3.681 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{11}} \, dx\)

Optimal. Leaf size=210 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{9 x^9 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{8 x^8 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 x^7 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)} \]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*x^10*(a + b*x)) - (a^2*(3*A*b + a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (3*a*b*(A*b + a*B)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (b^2*(A*b + 3*a*B)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(7*x^7*(a + b*x)) - (b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*x^6*(a +
b*x))

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Rubi [A]  time = 0.246216, 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 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{9 x^9 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{8 x^8 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 x^7 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^11,x]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*x^10*(a + b*x)) - (a^2*(3*A*b + a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (3*a*b*(A*b + a*B)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (b^2*(A*b + 3*a*B)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(7*x^7*(a + b*x)) - (b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*x^6*(a +
b*x))

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Rubi in Sympy [A]  time = 25.2406, size = 202, normalized size = 0.96 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{20 a x^{10}} - \frac{b^{2} \left (3 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2520 x^{7} \left (a + b x\right )} + \frac{b^{2} \left (3 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{360 a x^{7}} + \frac{b \left (a + b x\right ) \left (3 A b - 5 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{120 a x^{8}} + \frac{\left (3 A b - 5 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{45 a x^{9}} \]

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**11,x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(20*a*x**10) - b**2*(3*A*b
- 5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2520*x**7*(a + b*x)) + b**2*(3*A*b -
5*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(360*a*x**7) + b*(a + b*x)*(3*A*b - 5*B*
a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(120*a*x**8) + (3*A*b - 5*B*a)*(a**2 + 2*a*b
*x + b**2*x**2)**(3/2)/(45*a*x**9)

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Mathematica [A]  time = 0.0598074, size = 87, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (28 a^3 (9 A+10 B x)+105 a^2 b x (8 A+9 B x)+135 a b^2 x^2 (7 A+8 B x)+60 b^3 x^3 (6 A+7 B x)\right )}{2520 x^{10} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^11,x]

[Out]

-(Sqrt[(a + b*x)^2]*(60*b^3*x^3*(6*A + 7*B*x) + 135*a*b^2*x^2*(7*A + 8*B*x) + 10
5*a^2*b*x*(8*A + 9*B*x) + 28*a^3*(9*A + 10*B*x)))/(2520*x^10*(a + b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 0.4 \[ -{\frac{420\,B{x}^{4}{b}^{3}+360\,A{b}^{3}{x}^{3}+1080\,B{x}^{3}a{b}^{2}+945\,A{x}^{2}a{b}^{2}+945\,B{x}^{2}{a}^{2}b+840\,A{a}^{2}bx+280\,{a}^{3}Bx+252\,A{a}^{3}}{2520\,{x}^{10} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^11,x)

[Out]

-1/2520*(420*B*b^3*x^4+360*A*b^3*x^3+1080*B*a*b^2*x^3+945*A*a*b^2*x^2+945*B*a^2*
b*x^2+840*A*a^2*b*x+280*B*a^3*x+252*A*a^3)*((b*x+a)^2)^(3/2)/x^10/(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^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274534, size = 99, normalized size = 0.47 \[ -\frac{420 \, B b^{3} x^{4} + 252 \, A a^{3} + 360 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 945 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 280 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{10}} \]

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^11,x, algorithm="fricas")

[Out]

-1/2520*(420*B*b^3*x^4 + 252*A*a^3 + 360*(3*B*a*b^2 + A*b^3)*x^3 + 945*(B*a^2*b
+ A*a*b^2)*x^2 + 280*(B*a^3 + 3*A*a^2*b)*x)/x^10

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{11}}\, dx \]

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**11,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**11, x)

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GIAC/XCAS [A]  time = 0.271522, size = 201, normalized size = 0.96 \[ \frac{{\left (5 \, B a b^{9} - 3 \, A b^{10}\right )}{\rm sign}\left (b x + a\right )}{2520 \, a^{7}} - \frac{420 \, B b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 1080 \, B a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 360 \, A b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 945 \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + 945 \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 280 \, B a^{3} x{\rm sign}\left (b x + a\right ) + 840 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + 252 \, A a^{3}{\rm sign}\left (b x + a\right )}{2520 \, x^{10}} \]

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^11,x, algorithm="giac")

[Out]

1/2520*(5*B*a*b^9 - 3*A*b^10)*sign(b*x + a)/a^7 - 1/2520*(420*B*b^3*x^4*sign(b*x
+ a) + 1080*B*a*b^2*x^3*sign(b*x + a) + 360*A*b^3*x^3*sign(b*x + a) + 945*B*a^2
*b*x^2*sign(b*x + a) + 945*A*a*b^2*x^2*sign(b*x + a) + 280*B*a^3*x*sign(b*x + a)
+ 840*A*a^2*b*x*sign(b*x + a) + 252*A*a^3*sign(b*x + a))/x^10