = 193, normalized size = 1.74 \[ -\frac{2 \,{\left (c d f^{2} g + a d g^{3} - c f^{3} e - a f g^{2} e\right )}}{\sqrt{g x + f} g^{4}} + \frac{2 \,{\left (5 \,{\left (g x + f\right )}^{\frac{3}{2}} c d g^{17} - 30 \, \sqrt{g x + f} c d f g^{17} + 3 \,{\left (g x + f\right )}^{\frac{5}{2}} c g^{16} e - 15 \,{\left (g x + f\right )}^{\frac{3}{2}} c f g^{16} e + 45 \, \sqrt{g x + f} c f^{2} g^{16} e + 15 \, \sqrt{g x + f} a g^{18} e\right )}}{15 \, g^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(e*x + d)/(g*x + f)^(3/2),x, algorithm="giac")
[Out]
-2*(c*d*f^2*g + a*d*g^3 - c*f^3*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 + 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 + 15*sqrt(g*x +
f)*a*g^18*e)/g^20