3.13 \(\int \frac{A+B x+C x^2+D x^3}{(c+d x)^{3/2}} \, dx\)
Optimal. Leaf size=113 \[ -\frac{2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^4 \sqrt{c+d x}}-\frac{2 \sqrt{c+d x} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4}+\frac{2 (c+d x)^{3/2} (C d-3 c D)}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4} \]
[Out]
(-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^4*Sqrt[c + d*x]) - (2*(2*c*C*d - B*d
^2 - 3*c^2*D)*Sqrt[c + d*x])/d^4 + (2*(C*d - 3*c*D)*(c + d*x)^(3/2))/(3*d^4) + (
2*D*(c + d*x)^(5/2))/(5*d^4)
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Rubi [A] time = 0.143156, antiderivative size = 113, 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
\[ -\frac{2 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^4 \sqrt{c+d x}}-\frac{2 \sqrt{c+d x} \left (-B d^2-3 c^2 D+2 c C d\right )}{d^4}+\frac{2 (c+d x)^{3/2} (C d-3 c D)}{3 d^4}+\frac{2 D (c+d x)^{5/2}}{5 d^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(3/2),x]
[Out]
(-2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^4*Sqrt[c + d*x]) - (2*(2*c*C*d - B*d
^2 - 3*c^2*D)*Sqrt[c + d*x])/d^4 + (2*(C*d - 3*c*D)*(c + d*x)^(3/2))/(3*d^4) + (
2*D*(c + d*x)^(5/2))/(5*d^4)
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Rubi in Sympy [A] time = 27.8429, size = 114, normalized size = 1.01 \[ \frac{2 D \left (c + d x\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (C d - 3 D c\right )}{3 d^{4}} + \frac{2 \sqrt{c + d x} \left (B d^{2} - 2 C c d + 3 D c^{2}\right )}{d^{4}} - \frac{2 \left (A d^{3} - B c d^{2} + C c^{2} d - D c^{3}\right )}{d^{4} \sqrt{c + d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(3/2),x)
[Out]
2*D*(c + d*x)**(5/2)/(5*d**4) + 2*(c + d*x)**(3/2)*(C*d - 3*D*c)/(3*d**4) + 2*sq
rt(c + d*x)*(B*d**2 - 2*C*c*d + 3*D*c**2)/d**4 - 2*(A*d**3 - B*c*d**2 + C*c**2*d
- D*c**3)/(d**4*sqrt(c + d*x))
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Mathematica [A] time = 0.0889582, size = 82, normalized size = 0.73 \[ \frac{2 \left (d^3 \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+2 c d^2 (15 B-x (10 C+3 D x))+48 c^3 D-8 c^2 d (5 C-3 D x)\right )}{15 d^4 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x + C*x^2 + D*x^3)/(c + d*x)^(3/2),x]
[Out]
(2*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*D*x)) + d^3*(
-15*A + x*(15*B + 5*C*x + 3*D*x^2))))/(15*d^4*Sqrt[c + d*x])
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Maple [A] time = 0.009, size = 91, normalized size = 0.8 \[ -{\frac{-6\,D{x}^{3}{d}^{3}-10\,C{d}^{3}{x}^{2}+12\,Dc{d}^{2}{x}^{2}-30\,B{d}^{3}x+40\,Cc{d}^{2}x-48\,D{c}^{2}dx+30\,A{d}^{3}-60\,Bc{d}^{2}+80\,C{c}^{2}d-96\,D{c}^{3}}{15\,{d}^{4}}{\frac{1}{\sqrt{dx+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((D*x^3+C*x^2+B*x+A)/(d*x+c)^(3/2),x)
[Out]
-2/15/(d*x+c)^(1/2)*(-3*D*d^3*x^3-5*C*d^3*x^2+6*D*c*d^2*x^2-15*B*d^3*x+20*C*c*d^
2*x-24*D*c^2*d*x+15*A*d^3-30*B*c*d^2+40*C*c^2*d-48*D*c^3)/d^4
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Maxima [A] time = 1.35154, size = 138, normalized size = 1.22 \[ \frac{2 \,{\left (\frac{3 \,{\left (d x + c\right )}^{\frac{5}{2}} D - 5 \,{\left (3 \, D c - C d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, D c^{2} - 2 \, C c d + B d^{2}\right )} \sqrt{d x + c}}{d^{3}} + \frac{15 \,{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{\sqrt{d x + c} d^{3}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)/(d*x + c)^(3/2),x, algorithm="maxima")
[Out]
2/15*((3*(d*x + c)^(5/2)*D - 5*(3*D*c - C*d)*(d*x + c)^(3/2) + 15*(3*D*c^2 - 2*C
*c*d + B*d^2)*sqrt(d*x + c))/d^3 + 15*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)/(sqrt(
d*x + c)*d^3))/d
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Fricas [A] time = 0.218068, size = 122, normalized size = 1.08 \[ \frac{2 \,{\left (3 \, D d^{3} x^{3} + 48 \, D c^{3} - 40 \, C c^{2} d + 30 \, B c d^{2} - 15 \, A d^{3} -{\left (6 \, D c d^{2} - 5 \, C d^{3}\right )} x^{2} +{\left (24 \, D c^{2} d - 20 \, C c d^{2} + 15 \, B d^{3}\right )} x\right )}}{15 \, \sqrt{d x + c} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)/(d*x + c)^(3/2),x, algorithm="fricas")
[Out]
2/15*(3*D*d^3*x^3 + 48*D*c^3 - 40*C*c^2*d + 30*B*c*d^2 - 15*A*d^3 - (6*D*c*d^2 -
5*C*d^3)*x^2 + (24*D*c^2*d - 20*C*c*d^2 + 15*B*d^3)*x)/(sqrt(d*x + c)*d^4)
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Sympy [A] time = 19.3877, size = 2132, normalized size = 18.87 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x**3+C*x**2+B*x+A)/(d*x+c)**(3/2),x)
[Out]
-2*A/(d*sqrt(c + d*x)) + B*Piecewise((4*c/(d**2*sqrt(c + d*x)) + 2*x/(d*sqrt(c +
d*x)), Ne(d, 0)), (x**2/(2*c**(3/2)), True)) + C*(-16*c**(19/2)*sqrt(1 + d*x/c)
/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) + 16*c**(19
/2)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) - 40*c**
(17/2)*d*x*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c
**5*d**6*x**3) + 48*c**(17/2)*d*x/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**
2 + 3*c**5*d**6*x**3) - 30*c**(15/2)*d**2*x**2*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*
c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) + 48*c**(15/2)*d**2*x**2/(3*c
**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3) - 4*c**(13/2)*d*
*3*x**3*sqrt(1 + d*x/c)/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5
*d**6*x**3) + 16*c**(13/2)*d**3*x**3/(3*c**8*d**3 + 9*c**7*d**4*x + 9*c**6*d**5*
x**2 + 3*c**5*d**6*x**3) + 2*c**(11/2)*d**4*x**4*sqrt(1 + d*x/c)/(3*c**8*d**3 +
9*c**7*d**4*x + 9*c**6*d**5*x**2 + 3*c**5*d**6*x**3)) + D*(32*c**(45/2)*sqrt(1 +
d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x*
*3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 32*c**(45/2
)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 7
5*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 176*c**(43/2)*d*x
*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**1
7*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 19
2*c**(43/2)*d*x/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17
*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 396
*c**(41/2)*d**2*x**2*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*
d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c*
*14*d**10*x**6) - 480*c**(41/2)*d**2*x**2/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c
**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 +
5*c**14*d**10*x**6) + 462*c**(39/2)*d**3*x**3*sqrt(1 + d*x/c)/(5*c**20*d**4 + 3
0*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 +
30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 640*c**(39/2)*d**3*x**3/(5*c**20*d**
4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x
**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 290*c**(37/2)*d**4*x**4*sqrt(1
+ d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x
**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 480*c**(37
/2)*d**4*x**4/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d
**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 92*c*
*(35/2)*d**5*x**5*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**
6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14
*d**10*x**6) - 192*c**(35/2)*d**5*x**5/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**1
8*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*
c**14*d**10*x**6) + 16*c**(33/2)*d**6*x**6*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c*
*19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*
c**15*d**9*x**5 + 5*c**14*d**10*x**6) - 32*c**(33/2)*d**6*x**6/(5*c**20*d**4 + 3
0*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 75*c**16*d**8*x**4 +
30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 6*c**(31/2)*d**7*x**7*sqrt(1 + d*x/c
)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*c**17*d**7*x**3 + 7
5*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6) + 2*c**(29/2)*d**8*
x**8*sqrt(1 + d*x/c)/(5*c**20*d**4 + 30*c**19*d**5*x + 75*c**18*d**6*x**2 + 100*
c**17*d**7*x**3 + 75*c**16*d**8*x**4 + 30*c**15*d**9*x**5 + 5*c**14*d**10*x**6))
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GIAC/XCAS [A] time = 0.211352, size = 171, normalized size = 1.51 \[ \frac{2 \,{\left (D c^{3} - C c^{2} d + B c d^{2} - A d^{3}\right )}}{\sqrt{d x + c} d^{4}} + \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} D d^{16} - 15 \,{\left (d x + c\right )}^{\frac{3}{2}} D c d^{16} + 45 \, \sqrt{d x + c} D c^{2} d^{16} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} C d^{17} - 30 \, \sqrt{d x + c} C c d^{17} + 15 \, \sqrt{d x + c} B d^{18}\right )}}{15 \, d^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)/(d*x + c)^(3/2),x, algorithm="giac")
[Out]
2*(D*c^3 - C*c^2*d + B*c*d^2 - A*d^3)/(sqrt(d*x + c)*d^4) + 2/15*(3*(d*x + c)^(5
/2)*D*d^16 - 15*(d*x + c)^(3/2)*D*c*d^16 + 45*sqrt(d*x + c)*D*c^2*d^16 + 5*(d*x
+ c)^(3/2)*C*d^17 - 30*sqrt(d*x + c)*C*c*d^17 + 15*sqrt(d*x + c)*B*d^18)/d^20