3.680 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{10}} \, dx\)

Optimal. Leaf size=210 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{8 x^8 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 x^7 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 x^6 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)} \]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (a^2*(3*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (3*a*b*(A*b + a*B)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^2*(A*b + 3*a*B)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(6*x^6*(a + b*x)) - (b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b
*x))

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Rubi [A]  time = 0.241359, 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 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{8 x^8 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{7 x^7 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 x^6 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^10,x]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (a^2*(3*A*b + a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (3*a*b*(A*b + a*B)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^2*(A*b + 3*a*B)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(6*x^6*(a + b*x)) - (b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b
*x))

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Rubi in Sympy [A]  time = 25.145, size = 202, normalized size = 0.96 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{18 a x^{9}} - \frac{b^{2} \left (5 A b - 9 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2520 x^{6} \left (a + b x\right )} + \frac{b^{2} \left (5 A b - 9 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{420 a x^{6}} + \frac{b \left (a + b x\right ) \left (5 A b - 9 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{168 a x^{7}} + \frac{\left (5 A b - 9 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{72 a x^{8}} \]

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**10,x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(18*a*x**9) - b**2*(5*A*b -
9*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2520*x**6*(a + b*x)) + b**2*(5*A*b - 9
*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(420*a*x**6) + b*(a + b*x)*(5*A*b - 9*B*a
)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(168*a*x**7) + (5*A*b - 9*B*a)*(a**2 + 2*a*b*
x + b**2*x**2)**(3/2)/(72*a*x**8)

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Mathematica [A]  time = 0.0543658, size = 87, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (35 a^3 (8 A+9 B x)+135 a^2 b x (7 A+8 B x)+180 a b^2 x^2 (6 A+7 B x)+84 b^3 x^3 (5 A+6 B x)\right )}{2520 x^9 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^10,x]

[Out]

-(Sqrt[(a + b*x)^2]*(84*b^3*x^3*(5*A + 6*B*x) + 180*a*b^2*x^2*(6*A + 7*B*x) + 13
5*a^2*b*x*(7*A + 8*B*x) + 35*a^3*(8*A + 9*B*x)))/(2520*x^9*(a + b*x))

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Maple [A]  time = 0.011, size = 92, normalized size = 0.4 \[ -{\frac{504\,B{x}^{4}{b}^{3}+420\,A{b}^{3}{x}^{3}+1260\,B{x}^{3}a{b}^{2}+1080\,A{x}^{2}a{b}^{2}+1080\,B{x}^{2}{a}^{2}b+945\,A{a}^{2}bx+315\,{a}^{3}Bx+280\,A{a}^{3}}{2520\,{x}^{9} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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^10,x)

[Out]

-1/2520*(504*B*b^3*x^4+420*A*b^3*x^3+1260*B*a*b^2*x^3+1080*A*a*b^2*x^2+1080*B*a^
2*b*x^2+945*A*a^2*b*x+315*B*a^3*x+280*A*a^3)*((b*x+a)^2)^(3/2)/x^9/(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^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277008, size = 99, normalized size = 0.47 \[ -\frac{504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 315 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \]

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^10,x, algorithm="fricas")

[Out]

-1/2520*(504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b
+ A*a*b^2)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{10}}\, dx \]

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**10,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**10, x)

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GIAC/XCAS [A]  time = 0.274819, size = 201, normalized size = 0.96 \[ -\frac{{\left (9 \, B a b^{8} - 5 \, A b^{9}\right )}{\rm sign}\left (b x + a\right )}{2520 \, a^{6}} - \frac{504 \, B b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 1260 \, B a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 420 \, A b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 1080 \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + 1080 \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 315 \, B a^{3} x{\rm sign}\left (b x + a\right ) + 945 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + 280 \, A a^{3}{\rm sign}\left (b x + a\right )}{2520 \, x^{9}} \]

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^10,x, algorithm="giac")

[Out]

-1/2520*(9*B*a*b^8 - 5*A*b^9)*sign(b*x + a)/a^6 - 1/2520*(504*B*b^3*x^4*sign(b*x
+ a) + 1260*B*a*b^2*x^3*sign(b*x + a) + 420*A*b^3*x^3*sign(b*x + a) + 1080*B*a^
2*b*x^2*sign(b*x + a) + 1080*A*a*b^2*x^2*sign(b*x + a) + 315*B*a^3*x*sign(b*x +
a) + 945*A*a^2*b*x*sign(b*x + a) + 280*A*a^3*sign(b*x + a))/x^9