3.1218 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx\)
Optimal. Leaf size=122 \[ -\frac{2 (d+e x)^{3/2} (-A c e-b B e+3 B c d)}{3 e^4}+\frac{2 \sqrt{d+e x} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac{2 d (B d-A e) (c d-b e)}{e^4 \sqrt{d+e x}}+\frac{2 B c (d+e x)^{5/2}}{5 e^4} \]
[Out]
(2*d*(B*d - A*e)*(c*d - b*e))/(e^4*Sqrt[d + e*x]) + (2*(B*d*(3*c*d - 2*b*e) - A*
e*(2*c*d - b*e))*Sqrt[d + e*x])/e^4 - (2*(3*B*c*d - b*B*e - A*c*e)*(d + e*x)^(3/
2))/(3*e^4) + (2*B*c*(d + e*x)^(5/2))/(5*e^4)
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Rubi [A] time = 0.207361, antiderivative size = 122, 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
\[ -\frac{2 (d+e x)^{3/2} (-A c e-b B e+3 B c d)}{3 e^4}+\frac{2 \sqrt{d+e x} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac{2 d (B d-A e) (c d-b e)}{e^4 \sqrt{d+e x}}+\frac{2 B c (d+e x)^{5/2}}{5 e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2))/(d + e*x)^(3/2),x]
[Out]
(2*d*(B*d - A*e)*(c*d - b*e))/(e^4*Sqrt[d + e*x]) + (2*(B*d*(3*c*d - 2*b*e) - A*
e*(2*c*d - b*e))*Sqrt[d + e*x])/e^4 - (2*(3*B*c*d - b*B*e - A*c*e)*(d + e*x)^(3/
2))/(3*e^4) + (2*B*c*(d + e*x)^(5/2))/(5*e^4)
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Rubi in Sympy [A] time = 33.0276, size = 124, normalized size = 1.02 \[ \frac{2 B c \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{2 d \left (A e - B d\right ) \left (b e - c d\right )}{e^{4} \sqrt{d + e x}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A c e + B b e - 3 B c d\right )}{3 e^{4}} + \frac{2 \sqrt{d + e x} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)/(e*x+d)**(3/2),x)
[Out]
2*B*c*(d + e*x)**(5/2)/(5*e**4) + 2*d*(A*e - B*d)*(b*e - c*d)/(e**4*sqrt(d + e*x
)) + 2*(d + e*x)**(3/2)*(A*c*e + B*b*e - 3*B*c*d)/(3*e**4) + 2*sqrt(d + e*x)*(A*
b*e**2 - 2*A*c*d*e - 2*B*b*d*e + 3*B*c*d**2)/e**4
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Mathematica [A] time = 0.124331, size = 110, normalized size = 0.9 \[ \frac{2 \left (5 A e \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+B \left (5 b e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 c \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{15 e^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2))/(d + e*x)^(3/2),x]
[Out]
(2*(5*A*e*(3*b*e*(2*d + e*x) + c*(-8*d^2 - 4*d*e*x + e^2*x^2)) + B*(5*b*e*(-8*d^
2 - 4*d*e*x + e^2*x^2) + 3*c*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3))))/(15
*e^4*Sqrt[d + e*x])
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Maple [A] time = 0.008, size = 121, normalized size = 1. \[{\frac{6\,Bc{x}^{3}{e}^{3}+10\,Ac{e}^{3}{x}^{2}+10\,Bb{e}^{3}{x}^{2}-12\,Bcd{e}^{2}{x}^{2}+30\,Ab{e}^{3}x-40\,Acd{e}^{2}x-40\,Bbd{e}^{2}x+48\,Bc{d}^{2}ex+60\,Abd{e}^{2}-80\,Ac{d}^{2}e-80\,Bb{d}^{2}e+96\,Bc{d}^{3}}{15\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)/(e*x+d)^(3/2),x)
[Out]
2/15*(3*B*c*e^3*x^3+5*A*c*e^3*x^2+5*B*b*e^3*x^2-6*B*c*d*e^2*x^2+15*A*b*e^3*x-20*
A*c*d*e^2*x-20*B*b*d*e^2*x+24*B*c*d^2*e*x+30*A*b*d*e^2-40*A*c*d^2*e-40*B*b*d^2*e
+48*B*c*d^3)/(e*x+d)^(1/2)/e^4
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Maxima [A] time = 0.701602, size = 162, normalized size = 1.33 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c - 5 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )}}{\sqrt{e x + d} e^{3}}\right )}}{15 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
2/15*((3*(e*x + d)^(5/2)*B*c - 5*(3*B*c*d - (B*b + A*c)*e)*(e*x + d)^(3/2) + 15*
(3*B*c*d^2 + A*b*e^2 - 2*(B*b + A*c)*d*e)*sqrt(e*x + d))/e^3 + 15*(B*c*d^3 + A*b
*d*e^2 - (B*b + A*c)*d^2*e)/(sqrt(e*x + d)*e^3))/e
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Fricas [A] time = 0.309061, size = 146, normalized size = 1.2 \[ \frac{2 \,{\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} + 30 \, A b d e^{2} - 40 \,{\left (B b + A c\right )} d^{2} e -{\left (6 \, B c d e^{2} - 5 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e + 15 \, A b e^{3} - 20 \,{\left (B b + A c\right )} d e^{2}\right )} x\right )}}{15 \, \sqrt{e x + d} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
2/15*(3*B*c*e^3*x^3 + 48*B*c*d^3 + 30*A*b*d*e^2 - 40*(B*b + A*c)*d^2*e - (6*B*c*
d*e^2 - 5*(B*b + A*c)*e^3)*x^2 + (24*B*c*d^2*e + 15*A*b*e^3 - 20*(B*b + A*c)*d*e
^2)*x)/(sqrt(e*x + d)*e^4)
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Sympy [A] time = 23.3714, size = 2662, normalized size = 21.82 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)/(e*x+d)**(3/2),x)
[Out]
A*b*Piecewise((4*d/(e**2*sqrt(d + e*x)) + 2*x/(e*sqrt(d + e*x)), Ne(e, 0)), (x**
2/(2*d**(3/2)), True)) + A*c*(-16*d**(19/2)*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**
7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 16*d**(19/2)/(3*d**8*e**3 + 9*
d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 40*d**(17/2)*e*x*sqrt(1 + e
*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 48*d
**(17/2)*e*x/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3)
- 30*d**(15/2)*e**2*x**2*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*
e**5*x**2 + 3*d**5*e**6*x**3) + 48*d**(15/2)*e**2*x**2/(3*d**8*e**3 + 9*d**7*e**
4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 4*d**(13/2)*e**3*x**3*sqrt(1 + e*x/
d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 16*d**(
13/2)*e**3*x**3/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x*
*3) + 2*d**(11/2)*e**4*x**4*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**
6*e**5*x**2 + 3*d**5*e**6*x**3)) + B*b*(-16*d**(19/2)*sqrt(1 + e*x/d)/(3*d**8*e*
*3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 16*d**(19/2)/(3*d**8
*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 40*d**(17/2)*e*x*
sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x*
*3) + 48*d**(17/2)*e*x/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*
e**6*x**3) - 30*d**(15/2)*e**2*x**2*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x
+ 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 48*d**(15/2)*e**2*x**2/(3*d**8*e**3 +
9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 4*d**(13/2)*e**3*x**3*sqr
t(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3)
+ 16*d**(13/2)*e**3*x**3/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d*
*5*e**6*x**3) + 2*d**(11/2)*e**4*x**4*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4
*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3)) + B*c*(32*d**(45/2)*sqrt(1 + e*x/d)/(
5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d
**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 32*d**(45/2)/(5*d**2
0*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e
**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 176*d**(43/2)*e*x*sqrt(1 +
e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x*
*3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 192*d**(43/
2)*e*x/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**
3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 396*d**(41/2
)*e**2*x**2*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2
+ 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10
*x**6) - 480*d**(41/2)*e**2*x**2/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6
*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*
e**10*x**6) + 462*d**(39/2)*e**3*x**3*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e
**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15
*e**9*x**5 + 5*d**14*e**10*x**6) - 640*d**(39/2)*e**3*x**3/(5*d**20*e**4 + 30*d*
*19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*
d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 290*d**(37/2)*e**4*x**4*sqrt(1 + e*x/d)/
(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*
d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 480*d**(37/2)*e**4*
x**4/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3
+ 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 92*d**(35/2)*e
**5*x**5*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 +
100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x*
*6) - 192*d**(35/2)*e**5*x**5/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x*
*2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**
10*x**6) + 16*d**(33/2)*e**6*x**6*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*
x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**
9*x**5 + 5*d**14*e**10*x**6) - 32*d**(33/2)*e**6*x**6/(5*d**20*e**4 + 30*d**19*e
**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15
*e**9*x**5 + 5*d**14*e**10*x**6) + 6*d**(31/2)*e**7*x**7*sqrt(1 + e*x/d)/(5*d**2
0*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e
**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 2*d**(29/2)*e**8*x**8*sqrt
(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**
7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6))
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GIAC/XCAS [A] time = 0.277425, size = 225, normalized size = 1.84 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c e^{16} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B c d e^{16} + 45 \, \sqrt{x e + d} B c d^{2} e^{16} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} B b e^{17} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c e^{17} - 30 \, \sqrt{x e + d} B b d e^{17} - 30 \, \sqrt{x e + d} A c d e^{17} + 15 \, \sqrt{x e + d} A b e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B c d^{3} - B b d^{2} e - A c d^{2} e + A b d e^{2}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]
2/15*(3*(x*e + d)^(5/2)*B*c*e^16 - 15*(x*e + d)^(3/2)*B*c*d*e^16 + 45*sqrt(x*e +
d)*B*c*d^2*e^16 + 5*(x*e + d)^(3/2)*B*b*e^17 + 5*(x*e + d)^(3/2)*A*c*e^17 - 30*
sqrt(x*e + d)*B*b*d*e^17 - 30*sqrt(x*e + d)*A*c*d*e^17 + 15*sqrt(x*e + d)*A*b*e^
18)*e^(-20) + 2*(B*c*d^3 - B*b*d^2*e - A*c*d^2*e + A*b*d*e^2)*e^(-4)/sqrt(x*e +
d)