3.249 \(\int \frac{\left (a x^2+b x^3\right )^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=109 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{3/2}}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6} \]

[Out]

-(b*Sqrt[a*x^2 + b*x^3])/(4*x^3) - (b^2*Sqrt[a*x^2 + b*x^3])/(8*a*x^2) - (a*x^2
+ b*x^3)^(3/2)/(3*x^6) + (b^3*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/(8*a^(3/
2))

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Rubi [A]  time = 0.23508, antiderivative size = 109, 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 \[ \frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{3/2}}-\frac{b^2 \sqrt{a x^2+b x^3}}{8 a x^2}-\frac{b \sqrt{a x^2+b x^3}}{4 x^3}-\frac{\left (a x^2+b x^3\right )^{3/2}}{3 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^3)^(3/2)/x^7,x]

[Out]

-(b*Sqrt[a*x^2 + b*x^3])/(4*x^3) - (b^2*Sqrt[a*x^2 + b*x^3])/(8*a*x^2) - (a*x^2
+ b*x^3)^(3/2)/(3*x^6) + (b^3*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^3]])/(8*a^(3/
2))

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Rubi in Sympy [A]  time = 24.603, size = 94, normalized size = 0.86 \[ - \frac{b \sqrt{a x^{2} + b x^{3}}}{4 x^{3}} - \frac{\left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{3 x^{6}} - \frac{b^{2} \sqrt{a x^{2} + b x^{3}}}{8 a x^{2}} + \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a*x**2)**(3/2)/x**7,x)

[Out]

-b*sqrt(a*x**2 + b*x**3)/(4*x**3) - (a*x**2 + b*x**3)**(3/2)/(3*x**6) - b**2*sqr
t(a*x**2 + b*x**3)/(8*a*x**2) + b**3*atanh(sqrt(a)*x/sqrt(a*x**2 + b*x**3))/(8*a
**(3/2))

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Mathematica [A]  time = 0.0874827, size = 94, normalized size = 0.86 \[ \frac{\sqrt{x^2 (a+b x)} \left (3 b^3 x^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{a} \sqrt{a+b x} \left (8 a^2+14 a b x+3 b^2 x^2\right )\right )}{24 a^{3/2} x^4 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^2 + b*x^3)^(3/2)/x^7,x]

[Out]

(Sqrt[x^2*(a + b*x)]*(-(Sqrt[a]*Sqrt[a + b*x]*(8*a^2 + 14*a*b*x + 3*b^2*x^2)) +
3*b^3*x^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/(24*a^(3/2)*x^4*Sqrt[a + b*x])

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Maple [A]  time = 0.019, size = 87, normalized size = 0.8 \[ -{\frac{1}{24\,{x}^{6}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 3\, \left ( bx+a \right ) ^{5/2}{a}^{3/2}-3\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){x}^{3}a{b}^{3}+8\, \left ( bx+a \right ) ^{3/2}{a}^{5/2}-3\,\sqrt{bx+a}{a}^{7/2} \right ) \left ( bx+a \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^(3/2)/x^7,x)

[Out]

-1/24*(b*x^3+a*x^2)^(3/2)*(3*(b*x+a)^(5/2)*a^(3/2)-3*arctanh((b*x+a)^(1/2)/a^(1/
2))*x^3*a*b^3+8*(b*x+a)^(3/2)*a^(5/2)-3*(b*x+a)^(1/2)*a^(7/2))/x^6/(b*x+a)^(3/2)
/a^(5/2)

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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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241384, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{3} x^{4} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) - 2 \,{\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{48 \, a^{2} x^{4}}, \frac{3 \, \sqrt{-a} b^{3} x^{4} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) -{\left (3 \, a b^{2} x^{2} + 14 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{24 \, a^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x^2)^(3/2)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x**2)**(3/2)/x**7,x)

[Out]

Integral((x**2*(a + b*x))**(3/2)/x**7, x)

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GIAC/XCAS [A]  time = 0.266586, size = 124, normalized size = 1.14 \[ -\frac{\frac{3 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ){\rm sign}\left (x\right )}{\sqrt{-a} a} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4}{\rm sign}\left (x\right ) + 8 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4}{\rm sign}\left (x\right ) - 3 \, \sqrt{b x + a} a^{2} b^{4}{\rm sign}\left (x\right )}{a b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x^2)^(3/2)/x^7,x, algorithm="giac")

[Out]

-1/24*(3*b^4*arctan(sqrt(b*x + a)/sqrt(-a))*sign(x)/(sqrt(-a)*a) + (3*(b*x + a)^
(5/2)*b^4*sign(x) + 8*(b*x + a)^(3/2)*a*b^4*sign(x) - 3*sqrt(b*x + a)*a^2*b^4*si
gn(x))/(a*b^3*x^3))/b