3.679 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^9} \, dx\)

Optimal. Leaf size=210 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{7 x^7 (a+b x)}-\frac{a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{2 x^6 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{5 x^5 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.245126, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{7 x^7 (a+b x)}-\frac{a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{2 x^6 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{5 x^5 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 25.5058, size = 196, normalized size = 0.93 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{16 a x^{8}} - \frac{b^{2} \left (A b - 2 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{280 x^{5} \left (a + b x\right )} + \frac{b^{2} \left (A b - 2 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{56 a x^{5}} + \frac{b \left (a + b x\right ) \left (A b - 2 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{28 a x^{6}} + \frac{\left (A b - 2 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{14 a x^{7}} \]

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)**(3/2)/x**9,x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(16*a*x**8) - b**2*(A*b - 2
*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(280*x**5*(a + b*x)) + b**2*(A*b - 2*B*a)
*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(56*a*x**5) + b*(a + b*x)*(A*b - 2*B*a)*sqrt(a
**2 + 2*a*b*x + b**2*x**2)/(28*a*x**6) + (A*b - 2*B*a)*(a**2 + 2*a*b*x + b**2*x*
*2)**(3/2)/(14*a*x**7)

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Mathematica [A]  time = 0.0527601, size = 87, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (5 a^3 (7 A+8 B x)+20 a^2 b x (6 A+7 B x)+28 a b^2 x^2 (5 A+6 B x)+14 b^3 x^3 (4 A+5 B x)\right )}{280 x^8 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^9,x]

[Out]

-(Sqrt[(a + b*x)^2]*(14*b^3*x^3*(4*A + 5*B*x) + 28*a*b^2*x^2*(5*A + 6*B*x) + 20*
a^2*b*x*(6*A + 7*B*x) + 5*a^3*(7*A + 8*B*x)))/(280*x^8*(a + b*x))

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Maple [A]  time = 0.012, size = 92, normalized size = 0.4 \[ -{\frac{70\,B{x}^{4}{b}^{3}+56\,A{b}^{3}{x}^{3}+168\,B{x}^{3}a{b}^{2}+140\,A{x}^{2}a{b}^{2}+140\,B{x}^{2}{a}^{2}b+120\,A{a}^{2}bx+40\,{a}^{3}Bx+35\,A{a}^{3}}{280\,{x}^{8} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^9,x)

[Out]

-1/280*(70*B*b^3*x^4+56*A*b^3*x^3+168*B*a*b^2*x^3+140*A*a*b^2*x^2+140*B*a^2*b*x^
2+120*A*a^2*b*x+40*B*a^3*x+35*A*a^3)*((b*x+a)^2)^(3/2)/x^8/(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^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278667, size = 99, normalized size = 0.47 \[ -\frac{70 \, B b^{3} x^{4} + 35 \, A a^{3} + 56 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 140 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 40 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \]

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^9,x, algorithm="fricas")

[Out]

-1/280*(70*B*b^3*x^4 + 35*A*a^3 + 56*(3*B*a*b^2 + A*b^3)*x^3 + 140*(B*a^2*b + A*
a*b^2)*x^2 + 40*(B*a^3 + 3*A*a^2*b)*x)/x^8

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{9}}\, dx \]

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)**(3/2)/x**9,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**9, x)

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GIAC/XCAS [A]  time = 0.271745, size = 201, normalized size = 0.96 \[ \frac{{\left (2 \, B a b^{7} - A b^{8}\right )}{\rm sign}\left (b x + a\right )}{280 \, a^{5}} - \frac{70 \, B b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 168 \, B a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 56 \, A b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 140 \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + 140 \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 40 \, B a^{3} x{\rm sign}\left (b x + a\right ) + 120 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + 35 \, A a^{3}{\rm sign}\left (b x + a\right )}{280 \, x^{8}} \]

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^9,x, algorithm="giac")

[Out]

1/280*(2*B*a*b^7 - A*b^8)*sign(b*x + a)/a^5 - 1/280*(70*B*b^3*x^4*sign(b*x + a)
+ 168*B*a*b^2*x^3*sign(b*x + a) + 56*A*b^3*x^3*sign(b*x + a) + 140*B*a^2*b*x^2*s
ign(b*x + a) + 140*A*a*b^2*x^2*sign(b*x + a) + 40*B*a^3*x*sign(b*x + a) + 120*A*
a^2*b*x*sign(b*x + a) + 35*A*a^3*sign(b*x + a))/x^8