3.111 \(\int \frac{1}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=145 \[ \frac{35 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{9/2}}-\frac{35 c^3 \sqrt{x}}{8 b^4 \sqrt{b x+c x^2}}-\frac{35 c^2}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{7 c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}-\frac{1}{3 b x^{5/2} \sqrt{b x+c x^2}} \]

[Out]

-1/(3*b*x^(5/2)*Sqrt[b*x + c*x^2]) + (7*c)/(12*b^2*x^(3/2)*Sqrt[b*x + c*x^2]) -
(35*c^2)/(24*b^3*Sqrt[x]*Sqrt[b*x + c*x^2]) - (35*c^3*Sqrt[x])/(8*b^4*Sqrt[b*x +
c*x^2]) + (35*c^3*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(8*b^(9/2))

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Rubi [A]  time = 0.188577, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{35 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{9/2}}-\frac{35 c^3 \sqrt{x}}{8 b^4 \sqrt{b x+c x^2}}-\frac{35 c^2}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{7 c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}-\frac{1}{3 b x^{5/2} \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

-1/(3*b*x^(5/2)*Sqrt[b*x + c*x^2]) + (7*c)/(12*b^2*x^(3/2)*Sqrt[b*x + c*x^2]) -
(35*c^2)/(24*b^3*Sqrt[x]*Sqrt[b*x + c*x^2]) - (35*c^3*Sqrt[x])/(8*b^4*Sqrt[b*x +
c*x^2]) + (35*c^3*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(8*b^(9/2))

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Rubi in Sympy [A]  time = 21.8205, size = 136, normalized size = 0.94 \[ - \frac{1}{3 b x^{\frac{5}{2}} \sqrt{b x + c x^{2}}} + \frac{7 c}{12 b^{2} x^{\frac{3}{2}} \sqrt{b x + c x^{2}}} - \frac{35 c^{2}}{24 b^{3} \sqrt{x} \sqrt{b x + c x^{2}}} - \frac{35 c^{3} \sqrt{x}}{8 b^{4} \sqrt{b x + c x^{2}}} + \frac{35 c^{3} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

-1/(3*b*x**(5/2)*sqrt(b*x + c*x**2)) + 7*c/(12*b**2*x**(3/2)*sqrt(b*x + c*x**2))
- 35*c**2/(24*b**3*sqrt(x)*sqrt(b*x + c*x**2)) - 35*c**3*sqrt(x)/(8*b**4*sqrt(b
*x + c*x**2)) + 35*c**3*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(8*b**(9/2))

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Mathematica [A]  time = 0.069171, size = 96, normalized size = 0.66 \[ \frac{105 c^3 x^3 \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \left (8 b^3-14 b^2 c x+35 b c^2 x^2+105 c^3 x^3\right )}{24 b^{9/2} x^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(b*x + c*x^2)^(3/2)),x]

[Out]

(-(Sqrt[b]*(8*b^3 - 14*b^2*c*x + 35*b*c^2*x^2 + 105*c^3*x^3)) + 105*c^3*x^3*Sqrt
[b + c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(24*b^(9/2)*x^(5/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.015, size = 87, normalized size = 0.6 \[{\frac{1}{24\,cx+24\,b}\sqrt{x \left ( cx+b \right ) } \left ( 105\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{3}{c}^{3}-105\,{c}^{3}{x}^{3}\sqrt{b}-35\,{b}^{3/2}{x}^{2}{c}^{2}+14\,{b}^{5/2}xc-8\,{b}^{7/2} \right ){x}^{-{\frac{7}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(c*x^2+b*x)^(3/2),x)

[Out]

1/24*(x*(c*x+b))^(1/2)*(105*arctanh((c*x+b)^(1/2)/b^(1/2))*(c*x+b)^(1/2)*x^3*c^3
-105*c^3*x^3*b^(1/2)-35*b^(3/2)*x^2*c^2+14*b^(5/2)*x*c-8*b^(7/2))/x^(7/2)/(c*x+b
)/b^(9/2)

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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)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230151, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (105 \, c^{3} x^{3} + 35 \, b c^{2} x^{2} - 14 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x} - 105 \,{\left (c^{4} x^{5} + b c^{3} x^{4}\right )} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right )}{48 \,{\left (b^{4} c x^{5} + b^{5} x^{4}\right )} \sqrt{b}}, -\frac{{\left (105 \, c^{3} x^{3} + 35 \, b c^{2} x^{2} - 14 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x} - 105 \,{\left (c^{4} x^{5} + b c^{3} x^{4}\right )} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right )}{24 \,{\left (b^{4} c x^{5} + b^{5} x^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(3/2)*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/48*(2*(105*c^3*x^3 + 35*b*c^2*x^2 - 14*b^2*c*x + 8*b^3)*sqrt(c*x^2 + b*x)*sq
rt(b)*sqrt(x) - 105*(c^4*x^5 + b*c^3*x^4)*log(-(2*sqrt(c*x^2 + b*x)*b*sqrt(x) +
(c*x^2 + 2*b*x)*sqrt(b))/x^2))/((b^4*c*x^5 + b^5*x^4)*sqrt(b)), -1/24*((105*c^3*
x^3 + 35*b*c^2*x^2 - 14*b^2*c*x + 8*b^3)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x) - 10
5*(c^4*x^5 + b*c^3*x^4)*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)))/((b^4*c*x^5
+ b^5*x^4)*sqrt(-b))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{5}{2}} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Integral(1/(x**(5/2)*(x*(b + c*x))**(3/2)), x)

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GIAC/XCAS [A]  time = 0.337868, size = 113, normalized size = 0.78 \[ -\frac{1}{24} \, c^{3}{\left (\frac{105 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{4}} + \frac{48}{\sqrt{c x + b} b^{4}} + \frac{57 \,{\left (c x + b\right )}^{\frac{5}{2}} - 136 \,{\left (c x + b\right )}^{\frac{3}{2}} b + 87 \, \sqrt{c x + b} b^{2}}{b^{4} c^{3} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)^(3/2)*x^(5/2)),x, algorithm="giac")

[Out]

-1/24*c^3*(105*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^4) + 48/(sqrt(c*x + b)
*b^4) + (57*(c*x + b)^(5/2) - 136*(c*x + b)^(3/2)*b + 87*sqrt(c*x + b)*b^2)/(b^4
*c^3*x^3))