3.334 \(\int \frac{1}{(d+e x) \left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=230 \[ -\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac{2 \left (c x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}+\frac{e^4 \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]

[Out]

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(3*b^2*d*(c*d - b*e)*(b*x + c*x^2)^(3/2
)) + (2*(b*(c*d - b*e)*(8*c^2*d^2 - 4*b*c*d*e - 3*b^2*e^2) + c*(2*c*d - b*e)*(8*
c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2)*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[b*x + c*x^2])
+ (e^4*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*
x^2])])/(d^(5/2)*(c*d - b*e)^(5/2))

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Rubi [A]  time = 0.55106, antiderivative size = 230, 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 \[ -\frac{2 (c x (2 c d-b e)+b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}+\frac{2 \left (c x (2 c d-b e) \left (-3 b^2 e^2-8 b c d e+8 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}+\frac{e^4 \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(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)*(b*x + c*x^2)^(3/2
)) + (2*(b*(c*d - b*e)*(8*c^2*d^2 - 4*b*c*d*e - 3*b^2*e^2) + c*(2*c*d - b*e)*(8*
c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2)*x))/(3*b^4*d^2*(c*d - b*e)^2*Sqrt[b*x + c*x^2])
+ (e^4*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*
x^2])])/(d^(5/2)*(c*d - b*e)^(5/2))

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Rubi in Sympy [A]  time = 89.5808, size = 214, normalized size = 0.93 \[ \frac{e^{4} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{5}{2}} \left (b e - c d\right )^{\frac{5}{2}}} - \frac{2 \left (b \left (b e - c d\right ) + c x \left (b e - 2 c d\right )\right )}{3 b^{2} d \left (b e - c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (\frac{b \left (b e - c d\right ) \left (3 b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right )}{2} + \frac{c x \left (b e - 2 c d\right ) \left (3 b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right )}{2}\right )}{3 b^{4} d^{2} \left (b e - c d\right )^{2} \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+b*x)**(5/2),x)

[Out]

e**4*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)
))/(d**(5/2)*(b*e - c*d)**(5/2)) - 2*(b*(b*e - c*d) + c*x*(b*e - 2*c*d))/(3*b**2
*d*(b*e - c*d)*(b*x + c*x**2)**(3/2)) + 4*(b*(b*e - c*d)*(3*b**2*e**2 + 4*b*c*d*
e - 8*c**2*d**2)/2 + c*x*(b*e - 2*c*d)*(3*b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2)/2
)/(3*b**4*d**2*(b*e - c*d)**2*sqrt(b*x + c*x**2))

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Mathematica [A]  time = 1.25686, size = 180, normalized size = 0.78 \[ \frac{x^{5/2} \left (\frac{2 (b+c x)^3 \left (\frac{c^3 x^2 (8 c d-11 b e)}{(b+c x) (c d-b e)^2}+\frac{b c^3 x^2}{(b+c x)^2 (c d-b e)}+\frac{x (3 b e+8 c d)}{d^2}-\frac{b}{d}\right )}{3 b^4 x^{3/2}}+\frac{2 e^4 (b+c x)^{5/2} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{5/2} (b e-c d)^{5/2}}\right )}{(x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(b*x + c*x^2)^(5/2)),x]

[Out]

(x^(5/2)*((2*(b + c*x)^3*(-(b/d) + ((8*c*d + 3*b*e)*x)/d^2 + (b*c^3*x^2)/((c*d -
b*e)*(b + c*x)^2) + (c^3*(8*c*d - 11*b*e)*x^2)/((c*d - b*e)^2*(b + c*x))))/(3*b
^4*x^(3/2)) + (2*e^4*(b + c*x)^(5/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d
]*Sqrt[b + c*x])])/(d^(5/2)*(-(c*d) + b*e)^(5/2))))/(x*(b + c*x))^(5/2)

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Maple [B]  time = 0.014, size = 950, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+b*x)^(5/2),x)

[Out]

-2/3*e/d/(b*e-c*d)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)-2/3
*e/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)*c*x+4
/3/(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)*c^2*x
+2/3/(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)*c+16/
3*e/d/(b*e-c*d)*c^2/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-32/3/(b*e-c*d)*c^3/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+8/3*e/d/(b*e-c*d)*c/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)-16/3/(b*e-c*d)*c^2/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)+2*e^3/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)^(1/2)+2*e^3/d^2/(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)^(1/2)*x*c-4*e^2/d/(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)^(1/2)*x*c^2-2*e^2/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)^(1/2)*c-e^3/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)^(5/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230726, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (b^{4} c e^{4} x^{2} + b^{5} e^{4} x\right )} \sqrt{c x^{2} + b x} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right ) - 2 \,{\left (b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} -{\left (16 \, c^{5} d^{3} - 24 \, b c^{4} d^{2} e + 2 \, b^{2} c^{3} d e^{2} + 3 \, b^{3} c^{2} e^{3}\right )} x^{3} - 3 \,{\left (8 \, b c^{4} d^{3} - 12 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2} + 2 \, b^{4} c e^{3}\right )} x^{2} - 3 \,{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e}}{3 \, \sqrt{c d^{2} - b d e}{\left ({\left (b^{4} c^{3} d^{4} - 2 \, b^{5} c^{2} d^{3} e + b^{6} c d^{2} e^{2}\right )} x^{2} +{\left (b^{5} c^{2} d^{4} - 2 \, b^{6} c d^{3} e + b^{7} d^{2} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}, -\frac{2 \,{\left (3 \,{\left (b^{4} c e^{4} x^{2} + b^{5} e^{4} x\right )} \sqrt{c x^{2} + b x} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) +{\left (b^{3} c^{2} d^{3} - 2 \, b^{4} c d^{2} e + b^{5} d e^{2} -{\left (16 \, c^{5} d^{3} - 24 \, b c^{4} d^{2} e + 2 \, b^{2} c^{3} d e^{2} + 3 \, b^{3} c^{2} e^{3}\right )} x^{3} - 3 \,{\left (8 \, b c^{4} d^{3} - 12 \, b^{2} c^{3} d^{2} e + b^{3} c^{2} d e^{2} + 2 \, b^{4} c e^{3}\right )} x^{2} - 3 \,{\left (2 \, b^{2} c^{3} d^{3} - 3 \, b^{3} c^{2} d^{2} e + b^{5} e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e}\right )}}{3 \, \sqrt{-c d^{2} + b d e}{\left ({\left (b^{4} c^{3} d^{4} - 2 \, b^{5} c^{2} d^{3} e + b^{6} c d^{2} e^{2}\right )} x^{2} +{\left (b^{5} c^{2} d^{4} - 2 \, b^{6} c d^{3} e + b^{7} d^{2} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)),x, algorithm="fricas")

[Out]

[1/3*(3*(b^4*c*e^4*x^2 + b^5*e^4*x)*sqrt(c*x^2 + b*x)*log((2*(c*d^2 - b*d*e)*sqr
t(c*x^2 + b*x) + sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)) - 2*(b^
3*c^2*d^3 - 2*b^4*c*d^2*e + b^5*d*e^2 - (16*c^5*d^3 - 24*b*c^4*d^2*e + 2*b^2*c^3
*d*e^2 + 3*b^3*c^2*e^3)*x^3 - 3*(8*b*c^4*d^3 - 12*b^2*c^3*d^2*e + b^3*c^2*d*e^2
+ 2*b^4*c*e^3)*x^2 - 3*(2*b^2*c^3*d^3 - 3*b^3*c^2*d^2*e + b^5*e^3)*x)*sqrt(c*d^2
- b*d*e))/(sqrt(c*d^2 - b*d*e)*((b^4*c^3*d^4 - 2*b^5*c^2*d^3*e + b^6*c*d^2*e^2)
*x^2 + (b^5*c^2*d^4 - 2*b^6*c*d^3*e + b^7*d^2*e^2)*x)*sqrt(c*x^2 + b*x)), -2/3*(
3*(b^4*c*e^4*x^2 + b^5*e^4*x)*sqrt(c*x^2 + b*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqr
t(c*x^2 + b*x)/((c*d - b*e)*x)) + (b^3*c^2*d^3 - 2*b^4*c*d^2*e + b^5*d*e^2 - (16
*c^5*d^3 - 24*b*c^4*d^2*e + 2*b^2*c^3*d*e^2 + 3*b^3*c^2*e^3)*x^3 - 3*(8*b*c^4*d^
3 - 12*b^2*c^3*d^2*e + b^3*c^2*d*e^2 + 2*b^4*c*e^3)*x^2 - 3*(2*b^2*c^3*d^3 - 3*b
^3*c^2*d^2*e + b^5*e^3)*x)*sqrt(-c*d^2 + b*d*e))/(sqrt(-c*d^2 + b*d*e)*((b^4*c^3
*d^4 - 2*b^5*c^2*d^3*e + b^6*c*d^2*e^2)*x^2 + (b^5*c^2*d^4 - 2*b^6*c*d^3*e + b^7
*d^2*e^2)*x)*sqrt(c*x^2 + b*x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral(1/((x*(b + c*x))**(5/2)*(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.223924, size = 562, normalized size = 2.44 \[ -\frac{2 \, \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right ) e^{4}}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt{-c d^{2} + b d e}} - \frac{{\left (x{\left (\frac{{\left (16 \, c^{7} d^{10} - 56 \, b c^{6} d^{9} e + 66 \, b^{2} c^{5} d^{8} e^{2} - 25 \, b^{3} c^{4} d^{7} e^{3} - 4 \, b^{4} c^{3} d^{6} e^{4} + 3 \, b^{5} c^{2} d^{5} e^{5}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (8 \, b c^{6} d^{10} - 28 \, b^{2} c^{5} d^{9} e + 33 \, b^{3} c^{4} d^{8} e^{2} - 12 \, b^{4} c^{3} d^{7} e^{3} - 3 \, b^{5} c^{2} d^{6} e^{4} + 2 \, b^{6} c d^{5} e^{5}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (2 \, b^{2} c^{5} d^{10} - 7 \, b^{3} c^{4} d^{9} e + 8 \, b^{4} c^{3} d^{8} e^{2} - 2 \, b^{5} c^{2} d^{7} e^{3} - 2 \, b^{6} c d^{6} e^{4} + b^{7} d^{5} e^{5}\right )}}{b^{4} c^{2}}\right )} x - \frac{b^{3} c^{4} d^{10} - 4 \, b^{4} c^{3} d^{9} e + 6 \, b^{5} c^{2} d^{8} e^{2} - 4 \, b^{6} c d^{7} e^{3} + b^{7} d^{6} e^{4}}{b^{4} c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

-2*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))*
e^4/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) - 1/3*((x*((16*
c^7*d^10 - 56*b*c^6*d^9*e + 66*b^2*c^5*d^8*e^2 - 25*b^3*c^4*d^7*e^3 - 4*b^4*c^3*
d^6*e^4 + 3*b^5*c^2*d^5*e^5)*x/(b^4*c^2) + 3*(8*b*c^6*d^10 - 28*b^2*c^5*d^9*e +
33*b^3*c^4*d^8*e^2 - 12*b^4*c^3*d^7*e^3 - 3*b^5*c^2*d^6*e^4 + 2*b^6*c*d^5*e^5)/(
b^4*c^2)) + 3*(2*b^2*c^5*d^10 - 7*b^3*c^4*d^9*e + 8*b^4*c^3*d^8*e^2 - 2*b^5*c^2*
d^7*e^3 - 2*b^6*c*d^6*e^4 + b^7*d^5*e^5)/(b^4*c^2))*x - (b^3*c^4*d^10 - 4*b^4*c^
3*d^9*e + 6*b^5*c^2*d^8*e^2 - 4*b^6*c*d^7*e^3 + b^7*d^6*e^4)/(b^4*c^2))/(c*x^2 +
b*x)^(3/2)