3.90 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \]

[Out]

(-3*c*Sqrt[b*x + c*x^2])/(4*x^(3/2)) - (b*x + c*x^2)^(3/2)/(2*x^(7/2)) - (3*c^2*
ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(4*Sqrt[b])

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Rubi [A]  time = 0.10441, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^(9/2),x]

[Out]

(-3*c*Sqrt[b*x + c*x^2])/(4*x^(3/2)) - (b*x + c*x^2)^(3/2)/(2*x^(7/2)) - (3*c^2*
ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(4*Sqrt[b])

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Rubi in Sympy [A]  time = 12.7864, size = 76, normalized size = 0.92 \[ - \frac{3 c \sqrt{b x + c x^{2}}}{4 x^{\frac{3}{2}}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{2 x^{\frac{7}{2}}} - \frac{3 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**(9/2),x)

[Out]

-3*c*sqrt(b*x + c*x**2)/(4*x**(3/2)) - (b*x + c*x**2)**(3/2)/(2*x**(7/2)) - 3*c*
*2*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(4*sqrt(b))

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Mathematica [A]  time = 0.0693698, size = 82, normalized size = 0.99 \[ -\frac{\sqrt{x (b+c x)} \left (3 c^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )+\sqrt{b} \sqrt{b+c x} (2 b+5 c x)\right )}{4 \sqrt{b} x^{5/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^(9/2),x]

[Out]

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

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Maple [A]  time = 0.015, size = 72, normalized size = 0.9 \[ -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){c}^{2}{x}^{2}+5\,xc\sqrt{cx+b}\sqrt{b}+2\,{b}^{3/2}\sqrt{cx+b} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^(9/2),x)

[Out]

-1/4*(x*(c*x+b))^(1/2)*(3*arctanh((c*x+b)^(1/2)/b^(1/2))*c^2*x^2+5*x*c*(c*x+b)^(
1/2)*b^(1/2)+2*b^(3/2)*(c*x+b)^(1/2))/x^(5/2)/(c*x+b)^(1/2)/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^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232388, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{2} x^{3} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, \sqrt{c x^{2} + b x}{\left (5 \, c x + 2 \, b\right )} \sqrt{b} \sqrt{x}}{8 \, \sqrt{b} x^{3}}, -\frac{3 \, c^{2} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x}{\left (5 \, c x + 2 \, b\right )} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^(9/2),x, algorithm="fricas")

[Out]

[1/8*(3*c^2*x^3*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x^
2) - 2*sqrt(c*x^2 + b*x)*(5*c*x + 2*b)*sqrt(b)*sqrt(x))/(sqrt(b)*x^3), -1/4*(3*c
^2*x^3*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) + sqrt(c*x^2 + b*x)*(5*c*x + 2
*b)*sqrt(-b)*sqrt(x))/(sqrt(-b)*x^3)]

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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**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.236076, size = 74, normalized size = 0.89 \[ \frac{1}{4} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (c x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x + b} b}{c^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^(9/2),x, algorithm="giac")

[Out]

1/4*c^2*(3*arctan(sqrt(c*x + b)/sqrt(-b))/sqrt(-b) - (5*(c*x + b)^(3/2) - 3*sqrt
(c*x + b)*b)/(c^2*x^2))