3.101 \(\int \frac{1}{x^{7/2} \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{5 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.144866, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{5 c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}-\frac{5 c^2 \sqrt{b x+c x^2}}{8 b^3 x^{3/2}}+\frac{5 c \sqrt{b x+c x^2}}{12 b^2 x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 b x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 16.4318, size = 107, normalized size = 0.91 \[ - \frac{\sqrt{b x + c x^{2}}}{3 b x^{\frac{7}{2}}} + \frac{5 c \sqrt{b x + c x^{2}}}{12 b^{2} x^{\frac{5}{2}}} - \frac{5 c^{2} \sqrt{b x + c x^{2}}}{8 b^{3} x^{\frac{3}{2}}} + \frac{5 c^{3} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

-sqrt(b*x + c*x**2)/(3*b*x**(7/2)) + 5*c*sqrt(b*x + c*x**2)/(12*b**2*x**(5/2)) -
5*c**2*sqrt(b*x + c*x**2)/(8*b**3*x**(3/2)) + 5*c**3*atanh(sqrt(b*x + c*x**2)/(
sqrt(b)*sqrt(x)))/(8*b**(7/2))

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Mathematica [A]  time = 0.0703431, size = 96, normalized size = 0.82 \[ \frac{15 c^3 x^3 \sqrt{b+c x} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \left (8 b^3-2 b^2 c x+5 b c^2 x^2+15 c^3 x^3\right )}{24 b^{7/2} x^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*Sqrt[b*x + c*x^2]),x]

[Out]

(-(Sqrt[b]*(8*b^3 - 2*b^2*c*x + 5*b*c^2*x^2 + 15*c^3*x^3)) + 15*c^3*x^3*Sqrt[b +
c*x]*ArcTanh[Sqrt[b + c*x]/Sqrt[b]])/(24*b^(7/2)*x^(5/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.014, size = 90, normalized size = 0.8 \[{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-15\,{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+10\,x{b}^{3/2}c\sqrt{cx+b}-8\,{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(c*x^2+b*x)^(1/2),x)

[Out]

1/24*(x*(c*x+b))^(1/2)/b^(7/2)*(15*arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*c^3-15*x^2
*c^2*b^(1/2)*(c*x+b)^(1/2)+10*x*b^(3/2)*c*(c*x+b)^(1/2)-8*b^(5/2)*(c*x+b)^(1/2))
/x^(7/2)/(c*x+b)^(1/2)

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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/(sqrt(c*x^2 + b*x)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232584, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, c^{3} x^{4} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (15 \, c^{2} x^{2} - 10 \, b c x + 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{7}{2}} x^{4}}, \frac{15 \, c^{3} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (15 \, c^{2} x^{2} - 10 \, b c x + 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^(7/2)),x, algorithm="fricas")

[Out]

[1/48*(15*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) - 2*(15*c^2*x^2 - 10*b*c*x + 8*b^2)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/(b^
(7/2)*x^4), 1/24*(15*c^3*x^4*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - (15*c^
2*x^2 - 10*b*c*x + 8*b^2)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^3*x^4)
]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{\frac{7}{2}} \sqrt{x \left (b + c x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(7/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(x**(7/2)*sqrt(x*(b + c*x))), x)

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GIAC/XCAS [A]  time = 0.251137, size = 97, normalized size = 0.83 \[ -\frac{1}{24} \, c^{3}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (c x + b\right )}^{\frac{5}{2}} - 40 \,{\left (c x + b\right )}^{\frac{3}{2}} b + 33 \, \sqrt{c x + b} b^{2}}{b^{3} c^{3} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x)*x^(7/2)),x, algorithm="giac")

[Out]

-1/24*c^3*(15*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + (15*(c*x + b)^(5/2
) - 40*(c*x + b)^(3/2)*b + 33*sqrt(c*x + b)*b^2)/(b^3*c^3*x^3))