3.91 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx\)

Optimal. Leaf size=111 \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}} \]

[Out]

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

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Rubi [A]  time = 0.145275, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^(11/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.1847, size = 95, normalized size = 0.86 \[ - \frac{c \sqrt{b x + c x^{2}}}{4 x^{\frac{5}{2}}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 x^{\frac{9}{2}}} - \frac{c^{2} \sqrt{b x + c x^{2}}}{8 b x^{\frac{3}{2}}} + \frac{c^{3} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**(11/2),x)

[Out]

-c*sqrt(b*x + c*x**2)/(4*x**(5/2)) - (b*x + c*x**2)**(3/2)/(3*x**(9/2)) - c**2*s
qrt(b*x + c*x**2)/(8*b*x**(3/2)) + c**3*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x
)))/(8*b**(3/2))

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Mathematica [A]  time = 0.0945137, size = 94, normalized size = 0.85 \[ \frac{\sqrt{x (b+c x)} \left (3 c^3 x^3 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \sqrt{b+c x} \left (8 b^2+14 b c x+3 c^2 x^2\right )\right )}{24 b^{3/2} x^{7/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^(11/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(-(Sqrt[b]*Sqrt[b + c*x]*(8*b^2 + 14*b*c*x + 3*c^2*x^2)) + 3*
c^3*x^3*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(24*b^(3/2)*x^(7/2)*Sqrt[b + c*x])

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Maple [A]  time = 0.015, size = 90, normalized size = 0.8 \[{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-3\,{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}-14\,x{b}^{3/2}c\sqrt{cx+b}-8\,{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^(11/2),x)

[Out]

1/24*(x*(c*x+b))^(1/2)/b^(3/2)*(3*arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*c^3-3*x^2*c
^2*b^(1/2)*(c*x+b)^(1/2)-14*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((c*x^2 + b*x)^(3/2)/x^(11/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237189, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{3} x^{4} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (3 \, c^{2} x^{2} + 14 \, b c x + 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{3}{2}} x^{4}}, \frac{3 \, c^{3} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (3 \, c^{2} x^{2} + 14 \, b c x + 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^(11/2),x, algorithm="fricas")

[Out]

[1/48*(3*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*(3*c^2*x^2 + 14*b*c*x + 8*b^2)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/(b^(3
/2)*x^4), 1/24*(3*c^3*x^4*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - (3*c^2*x^
2 + 14*b*c*x + 8*b^2)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b*x^4)]

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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)**(3/2)/x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.244208, size = 97, normalized size = 0.87 \[ -\frac{1}{24} \, c^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} + 8 \,{\left (c x + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{c x + b} b^{2}}{b c^{3} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^(11/2),x, algorithm="giac")

[Out]

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