3.175 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=37 \[ -\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6} \]

[Out]

-((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*a*x^6)

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Rubi [A]  time = 0.052523, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^7,x]

[Out]

-((a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*a*x^6)

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Rubi in Sympy [A]  time = 6.3698, size = 36, normalized size = 0.97 \[ - \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 a x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/x**7,x)

[Out]

-(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*a*x**6)

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Mathematica [B]  time = 0.0262002, size = 75, normalized size = 2.03 \[ -\frac{\sqrt{(a+b x)^2} \left (a^5+6 a^4 b x+15 a^3 b^2 x^2+20 a^2 b^3 x^3+15 a b^4 x^4+6 b^5 x^5\right )}{6 x^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^7,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^5 + 6*a^4*b*x + 15*a^3*b^2*x^2 + 20*a^2*b^3*x^3 + 15*a*b^
4*x^4 + 6*b^5*x^5))/(6*x^6*(a + b*x))

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Maple [B]  time = 0.009, size = 72, normalized size = 2. \[ -{\frac{6\,{b}^{5}{x}^{5}+15\,a{b}^{4}{x}^{4}+20\,{a}^{2}{b}^{3}{x}^{3}+15\,{a}^{3}{b}^{2}{x}^{2}+6\,{a}^{4}bx+{a}^{5}}{6\,{x}^{6} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x)

[Out]

-1/6*(6*b^5*x^5+15*a*b^4*x^4+20*a^2*b^3*x^3+15*a^3*b^2*x^2+6*a^4*b*x+a^5)*((b*x+
a)^2)^(5/2)/x^6/(b*x+a)^5

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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^2*x^2 + 2*a*b*x + a^2)^(5/2)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223842, size = 74, normalized size = 2. \[ -\frac{6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/x^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a
^5)/x^6

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{7}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/x**7,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**7, x)

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GIAC/XCAS [A]  time = 0.209646, size = 144, normalized size = 3.89 \[ -\frac{b^{6}{\rm sign}\left (b x + a\right )}{6 \, a} - \frac{6 \, b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 15 \, a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 20 \, a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 15 \, a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 6 \, a^{4} b x{\rm sign}\left (b x + a\right ) + a^{5}{\rm sign}\left (b x + a\right )}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)/x^7,x, algorithm="giac")

[Out]

-1/6*b^6*sign(b*x + a)/a - 1/6*(6*b^5*x^5*sign(b*x + a) + 15*a*b^4*x^4*sign(b*x
+ a) + 20*a^2*b^3*x^3*sign(b*x + a) + 15*a^3*b^2*x^2*sign(b*x + a) + 6*a^4*b*x*s
ign(b*x + a) + a^5*sign(b*x + a))/x^6