3.92 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^{13/2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{3 c^4 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{5/2}}+\frac{3 c^3 \sqrt{b x+c x^2}}{64 b^2 x^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{32 b x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{8 x^{7/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}} \]

[Out]

-(c*Sqrt[b*x + c*x^2])/(8*x^(7/2)) - (c^2*Sqrt[b*x + c*x^2])/(32*b*x^(5/2)) + (3
*c^3*Sqrt[b*x + c*x^2])/(64*b^2*x^(3/2)) - (b*x + c*x^2)^(3/2)/(4*x^(11/2)) - (3
*c^4*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(64*b^(5/2))

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Rubi [A]  time = 0.189204, antiderivative size = 139, 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 \[ -\frac{3 c^4 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{64 b^{5/2}}+\frac{3 c^3 \sqrt{b x+c x^2}}{64 b^2 x^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{32 b x^{5/2}}-\frac{c \sqrt{b x+c x^2}}{8 x^{7/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{4 x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(3/2)/x^(13/2),x]

[Out]

-(c*Sqrt[b*x + c*x^2])/(8*x^(7/2)) - (c^2*Sqrt[b*x + c*x^2])/(32*b*x^(5/2)) + (3
*c^3*Sqrt[b*x + c*x^2])/(64*b^2*x^(3/2)) - (b*x + c*x^2)^(3/2)/(4*x^(11/2)) - (3
*c^4*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x])])/(64*b^(5/2))

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Rubi in Sympy [A]  time = 22.568, size = 124, normalized size = 0.89 \[ - \frac{c \sqrt{b x + c x^{2}}}{8 x^{\frac{7}{2}}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 x^{\frac{11}{2}}} - \frac{c^{2} \sqrt{b x + c x^{2}}}{32 b x^{\frac{5}{2}}} + \frac{3 c^{3} \sqrt{b x + c x^{2}}}{64 b^{2} x^{\frac{3}{2}}} - \frac{3 c^{4} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{64 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**(13/2),x)

[Out]

-c*sqrt(b*x + c*x**2)/(8*x**(7/2)) - (b*x + c*x**2)**(3/2)/(4*x**(11/2)) - c**2*
sqrt(b*x + c*x**2)/(32*b*x**(5/2)) + 3*c**3*sqrt(b*x + c*x**2)/(64*b**2*x**(3/2)
) - 3*c**4*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(64*b**(5/2))

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Mathematica [A]  time = 0.0913756, size = 104, normalized size = 0.75 \[ -\frac{\sqrt{x (b+c x)} \left (\sqrt{b} \sqrt{b+c x} \left (16 b^3+24 b^2 c x+2 b c^2 x^2-3 c^3 x^3\right )+3 c^4 x^4 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{64 b^{5/2} x^{9/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(3/2)/x^(13/2),x]

[Out]

-(Sqrt[x*(b + c*x)]*(Sqrt[b]*Sqrt[b + c*x]*(16*b^3 + 24*b^2*c*x + 2*b*c^2*x^2 -
3*c^3*x^3) + 3*c^4*x^4*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(64*b^(5/2)*x^(9/2)*Sqrt
[b + c*x])

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Maple [A]  time = 0.026, size = 108, normalized size = 0.8 \[ -{\frac{1}{64}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){c}^{4}{x}^{4}-3\,{x}^{3}{c}^{3}\sqrt{b}\sqrt{cx+b}+2\,{x}^{2}{b}^{3/2}{c}^{2}\sqrt{cx+b}+24\,x{b}^{5/2}c\sqrt{cx+b}+16\,{b}^{7/2}\sqrt{cx+b} \right ){b}^{-{\frac{5}{2}}}{x}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^(13/2),x)

[Out]

-1/64*(x*(c*x+b))^(1/2)/b^(5/2)*(3*arctanh((c*x+b)^(1/2)/b^(1/2))*c^4*x^4-3*x^3*
c^3*b^(1/2)*(c*x+b)^(1/2)+2*x^2*b^(3/2)*c^2*(c*x+b)^(1/2)+24*x*b^(5/2)*c*(c*x+b)
^(1/2)+16*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((c*x^2 + b*x)^(3/2)/x^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232673, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{4} x^{5} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (3 \, c^{3} x^{3} - 2 \, b c^{2} x^{2} - 24 \, b^{2} c x - 16 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{128 \, b^{\frac{5}{2}} x^{5}}, -\frac{3 \, c^{4} x^{5} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (3 \, c^{3} x^{3} - 2 \, b c^{2} x^{2} - 24 \, b^{2} c x - 16 \, b^{3}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{64 \, \sqrt{-b} b^{2} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^(13/2),x, algorithm="fricas")

[Out]

[1/128*(3*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*(3*c^3*x^3 - 2*b*c^2*x^2 - 24*b^2*c*x - 16*b^3)*sqrt(c*x^2 + b*x)*sqrt(
b)*sqrt(x))/(b^(5/2)*x^5), -1/64*(3*c^4*x^5*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 +
b*x)) - (3*c^3*x^3 - 2*b*c^2*x^2 - 24*b^2*c*x - 16*b^3)*sqrt(c*x^2 + b*x)*sqrt(
-b)*sqrt(x))/(sqrt(-b)*b^2*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)**(3/2)/x**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.267362, size = 113, normalized size = 0.81 \[ \frac{1}{64} \, c^{4}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{7}{2}} - 11 \,{\left (c x + b\right )}^{\frac{5}{2}} b - 11 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} + 3 \, \sqrt{c x + b} b^{3}}{b^{2} c^{4} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)/x^(13/2),x, algorithm="giac")

[Out]

1/64*c^4*(3*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (3*(c*x + b)^(7/2) -
11*(c*x + b)^(5/2)*b - 11*(c*x + b)^(3/2)*b^2 + 3*sqrt(c*x + b)*b^3)/(b^2*c^4*x
^4))