3.81 \(\int \frac{\sqrt{b x+c x^2}}{x^{11/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{5 c^4 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{7/2}}-\frac{5 c^3 \sqrt{b x+c x^2}}{64 b^3 x^{3/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}} \]

[Out]

-Sqrt[b*x + c*x^2]/(4*x^(9/2)) - (c*Sqrt[b*x + c*x^2])/(24*b*x^(7/2)) + (5*c^2*S
qrt[b*x + c*x^2])/(96*b^2*x^(5/2)) - (5*c^3*Sqrt[b*x + c*x^2])/(64*b^3*x^(3/2))
+ (5*c^4*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(64*b^(7/2))

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Rubi [A]  time = 0.188971, antiderivative size = 142, 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 \[ \frac{5 c^4 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{7/2}}-\frac{5 c^3 \sqrt{b x+c x^2}}{64 b^3 x^{3/2}}+\frac{5 c^2 \sqrt{b x+c x^2}}{96 b^2 x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{24 b x^{7/2}}-\frac{\sqrt{b x+c x^2}}{4 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^(11/2),x]

[Out]

-Sqrt[b*x + c*x^2]/(4*x^(9/2)) - (c*Sqrt[b*x + c*x^2])/(24*b*x^(7/2)) + (5*c^2*S
qrt[b*x + c*x^2])/(96*b^2*x^(5/2)) - (5*c^3*Sqrt[b*x + c*x^2])/(64*b^3*x^(3/2))
+ (5*c^4*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(64*b^(7/2))

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Rubi in Sympy [A]  time = 21.8548, size = 129, normalized size = 0.91 \[ - \frac{\sqrt{b x + c x^{2}}}{4 x^{\frac{9}{2}}} - \frac{c \sqrt{b x + c x^{2}}}{24 b x^{\frac{7}{2}}} + \frac{5 c^{2} \sqrt{b x + c x^{2}}}{96 b^{2} x^{\frac{5}{2}}} - \frac{5 c^{3} \sqrt{b x + c x^{2}}}{64 b^{3} x^{\frac{3}{2}}} + \frac{5 c^{4} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{64 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/x**(11/2),x)

[Out]

-sqrt(b*x + c*x**2)/(4*x**(9/2)) - c*sqrt(b*x + c*x**2)/(24*b*x**(7/2)) + 5*c**2
*sqrt(b*x + c*x**2)/(96*b**2*x**(5/2)) - 5*c**3*sqrt(b*x + c*x**2)/(64*b**3*x**(
3/2)) + 5*c**4*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(64*b**(7/2))

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Mathematica [A]  time = 0.109301, size = 105, normalized size = 0.74 \[ \frac{\sqrt{x (b+c x)} \left (15 c^4 x^4 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \sqrt{b+c x} \left (48 b^3+8 b^2 c x-10 b c^2 x^2+15 c^3 x^3\right )\right )}{192 b^{7/2} x^{9/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/x^(11/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(-(Sqrt[b]*Sqrt[b + c*x]*(48*b^3 + 8*b^2*c*x - 10*b*c^2*x^2 +
15*c^3*x^3)) + 15*c^4*x^4*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(192*b^(7/2)*x^(9/2)
*Sqrt[b + c*x])

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Maple [A]  time = 0.024, size = 108, normalized size = 0.8 \[{\frac{1}{192}\sqrt{x \left ( cx+b \right ) } \left ( 15\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){c}^{4}{x}^{4}-15\,{x}^{3}{c}^{3}\sqrt{b}\sqrt{cx+b}+10\,{x}^{2}{b}^{3/2}{c}^{2}\sqrt{cx+b}-8\,x{b}^{5/2}c\sqrt{cx+b}-48\,{b}^{7/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^(11/2),x)

[Out]

1/192*(x*(c*x+b))^(1/2)/b^(7/2)*(15*arctanh((c*x+b)^(1/2)/b^(1/2))*c^4*x^4-15*x^
3*c^3*b^(1/2)*(c*x+b)^(1/2)+10*x^2*b^(3/2)*c^2*(c*x+b)^(1/2)-8*x*b^(5/2)*c*(c*x+
b)^(1/2)-48*b^(7/2)*(c*x+b)^(1/2))/x^(9/2)/(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(sqrt(c*x^2 + b*x)/x^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23073, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, c^{4} x^{5} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (15 \, c^{3} x^{3} - 10 \, b c^{2} x^{2} + 8 \, b^{2} c x + 48 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{384 \, b^{\frac{7}{2}} x^{5}}, \frac{15 \, c^{4} x^{5} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (15 \, c^{3} x^{3} - 10 \, b c^{2} x^{2} + 8 \, b^{2} c x + 48 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{192 \, \sqrt{-b} b^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(11/2),x, algorithm="fricas")

[Out]

[1/384*(15*c^4*x^5*log(-(2*sqrt(c*x^2 + b*x)*b*sqrt(x) + (c*x^2 + 2*b*x)*sqrt(b)
)/x^2) - 2*(15*c^3*x^3 - 10*b*c^2*x^2 + 8*b^2*c*x + 48*b^3)*sqrt(c*x^2 + b*x)*sq
rt(b)*sqrt(x))/(b^(7/2)*x^5), 1/192*(15*c^4*x^5*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x
^2 + b*x)) - (15*c^3*x^3 - 10*b*c^2*x^2 + 8*b^2*c*x + 48*b^3)*sqrt(c*x^2 + b*x)*
sqrt(-b)*sqrt(x))/(sqrt(-b)*b^3*x^5)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.256102, size = 113, normalized size = 0.8 \[ -\frac{1}{192} \, c^{4}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 55 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 73 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} + 15 \, \sqrt{c x + b} b^{3}}{b^{3} c^{4} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(11/2),x, algorithm="giac")

[Out]

-1/192*c^4*(15*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^3) + (15*(c*x + b)^(7/
2) - 55*(c*x + b)^(5/2)*b + 73*(c*x + b)^(3/2)*b^2 + 15*sqrt(c*x + b)*b^3)/(b^3*
c^4*x^4))