3.701 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=306 \[ -\frac{5 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^8 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{6 x^6 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 x^7 (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{10 x^{10} (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 x^9 (a+b x)} \]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^4*(5*A*b + a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*x^10*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (5*a^2*b^2*(A*b + a*B)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*x^8*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(6*x^6*(a + b*x)) - (b^5*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 = 0.370289, antiderivative size = 306, 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 \[ -\frac{5 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^8 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{6 x^6 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 x^7 (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{10 x^{10} (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 x^9 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^12,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^4*(5*A*b + a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*x^10*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (5*a^2*b^2*(A*b + a*B)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(4*x^8*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(6*x^6*(a + b*x)) - (b^5*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 = 35.291, size = 289, normalized size = 0.94 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{22 a x^{11}} - \frac{b^{4} \left (5 A b - 11 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{13860 x^{6} \left (a + b x\right )} + \frac{b^{4} \left (5 A b - 11 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2310 a x^{6}} + \frac{b^{3} \left (a + b x\right ) \left (5 A b - 11 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{924 a x^{7}} + \frac{b^{2} \left (5 A b - 11 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{396 a x^{8}} + \frac{b \left (a + b x\right ) \left (5 A b - 11 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{198 a x^{9}} + \frac{\left (5 A b - 11 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{110 a x^{10}} \]

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**12,x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(22*a*x**11) - b**4*(5*A*b
- 11*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(13860*x**6*(a + b*x)) + b**4*(5*A*b
- 11*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2310*a*x**6) + b**3*(a + b*x)*(5*A*b
- 11*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(924*a*x**7) + b**2*(5*A*b - 11*B*a)
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(396*a*x**8) + b*(a + b*x)*(5*A*b - 11*B*a)
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(198*a*x**9) + (5*A*b - 11*B*a)*(a**2 + 2*a
*b*x + b**2*x**2)**(5/2)/(110*a*x**10)

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Mathematica [A]  time = 0.0770154, size = 125, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (126 a^5 (10 A+11 B x)+770 a^4 b x (9 A+10 B x)+1925 a^3 b^2 x^2 (8 A+9 B x)+2475 a^2 b^3 x^3 (7 A+8 B x)+1650 a b^4 x^4 (6 A+7 B x)+462 b^5 x^5 (5 A+6 B x)\right )}{13860 x^{11} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(462*b^5*x^5*(5*A + 6*B*x) + 1650*a*b^4*x^4*(6*A + 7*B*x) +
2475*a^2*b^3*x^3*(7*A + 8*B*x) + 1925*a^3*b^2*x^2*(8*A + 9*B*x) + 770*a^4*b*x*(9
*A + 10*B*x) + 126*a^5*(10*A + 11*B*x)))/(13860*x^11*(a + b*x))

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Maple [A]  time = 0.011, size = 140, normalized size = 0.5 \[ -{\frac{2772\,B{b}^{5}{x}^{6}+2310\,A{x}^{5}{b}^{5}+11550\,B{x}^{5}a{b}^{4}+9900\,A{x}^{4}a{b}^{4}+19800\,B{x}^{4}{a}^{2}{b}^{3}+17325\,A{x}^{3}{a}^{2}{b}^{3}+17325\,B{x}^{3}{a}^{3}{b}^{2}+15400\,A{x}^{2}{a}^{3}{b}^{2}+7700\,B{x}^{2}{a}^{4}b+6930\,Ax{a}^{4}b+1386\,Bx{a}^{5}+1260\,A{a}^{5}}{13860\,{x}^{11} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^12,x)

[Out]

-1/13860*(2772*B*b^5*x^6+2310*A*b^5*x^5+11550*B*a*b^4*x^5+9900*A*a*b^4*x^4+19800
*B*a^2*b^3*x^4+17325*A*a^2*b^3*x^3+17325*B*a^3*b^2*x^3+15400*A*a^3*b^2*x^2+7700*
B*a^4*b*x^2+6930*A*a^4*b*x+1386*B*a^5*x+1260*A*a^5)*((b*x+a)^2)^(5/2)/x^11/(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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302355, size = 161, normalized size = 0.53 \[ -\frac{2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \]

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^12,x, algorithm="fricas")

[Out]

-1/13860*(2772*B*b^5*x^6 + 1260*A*a^5 + 2310*(5*B*a*b^4 + A*b^5)*x^5 + 9900*(2*B
*a^2*b^3 + A*a*b^4)*x^4 + 17325*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 7700*(B*a^4*b + 2*
A*a^3*b^2)*x^2 + 1386*(B*a^5 + 5*A*a^4*b)*x)/x^11

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{12}}\, dx \]

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**12,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**12, x)

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GIAC/XCAS [A]  time = 0.27263, size = 298, normalized size = 0.97 \[ -\frac{{\left (11 \, B a b^{10} - 5 \, A b^{11}\right )}{\rm sign}\left (b x + a\right )}{13860 \, a^{6}} - \frac{2772 \, B b^{5} x^{6}{\rm sign}\left (b x + a\right ) + 11550 \, B a b^{4} x^{5}{\rm sign}\left (b x + a\right ) + 2310 \, A b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 19800 \, B a^{2} b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 9900 \, A a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 17325 \, B a^{3} b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 17325 \, A a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 7700 \, B a^{4} b x^{2}{\rm sign}\left (b x + a\right ) + 15400 \, A a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 1386 \, B a^{5} x{\rm sign}\left (b x + a\right ) + 6930 \, A a^{4} b x{\rm sign}\left (b x + a\right ) + 1260 \, A a^{5}{\rm sign}\left (b x + a\right )}{13860 \, x^{11}} \]

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^12,x, algorithm="giac")

[Out]

-1/13860*(11*B*a*b^10 - 5*A*b^11)*sign(b*x + a)/a^6 - 1/13860*(2772*B*b^5*x^6*si
gn(b*x + a) + 11550*B*a*b^4*x^5*sign(b*x + a) + 2310*A*b^5*x^5*sign(b*x + a) + 1
9800*B*a^2*b^3*x^4*sign(b*x + a) + 9900*A*a*b^4*x^4*sign(b*x + a) + 17325*B*a^3*
b^2*x^3*sign(b*x + a) + 17325*A*a^2*b^3*x^3*sign(b*x + a) + 7700*B*a^4*b*x^2*sig
n(b*x + a) + 15400*A*a^3*b^2*x^2*sign(b*x + a) + 1386*B*a^5*x*sign(b*x + a) + 69
30*A*a^4*b*x*sign(b*x + a) + 1260*A*a^5*sign(b*x + a))/x^11