3.335 \(\int \frac{1}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=348 \[ -\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}+\frac{2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}+\frac{e \sqrt{b x+c x^2} \left (-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \]
[Out]
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(d + e*x)*(b*x + c
*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*c^2*d^2 - 2*b*c*d*e - 5*b^2*e^2) + c*(2*c*d
- b*e)*(8*c^2*d^2 - 8*b*c*d*e - 5*b^2*e^2)*x))/(3*b^4*d^2*(c*d - b*e)^2*(d + e*x
)*Sqrt[b*x + c*x^2]) + (e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 12*b^2*c^2*d^2*e^2 + 20
*b^3*c*d*e^3 - 15*b^4*e^4)*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)
) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b
*e]*Sqrt[b*x + c*x^2])])/(2*d^(7/2)*(c*d - b*e)^(7/2))
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Rubi [A] time = 1.03511, antiderivative size = 348, 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
\[ -\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}+\frac{2 \left (c x (2 c d-b e) \left (-5 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-5 b^2 e^2-2 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}+\frac{e \sqrt{b x+c x^2} \left (-15 b^4 e^4+20 b^3 c d e^3+12 b^2 c^2 d^2 e^2-64 b c^3 d^3 e+32 c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]
[Out]
(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(d + e*x)*(b*x + c
*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*c^2*d^2 - 2*b*c*d*e - 5*b^2*e^2) + c*(2*c*d
- b*e)*(8*c^2*d^2 - 8*b*c*d*e - 5*b^2*e^2)*x))/(3*b^4*d^2*(c*d - b*e)^2*(d + e*x
)*Sqrt[b*x + c*x^2]) + (e*(32*c^4*d^4 - 64*b*c^3*d^3*e + 12*b^2*c^2*d^2*e^2 + 20
*b^3*c*d*e^3 - 15*b^4*e^4)*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)
) + (5*e^4*(2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b
*e]*Sqrt[b*x + c*x^2])])/(2*d^(7/2)*(c*d - b*e)^(7/2))
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Rubi in Sympy [A] time = 169.534, size = 326, normalized size = 0.94 \[ \frac{5 e^{4} \left (\frac{b e}{2} - c d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{7}{2}} \left (b e - c d\right )^{\frac{7}{2}}} - \frac{2 \left (b \left (b e - c d\right ) + c x \left (b e - 2 c d\right )\right )}{3 b^{2} d \left (d + e x\right ) \left (b e - c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (\frac{b \left (b e - c d\right ) \left (5 b^{2} e^{2} + 2 b c d e - 8 c^{2} d^{2}\right )}{2} + \frac{c x \left (b e - 2 c d\right ) \left (5 b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right )}{2}\right )}{3 b^{4} d^{2} \left (d + e x\right ) \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} + \frac{e \sqrt{b x + c x^{2}} \left (15 b^{4} e^{4} - 20 b^{3} c d e^{3} - 12 b^{2} c^{2} d^{2} e^{2} + 64 b c^{3} d^{3} e - 32 c^{4} d^{4}\right )}{3 b^{4} d^{3} \left (d + e x\right ) \left (b e - c d\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(c*x**2+b*x)**(5/2),x)
[Out]
5*e**4*(b*e/2 - c*d)*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sq
rt(b*x + c*x**2)))/(d**(7/2)*(b*e - c*d)**(7/2)) - 2*(b*(b*e - c*d) + c*x*(b*e -
2*c*d))/(3*b**2*d*(d + e*x)*(b*e - c*d)*(b*x + c*x**2)**(3/2)) + 4*(b*(b*e - c*
d)*(5*b**2*e**2 + 2*b*c*d*e - 8*c**2*d**2)/2 + c*x*(b*e - 2*c*d)*(5*b**2*e**2 +
8*b*c*d*e - 8*c**2*d**2)/2)/(3*b**4*d**2*(d + e*x)*(b*e - c*d)**2*sqrt(b*x + c*x
**2)) + e*sqrt(b*x + c*x**2)*(15*b**4*e**4 - 20*b**3*c*d*e**3 - 12*b**2*c**2*d**
2*e**2 + 64*b*c**3*d**3*e - 32*c**4*d**4)/(3*b**4*d**3*(d + e*x)*(b*e - c*d)**3)
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Mathematica [A] time = 1.45666, size = 227, normalized size = 0.65 \[ \frac{x^{5/2} \left (\frac{(b+c x)^3 \left (\frac{4 c^4 x^2 (7 b e-4 c d)}{b^4 (b+c x) (b e-c d)^3}+\frac{4 x (3 b e+4 c d)}{b^4 d^3}+\frac{2 c^4 x^2}{b^3 (b+c x)^2 (c d-b e)^2}-\frac{2}{b^3 d^2}-\frac{3 e^5 x^2}{d^3 (d+e x) (c d-b e)^3}\right )}{3 x^{3/2}}-\frac{5 e^4 (b+c x)^{5/2} (2 c d-b e) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{7/2} (b e-c d)^{7/2}}\right )}{(x (b+c x))^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(5/2)),x]
[Out]
(x^(5/2)*(((b + c*x)^3*(-2/(b^3*d^2) + (4*(4*c*d + 3*b*e)*x)/(b^4*d^3) + (2*c^4*
x^2)/(b^3*(c*d - b*e)^2*(b + c*x)^2) + (4*c^4*(-4*c*d + 7*b*e)*x^2)/(b^4*(-(c*d)
+ b*e)^3*(b + c*x)) - (3*e^5*x^2)/(d^3*(c*d - b*e)^3*(d + e*x))))/(3*x^(3/2)) -
(5*e^4*(2*c*d - b*e)*(b + c*x)^(5/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[
d]*Sqrt[b + c*x])])/(d^(7/2)*(-(c*d) + b*e)^(7/2))))/(x*(b + c*x))^(5/2)
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Maple [B] time = 0.02, size = 1857, normalized size = 5.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(c*x^2+b*x)^(5/2),x)
[Out]
-160/3*e/d/(b*e-c*d)^2*c^3/b^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-20*e^3/d^2/(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^2+20*e^2/d/(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^3+20/3*e/d/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(
d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2*x+40/3*e^2/d^2/(b*e-c*d)^2*c^2/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-16/3*c^2/d/(b*e-c*d)/b^2/(c*(d/
e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*x+128/3*c^3/d/(b*e-c*d)/b^4/
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x+20/3*e^2/d^2/(b*e-c*
d)^2*c/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)+5*e^3/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))*c-5/3*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(3/2)*c*x+5*e^4/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-8/3*c/d/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
-d*(b*e-c*d)/e^2)^(3/2)+80/3/(b*e-c*d)^2*c^3/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(1/2)+1/d/(b*e-c*d)/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(3/2)-10/3/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
-d*(b*e-c*d)/e^2)^(3/2)*c^2-80/3*e/d/(b*e-c*d)^2*c^2/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)+10*e^2/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)*c^2-5/2*e^4/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+64/3*c^2/
d/(b*e-c*d)/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-20/3/(
b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^3*x+1
60/3/(b*e-c*d)^2*c^4/b^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/
2)*x-5/3*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)
^(3/2)*b+5*e/d/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(
3/2)*c+5*e^4/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-15*e^3/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)*c
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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)^(5/2)*(e*x + d)^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.238781, 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)^(5/2)*(e*x + d)^2),x, algorithm="fricas")
[Out]
[1/6*(15*((2*b^4*c^2*d*e^5 - b^5*c*e^6)*x^3 + (2*b^4*c^2*d^2*e^4 + b^5*c*d*e^5 -
b^6*e^6)*x^2 + (2*b^5*c*d^2*e^4 - b^6*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*(2*b^3*c^3*d^5 - 6*b^4*c^2*d^4*e + 6*b^5*c*d^3*e^2 - 2*b^6*d^2*e^3 - (
32*c^6*d^4*e - 64*b*c^5*d^3*e^2 + 12*b^2*c^4*d^2*e^3 + 20*b^3*c^3*d*e^4 - 15*b^4
*c^2*e^5)*x^4 - 2*(16*c^6*d^5 - 8*b*c^5*d^4*e - 42*b^2*c^4*d^3*e^2 + 19*b^3*c^3*
d^2*e^3 + 15*b^4*c^2*d*e^4 - 15*b^5*c*e^5)*x^3 - 3*(16*b*c^5*d^5 - 28*b^2*c^4*d^
4*e - 2*b^3*c^3*d^3*e^2 + 14*b^4*c^2*d^2*e^3 - 5*b^6*e^5)*x^2 - 2*(6*b^2*c^4*d^5
- 13*b^3*c^3*d^4*e + 3*b^4*c^2*d^3*e^2 + 9*b^5*c*d^2*e^3 - 5*b^6*d*e^4)*x)*sqrt
(c*d^2 - b*d*e))/(((b^4*c^4*d^6*e - 3*b^5*c^3*d^5*e^2 + 3*b^6*c^2*d^4*e^3 - b^7*
c*d^3*e^4)*x^3 + (b^4*c^4*d^7 - 2*b^5*c^3*d^6*e + 2*b^7*c*d^4*e^3 - b^8*d^3*e^4)
*x^2 + (b^5*c^3*d^7 - 3*b^6*c^2*d^6*e + 3*b^7*c*d^5*e^2 - b^8*d^4*e^3)*x)*sqrt(c
*d^2 - b*d*e)*sqrt(c*x^2 + b*x)), -1/3*(15*((2*b^4*c^2*d*e^5 - b^5*c*e^6)*x^3 +
(2*b^4*c^2*d^2*e^4 + b^5*c*d*e^5 - b^6*e^6)*x^2 + (2*b^5*c*d^2*e^4 - b^6*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)) + (2*b^3*c^3*d^5 - 6*b^4*c^2*d^4*e + 6*b^5*c*d^3*e^2 - 2*b^6*d^2*e^3 - (32*
c^6*d^4*e - 64*b*c^5*d^3*e^2 + 12*b^2*c^4*d^2*e^3 + 20*b^3*c^3*d*e^4 - 15*b^4*c^
2*e^5)*x^4 - 2*(16*c^6*d^5 - 8*b*c^5*d^4*e - 42*b^2*c^4*d^3*e^2 + 19*b^3*c^3*d^2
*e^3 + 15*b^4*c^2*d*e^4 - 15*b^5*c*e^5)*x^3 - 3*(16*b*c^5*d^5 - 28*b^2*c^4*d^4*e
- 2*b^3*c^3*d^3*e^2 + 14*b^4*c^2*d^2*e^3 - 5*b^6*e^5)*x^2 - 2*(6*b^2*c^4*d^5 -
13*b^3*c^3*d^4*e + 3*b^4*c^2*d^3*e^2 + 9*b^5*c*d^2*e^3 - 5*b^6*d*e^4)*x)*sqrt(-c
*d^2 + b*d*e))/(((b^4*c^4*d^6*e - 3*b^5*c^3*d^5*e^2 + 3*b^6*c^2*d^4*e^3 - b^7*c*
d^3*e^4)*x^3 + (b^4*c^4*d^7 - 2*b^5*c^3*d^6*e + 2*b^7*c*d^4*e^3 - b^8*d^3*e^4)*x
^2 + (b^5*c^3*d^7 - 3*b^6*c^2*d^6*e + 3*b^7*c*d^5*e^2 - b^8*d^4*e^3)*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. \[ \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**2/(c*x**2+b*x)**(5/2),x)
[Out]
Integral(1/((x*(b + c*x))**(5/2)*(d + e*x)**2), x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)^2),x, algorithm="giac")
[Out]
integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)^2), x)