3.698 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=130 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-4 a B)}{28 a^2 x^7}-\frac{A \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-4 a B)}{168 a^3 x^6} \]

[Out]

-(A*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*a*x^8) + ((A*b - 4*a*B)*(a + b
*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(28*a^2*x^7) - (b*(A*b - 4*a*B)*(a + b*x)^5
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(168*a^3*x^6)

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Rubi [A]  time = 0.232332, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-4 a B)}{28 a^2 x^7}-\frac{A \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{8 a x^8}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-4 a B)}{168 a^3 x^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^9,x]

[Out]

-(A*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*a*x^8) + ((A*b - 4*a*B)*(a + b
*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(28*a^2*x^7) - (b*(A*b - 4*a*B)*(a + b*x)^5
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(168*a^3*x^6)

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Rubi in Sympy [A]  time = 15.0274, size = 119, normalized size = 0.92 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{16 a x^{8}} + \frac{\left (2 a + 2 b x\right ) \left (A b - 4 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{48 a^{2} x^{7}} - \frac{\left (A b - 4 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{168 a^{3} x^{7}} \]

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)**(5/2)/x**9,x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(16*a*x**8) + (2*a + 2*b*x)
*(A*b - 4*B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(48*a**2*x**7) - (A*b - 4*B*a
)*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(168*a**3*x**7)

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Mathematica [A]  time = 0.0765227, size = 125, normalized size = 0.96 \[ -\frac{\sqrt{(a+b x)^2} \left (3 a^5 (7 A+8 B x)+20 a^4 b x (6 A+7 B x)+56 a^3 b^2 x^2 (5 A+6 B x)+84 a^2 b^3 x^3 (4 A+5 B x)+70 a b^4 x^4 (3 A+4 B x)+28 b^5 x^5 (2 A+3 B x)\right )}{168 x^8 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(28*b^5*x^5*(2*A + 3*B*x) + 70*a*b^4*x^4*(3*A + 4*B*x) + 84*
a^2*b^3*x^3*(4*A + 5*B*x) + 56*a^3*b^2*x^2*(5*A + 6*B*x) + 20*a^4*b*x*(6*A + 7*B
*x) + 3*a^5*(7*A + 8*B*x)))/(168*x^8*(a + b*x))

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Maple [A]  time = 0.012, size = 140, normalized size = 1.1 \[ -{\frac{84\,B{b}^{5}{x}^{6}+56\,A{x}^{5}{b}^{5}+280\,B{x}^{5}a{b}^{4}+210\,A{x}^{4}a{b}^{4}+420\,B{x}^{4}{a}^{2}{b}^{3}+336\,A{x}^{3}{a}^{2}{b}^{3}+336\,B{x}^{3}{a}^{3}{b}^{2}+280\,A{x}^{2}{a}^{3}{b}^{2}+140\,B{x}^{2}{a}^{4}b+120\,Ax{a}^{4}b+24\,Bx{a}^{5}+21\,A{a}^{5}}{168\,{x}^{8} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^9,x)

[Out]

-1/168*(84*B*b^5*x^6+56*A*b^5*x^5+280*B*a*b^4*x^5+210*A*a*b^4*x^4+420*B*a^2*b^3*
x^4+336*A*a^2*b^3*x^3+336*B*a^3*b^2*x^3+280*A*a^3*b^2*x^2+140*B*a^4*b*x^2+120*A*
a^4*b*x+24*B*a^5*x+21*A*a^5)*((b*x+a)^2)^(5/2)/x^8/(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)*(B*x + A)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282594, size = 161, normalized size = 1.24 \[ -\frac{84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^9,x, algorithm="fricas")

[Out]

-1/168*(84*B*b^5*x^6 + 21*A*a^5 + 56*(5*B*a*b^4 + A*b^5)*x^5 + 210*(2*B*a^2*b^3
+ A*a*b^4)*x^4 + 336*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*x
^2 + 24*(B*a^5 + 5*A*a^4*b)*x)/x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{9}}\, dx \]

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)**(5/2)/x**9,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**9, x)

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GIAC/XCAS [A]  time = 0.27388, size = 298, normalized size = 2.29 \[ \frac{{\left (4 \, B a b^{7} - A b^{8}\right )}{\rm sign}\left (b x + a\right )}{168 \, a^{3}} - \frac{84 \, B b^{5} x^{6}{\rm sign}\left (b x + a\right ) + 280 \, B a b^{4} x^{5}{\rm sign}\left (b x + a\right ) + 56 \, A b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 420 \, B a^{2} b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 210 \, A a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 336 \, B a^{3} b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 336 \, A a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 140 \, B a^{4} b x^{2}{\rm sign}\left (b x + a\right ) + 280 \, A a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 24 \, B a^{5} x{\rm sign}\left (b x + a\right ) + 120 \, A a^{4} b x{\rm sign}\left (b x + a\right ) + 21 \, A a^{5}{\rm sign}\left (b x + a\right )}{168 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^9,x, algorithm="giac")

[Out]

1/168*(4*B*a*b^7 - A*b^8)*sign(b*x + a)/a^3 - 1/168*(84*B*b^5*x^6*sign(b*x + a)
+ 280*B*a*b^4*x^5*sign(b*x + a) + 56*A*b^5*x^5*sign(b*x + a) + 420*B*a^2*b^3*x^4
*sign(b*x + a) + 210*A*a*b^4*x^4*sign(b*x + a) + 336*B*a^3*b^2*x^3*sign(b*x + a)
+ 336*A*a^2*b^3*x^3*sign(b*x + a) + 140*B*a^4*b*x^2*sign(b*x + a) + 280*A*a^3*b
^2*x^2*sign(b*x + a) + 24*B*a^5*x*sign(b*x + a) + 120*A*a^4*b*x*sign(b*x + a) +
21*A*a^5*sign(b*x + a))/x^8