3.328 \(\int \frac{1}{(d+e x)^3 \left (b x+c x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=296 \[ -\frac{e \sqrt{b x+c x^2} \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}+\frac{3 e^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{e \sqrt{b x+c x^2} (2 c d-b e) \left (15 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)} \]
[Out]
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(d + e*x)^2*Sqrt[b*x
+ c*x^2]) - (e*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2)*Sqrt[b*x + c*x^2])/(2*b^2*d^
2*(c*d - b*e)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 15*b^2*
e^2)*Sqrt[b*x + c*x^2])/(4*b^2*d^3*(c*d - b*e)^3*(d + e*x)) + (3*e^2*(16*c^2*d^2
- 16*b*c*d*e + 5*b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d -
b*e]*Sqrt[b*x + c*x^2])])/(8*d^(7/2)*(c*d - b*e)^(7/2))
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Rubi [A] time = 0.991379, antiderivative size = 296, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238
\[ -\frac{e \sqrt{b x+c x^2} \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}+\frac{3 e^2 \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{e \sqrt{b x+c x^2} (2 c d-b e) \left (15 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3}-\frac{2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^3*(b*x + c*x^2)^(3/2)),x]
[Out]
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(d + e*x)^2*Sqrt[b*x
+ c*x^2]) - (e*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2)*Sqrt[b*x + c*x^2])/(2*b^2*d^
2*(c*d - b*e)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(8*c^2*d^2 - 8*b*c*d*e + 15*b^2*
e^2)*Sqrt[b*x + c*x^2])/(4*b^2*d^3*(c*d - b*e)^3*(d + e*x)) + (3*e^2*(16*c^2*d^2
- 16*b*c*d*e + 5*b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d -
b*e]*Sqrt[b*x + c*x^2])])/(8*d^(7/2)*(c*d - b*e)^(7/2))
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Rubi in Sympy [A] time = 120.164, size = 279, normalized size = 0.94 \[ - \frac{3 e^{2} \left (5 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{8 d^{\frac{7}{2}} \left (b e - c d\right )^{\frac{7}{2}}} - \frac{2 \left (b \left (b e - c d\right ) + c x \left (b e - 2 c d\right )\right )}{b^{2} d \left (d + e x\right )^{2} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} - \frac{e \sqrt{b x + c x^{2}} \left (5 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{2 b^{2} d^{2} \left (d + e x\right )^{2} \left (b e - c d\right )^{2}} - \frac{e \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}} \left (15 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right )}{4 b^{2} d^{3} \left (d + e x\right ) \left (b e - c d\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**3/(c*x**2+b*x)**(3/2),x)
[Out]
-3*e**2*(5*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)*atan((-b*d + x*(b*e - 2*c*d))/
(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)))/(8*d**(7/2)*(b*e - c*d)**(7/2))
- 2*(b*(b*e - c*d) + c*x*(b*e - 2*c*d))/(b**2*d*(d + e*x)**2*(b*e - c*d)*sqrt(b*
x + c*x**2)) - e*sqrt(b*x + c*x**2)*(5*b**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)/(2*b
**2*d**2*(d + e*x)**2*(b*e - c*d)**2) - e*(b*e - 2*c*d)*sqrt(b*x + c*x**2)*(15*b
**2*e**2 - 8*b*c*d*e + 8*c**2*d**2)/(4*b**2*d**3*(d + e*x)*(b*e - c*d)**3)
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Mathematica [A] time = 1.17178, size = 216, normalized size = 0.73 \[ \frac{x^{3/2} \left (\frac{(b+c x)^2 \left (\frac{8 c^4 x}{b^2 (b+c x) (b e-c d)^3}-\frac{8}{b^2 d^3}+\frac{7 e^3 x (b e-2 c d)}{d^3 (d+e x) (c d-b e)^3}-\frac{2 e^3 x}{d^2 (d+e x)^2 (c d-b e)^2}\right )}{\sqrt{x}}-\frac{3 e^2 (b+c x)^{3/2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{7/2} (b e-c d)^{7/2}}\right )}{4 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^3*(b*x + c*x^2)^(3/2)),x]
[Out]
(x^(3/2)*(((b + c*x)^2*(-8/(b^2*d^3) + (8*c^4*x)/(b^2*(-(c*d) + b*e)^3*(b + c*x)
) - (2*e^3*x)/(d^2*(c*d - b*e)^2*(d + e*x)^2) + (7*e^3*(-2*c*d + b*e)*x)/(d^3*(c
*d - b*e)^3*(d + e*x))))/Sqrt[x] - (3*e^2*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2)*
(b + c*x)^(3/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(d
^(7/2)*(-(c*d) + b*e)^(7/2))))/(4*(x*(b + c*x))^(3/2))
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Maple [B] time = 0.019, size = 1612, normalized size = 5.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^3/(c*x^2+b*x)^(3/2),x)
[Out]
1/2/e/d/(b*e-c*d)/(d/e+x)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(1/2)+5/4*e/d^2/(b*e-c*d)^2/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*
d)/e^2)^(1/2)*b-5/2/d/(b*e-c*d)^2/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(
b*e-c*d)/e^2)^(1/2)*c-15/4*e^3/d^3/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)-d*(b*e-c*d)/e^2)^(1/2)*b^2+75/4*e^2/d^2/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e
*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*c-30*e/d/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2-15/4*e^3/d^3/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c*b+45/2*e^2/d^2/(b*e-c*d)^3/(c*(d/e+x
)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2-45*e/d/(b*e-c*d)^3/b/(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^3+30/(b*e-c*d)^3/b^2/(
c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^4+15/(b*e-c*d)^3/b/
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^3+15/8*e^3/d^3/(b*e-
c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(
-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2
))/(d/e+x))*b^2-15/2*e^2/d^2/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-
c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*
d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*c+15/2*e/d/(b*e-c*d)^3/(-d*(b*e-
c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2
)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^2-
13*e/d^2/(b*e-c*d)^2*c^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(
1/2)*x+26/d/(b*e-c*d)^2*c^3/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e
^2)^(1/2)*x-8*e/d^2/(b*e-c*d)^2*c/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)
/e^2)^(1/2)+13/d/(b*e-c*d)^2*c^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(1/2)+3/2*e*c/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)
/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e
*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.244903, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^3),x, algorithm="fricas")
[Out]
[-1/8*(3*(16*b^2*c^2*d^4*e^2 - 16*b^3*c*d^3*e^3 + 5*b^4*d^2*e^4 + (16*b^2*c^2*d^
2*e^4 - 16*b^3*c*d*e^5 + 5*b^4*e^6)*x^2 + 2*(16*b^2*c^2*d^3*e^3 - 16*b^3*c*d^2*e
^4 + 5*b^4*d*e^5)*x)*sqrt(c*x^2 + b*x)*log(-(2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)
- sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)) + 2*(8*b*c^3*d^5 - 24
*b^2*c^2*d^4*e + 24*b^3*c*d^3*e^2 - 8*b^4*d^2*e^3 + (16*c^4*d^3*e^2 - 24*b*c^3*d
^2*e^3 + 38*b^2*c^2*d*e^4 - 15*b^3*c*e^5)*x^3 + (32*c^4*d^4*e - 40*b*c^3*d^3*e^2
+ 40*b^2*c^2*d^2*e^3 + 13*b^3*c*d*e^4 - 15*b^4*e^5)*x^2 + (16*c^4*d^5 - 8*b*c^3
*d^4*e - 24*b^2*c^2*d^3*e^2 + 56*b^3*c*d^2*e^3 - 25*b^4*d*e^4)*x)*sqrt(c*d^2 - b
*d*e))/((b^2*c^3*d^8 - 3*b^3*c^2*d^7*e + 3*b^4*c*d^6*e^2 - b^5*d^5*e^3 + (b^2*c^
3*d^6*e^2 - 3*b^3*c^2*d^5*e^3 + 3*b^4*c*d^4*e^4 - b^5*d^3*e^5)*x^2 + 2*(b^2*c^3*
d^7*e - 3*b^3*c^2*d^6*e^2 + 3*b^4*c*d^5*e^3 - b^5*d^4*e^4)*x)*sqrt(c*d^2 - b*d*e
)*sqrt(c*x^2 + b*x)), -1/4*(3*(16*b^2*c^2*d^4*e^2 - 16*b^3*c*d^3*e^3 + 5*b^4*d^2
*e^4 + (16*b^2*c^2*d^2*e^4 - 16*b^3*c*d*e^5 + 5*b^4*e^6)*x^2 + 2*(16*b^2*c^2*d^3
*e^3 - 16*b^3*c*d^2*e^4 + 5*b^4*d*e^5)*x)*sqrt(c*x^2 + b*x)*arctan(-sqrt(-c*d^2
+ b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + (8*b*c^3*d^5 - 24*b^2*c^2*d^4*e +
24*b^3*c*d^3*e^2 - 8*b^4*d^2*e^3 + (16*c^4*d^3*e^2 - 24*b*c^3*d^2*e^3 + 38*b^2*c
^2*d*e^4 - 15*b^3*c*e^5)*x^3 + (32*c^4*d^4*e - 40*b*c^3*d^3*e^2 + 40*b^2*c^2*d^2
*e^3 + 13*b^3*c*d*e^4 - 15*b^4*e^5)*x^2 + (16*c^4*d^5 - 8*b*c^3*d^4*e - 24*b^2*c
^2*d^3*e^2 + 56*b^3*c*d^2*e^3 - 25*b^4*d*e^4)*x)*sqrt(-c*d^2 + b*d*e))/((b^2*c^3
*d^8 - 3*b^3*c^2*d^7*e + 3*b^4*c*d^6*e^2 - b^5*d^5*e^3 + (b^2*c^3*d^6*e^2 - 3*b^
3*c^2*d^5*e^3 + 3*b^4*c*d^4*e^4 - b^5*d^3*e^5)*x^2 + 2*(b^2*c^3*d^7*e - 3*b^3*c^
2*d^6*e^2 + 3*b^4*c*d^5*e^3 - b^5*d^4*e^4)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 +
b*x))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**3/(c*x**2+b*x)**(3/2),x)
[Out]
Integral(1/((x*(b + c*x))**(3/2)*(d + e*x)**3), x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^(3/2)*(e*x + d)^3),x, algorithm="giac")
[Out]
Exception raised: TypeError