3.676 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=77 \[ \frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{4 a^2 x^4}-\frac{A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5} \]

[Out]

((A*b - a*B)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*a^2*x^4) - (A*(a^2 +
2*a*b*x + b^2*x^2)^(5/2))/(5*a^2*x^5)

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Rubi [A]  time = 0.154753, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{4 a^2 x^4}-\frac{A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^6,x]

[Out]

((A*b - a*B)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*a^2*x^4) - (A*(a^2 +
2*a*b*x + b^2*x^2)^(5/2))/(5*a^2*x^5)

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Rubi in Sympy [A]  time = 23.8648, size = 73, normalized size = 0.95 \[ - \frac{A \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 a^{2} x^{5}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/x**6,x)

[Out]

-A*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*a**2*x**5) + (2*a + 2*b*x)*(A*b - B*a)
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(8*a**2*x**4)

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Mathematica [A]  time = 0.0525998, size = 84, normalized size = 1.09 \[ -\frac{\sqrt{(a+b x)^2} \left (a^3 (4 A+5 B x)+5 a^2 b x (3 A+4 B x)+10 a b^2 x^2 (2 A+3 B x)+10 b^3 x^3 (A+2 B x)\right )}{20 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(10*b^3*x^3*(A + 2*B*x) + 10*a*b^2*x^2*(2*A + 3*B*x) + 5*a^2
*b*x*(3*A + 4*B*x) + a^3*(4*A + 5*B*x)))/(20*x^5*(a + b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 1.2 \[ -{\frac{20\,B{x}^{4}{b}^{3}+10\,A{b}^{3}{x}^{3}+30\,B{x}^{3}a{b}^{2}+20\,A{x}^{2}a{b}^{2}+20\,B{x}^{2}{a}^{2}b+15\,A{a}^{2}bx+5\,{a}^{3}Bx+4\,A{a}^{3}}{20\,{x}^{5} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^6,x)

[Out]

-1/20*(20*B*b^3*x^4+10*A*b^3*x^3+30*B*a*b^2*x^3+20*A*a*b^2*x^2+20*B*a^2*b*x^2+15
*A*a^2*b*x+5*B*a^3*x+4*A*a^3)*((b*x+a)^2)^(3/2)/x^5/(b*x+a)^3

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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)^(3/2)*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274639, size = 99, normalized size = 1.29 \[ -\frac{20 \, B b^{3} x^{4} + 4 \, A a^{3} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 20 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

-1/20*(20*B*b^3*x^4 + 4*A*a^3 + 10*(3*B*a*b^2 + A*b^3)*x^3 + 20*(B*a^2*b + A*a*b
^2)*x^2 + 5*(B*a^3 + 3*A*a^2*b)*x)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/x**6,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**6, x)

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GIAC/XCAS [A]  time = 0.273528, size = 201, normalized size = 2.61 \[ -\frac{{\left (5 \, B a b^{4} - A b^{5}\right )}{\rm sign}\left (b x + a\right )}{20 \, a^{2}} - \frac{20 \, B b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 30 \, B a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 10 \, A b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 20 \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + 20 \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 5 \, B a^{3} x{\rm sign}\left (b x + a\right ) + 15 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + 4 \, A a^{3}{\rm sign}\left (b x + a\right )}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

-1/20*(5*B*a*b^4 - A*b^5)*sign(b*x + a)/a^2 - 1/20*(20*B*b^3*x^4*sign(b*x + a) +
30*B*a*b^2*x^3*sign(b*x + a) + 10*A*b^3*x^3*sign(b*x + a) + 20*B*a^2*b*x^2*sign
(b*x + a) + 20*A*a*b^2*x^2*sign(b*x + a) + 5*B*a^3*x*sign(b*x + a) + 15*A*a^2*b*
x*sign(b*x + a) + 4*A*a^3*sign(b*x + a))/x^5