3.250 \(\int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^8} \, dx\)

Optimal. Leaf size=137 \[ -\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{64 a^{5/2}}+\frac{3 b^3 \sqrt{a x^2+b x^3}}{64 a^2 x^2}-\frac{b^2 \sqrt{a x^2+b x^3}}{32 a x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{4 x^7}-\frac{b \sqrt{a x^2+b x^3}}{8 x^4} \]

[Out]

-(b*Sqrt[a*x^2 + b*x^3])/(8*x^4) - (b^2*Sqrt[a*x^2 + b*x^3])/(32*a*x^3) + (3*b^3
*Sqrt[a*x^2 + b*x^3])/(64*a^2*x^2) - (a*x^2 + b*x^3)^(3/2)/(4*x^7) - (3*b^4*ArcT
anh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/(64*a^(5/2))

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Rubi [A]  time = 0.319166, antiderivative size = 137, 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 \[ -\frac{3 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{64 a^{5/2}}+\frac{3 b^3 \sqrt{a x^2+b x^3}}{64 a^2 x^2}-\frac{b^2 \sqrt{a x^2+b x^3}}{32 a x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{4 x^7}-\frac{b \sqrt{a x^2+b x^3}}{8 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^3)^(3/2)/x^8,x]

[Out]

-(b*Sqrt[a*x^2 + b*x^3])/(8*x^4) - (b^2*Sqrt[a*x^2 + b*x^3])/(32*a*x^3) + (3*b^3
*Sqrt[a*x^2 + b*x^3])/(64*a^2*x^2) - (a*x^2 + b*x^3)^(3/2)/(4*x^7) - (3*b^4*ArcT
anh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/(64*a^(5/2))

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Rubi in Sympy [A]  time = 32.6452, size = 122, normalized size = 0.89 \[ - \frac{b \sqrt{a x^{2} + b x^{3}}}{8 x^{4}} - \frac{\left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{4 x^{7}} - \frac{b^{2} \sqrt{a x^{2} + b x^{3}}}{32 a x^{3}} + \frac{3 b^{3} \sqrt{a x^{2} + b x^{3}}}{64 a^{2} x^{2}} - \frac{3 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{64 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a*x**2)**(3/2)/x**8,x)

[Out]

-b*sqrt(a*x**2 + b*x**3)/(8*x**4) - (a*x**2 + b*x**3)**(3/2)/(4*x**7) - b**2*sqr
t(a*x**2 + b*x**3)/(32*a*x**3) + 3*b**3*sqrt(a*x**2 + b*x**3)/(64*a**2*x**2) - 3
*b**4*atanh(sqrt(a)*x/sqrt(a*x**2 + b*x**3))/(64*a**(5/2))

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Mathematica [A]  time = 0.0931592, size = 104, normalized size = 0.76 \[ -\frac{\sqrt{x^2 (a+b x)} \left (\sqrt{a} \sqrt{a+b x} \left (16 a^3+24 a^2 b x+2 a b^2 x^2-3 b^3 x^3\right )+3 b^4 x^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right )}{64 a^{5/2} x^5 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^2 + b*x^3)^(3/2)/x^8,x]

[Out]

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

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Maple [A]  time = 0.018, size = 101, normalized size = 0.7 \[{\frac{1}{64\,{x}^{7}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\, \left ( bx+a \right ) ^{7/2}{a}^{5/2}-11\, \left ( bx+a \right ) ^{5/2}{a}^{7/2}-3\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){a}^{2}{x}^{4}{b}^{4}-11\, \left ( bx+a \right ) ^{3/2}{a}^{9/2}+3\,\sqrt{bx+a}{a}^{11/2} \right ) \left ( bx+a \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^(3/2)/x^8,x)

[Out]

1/64*(b*x^3+a*x^2)^(3/2)*(3*(b*x+a)^(7/2)*a^(5/2)-11*(b*x+a)^(5/2)*a^(7/2)-3*arc
tanh((b*x+a)^(1/2)/a^(1/2))*a^2*x^4*b^4-11*(b*x+a)^(3/2)*a^(9/2)+3*(b*x+a)^(1/2)
*a^(11/2))/x^7/(b*x+a)^(3/2)/a^(9/2)

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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((b*x^3 + a*x^2)^(3/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236194, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{4} x^{5} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) + 2 \,{\left (3 \, a b^{3} x^{3} - 2 \, a^{2} b^{2} x^{2} - 24 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt{b x^{3} + a x^{2}}}{128 \, a^{3} x^{5}}, -\frac{3 \, \sqrt{-a} b^{4} x^{5} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) -{\left (3 \, a b^{3} x^{3} - 2 \, a^{2} b^{2} x^{2} - 24 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt{b x^{3} + a x^{2}}}{64 \, a^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x^2)^(3/2)/x^8,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x**2)**(3/2)/x**8,x)

[Out]

Integral((x**2*(a + b*x))**(3/2)/x**8, x)

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GIAC/XCAS [A]  time = 0.262168, size = 147, normalized size = 1.07 \[ \frac{\frac{3 \, b^{5} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{5}{\rm sign}\left (x\right ) - 11 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{5}{\rm sign}\left (x\right ) - 11 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{5}{\rm sign}\left (x\right ) + 3 \, \sqrt{b x + a} a^{3} b^{5}{\rm sign}\left (x\right )}{a^{2} b^{4} x^{4}}}{64 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x^2)^(3/2)/x^8,x, algorithm="giac")

[Out]

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