3.703 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{14}} \, dx\)

Optimal. Leaf size=304 \[ -\frac{a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{10} (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{8 x^8 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 x^9 (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{12 x^{12} (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 x^{11} (a+b x)} \]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*x^13*(a + b*x)) - (a^4*(5*A*b + a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*x^12*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^2*b^2*(A*b + a*B)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(x^10*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(9*x^9*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(8*x^8*(a + b*x)) - (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x))

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Rubi [A]  time = 0.373877, antiderivative size = 304, 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 \[ -\frac{a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{10} (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{8 x^8 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 x^9 (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{12 x^{12} (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 x^{11} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^14,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*x^13*(a + b*x)) - (a^4*(5*A*b + a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*x^12*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^2*b^2*(A*b + a*B)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(x^10*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(9*x^9*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(8*x^8*(a + b*x)) - (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x))

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Rubi in Sympy [A]  time = 35.4518, size = 291, normalized size = 0.96 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{26 a x^{13}} - \frac{b^{4} \left (7 A b - 13 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{72072 x^{8} \left (a + b x\right )} + \frac{b^{4} \left (7 A b - 13 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 a x^{8}} + \frac{b^{3} \left (a + b x\right ) \left (7 A b - 13 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2574 a x^{9}} + \frac{b^{2} \left (7 A b - 13 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{858 a x^{10}} + \frac{5 b \left (a + b x\right ) \left (7 A b - 13 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{1716 a x^{11}} + \frac{\left (7 A b - 13 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{156 a x^{12}} \]

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**14,x)

[Out]

-A*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(26*a*x**13) - b**4*(7*A*b
- 13*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(72072*x**8*(a + b*x)) + b**4*(7*A*b
- 13*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(9009*a*x**8) + b**3*(a + b*x)*(7*A*b
- 13*B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(2574*a*x**9) + b**2*(7*A*b - 13*B*a
)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(858*a*x**10) + 5*b*(a + b*x)*(7*A*b - 13*
B*a)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(1716*a*x**11) + (7*A*b - 13*B*a)*(a**2
+ 2*a*b*x + b**2*x**2)**(5/2)/(156*a*x**12)

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Mathematica [A]  time = 0.0785955, size = 125, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (462 a^5 (12 A+13 B x)+2730 a^4 b x (11 A+12 B x)+6552 a^3 b^2 x^2 (10 A+11 B x)+8008 a^2 b^3 x^3 (9 A+10 B x)+5005 a b^4 x^4 (8 A+9 B x)+1287 b^5 x^5 (7 A+8 B x)\right )}{72072 x^{13} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^14,x]

[Out]

-(Sqrt[(a + b*x)^2]*(1287*b^5*x^5*(7*A + 8*B*x) + 5005*a*b^4*x^4*(8*A + 9*B*x) +
8008*a^2*b^3*x^3*(9*A + 10*B*x) + 6552*a^3*b^2*x^2*(10*A + 11*B*x) + 2730*a^4*b
*x*(11*A + 12*B*x) + 462*a^5*(12*A + 13*B*x)))/(72072*x^13*(a + b*x))

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Maple [A]  time = 0.012, size = 140, normalized size = 0.5 \[ -{\frac{10296\,B{b}^{5}{x}^{6}+9009\,A{x}^{5}{b}^{5}+45045\,B{x}^{5}a{b}^{4}+40040\,A{x}^{4}a{b}^{4}+80080\,B{x}^{4}{a}^{2}{b}^{3}+72072\,A{x}^{3}{a}^{2}{b}^{3}+72072\,B{x}^{3}{a}^{3}{b}^{2}+65520\,A{x}^{2}{a}^{3}{b}^{2}+32760\,B{x}^{2}{a}^{4}b+30030\,Ax{a}^{4}b+6006\,Bx{a}^{5}+5544\,A{a}^{5}}{72072\,{x}^{13} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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^14,x)

[Out]

-1/72072*(10296*B*b^5*x^6+9009*A*b^5*x^5+45045*B*a*b^4*x^5+40040*A*a*b^4*x^4+800
80*B*a^2*b^3*x^4+72072*A*a^2*b^3*x^3+72072*B*a^3*b^2*x^3+65520*A*a^3*b^2*x^2+327
60*B*a^4*b*x^2+30030*A*a^4*b*x+6006*B*a^5*x+5544*A*a^5)*((b*x+a)^2)^(5/2)/x^13/(
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^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275721, size = 161, normalized size = 0.53 \[ -\frac{10296 \, B b^{5} x^{6} + 5544 \, A a^{5} + 9009 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 40040 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 72072 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 32760 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 6006 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{72072 \, x^{13}} \]

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^14,x, algorithm="fricas")

[Out]

-1/72072*(10296*B*b^5*x^6 + 5544*A*a^5 + 9009*(5*B*a*b^4 + A*b^5)*x^5 + 40040*(2
*B*a^2*b^3 + A*a*b^4)*x^4 + 72072*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 32760*(B*a^4*b +
2*A*a^3*b^2)*x^2 + 6006*(B*a^5 + 5*A*a^4*b)*x)/x^13

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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**14,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275761, size = 298, normalized size = 0.98 \[ -\frac{{\left (13 \, B a b^{12} - 7 \, A b^{13}\right )}{\rm sign}\left (b x + a\right )}{72072 \, a^{8}} - \frac{10296 \, B b^{5} x^{6}{\rm sign}\left (b x + a\right ) + 45045 \, B a b^{4} x^{5}{\rm sign}\left (b x + a\right ) + 9009 \, A b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 80080 \, B a^{2} b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 40040 \, A a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 72072 \, B a^{3} b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 72072 \, A a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 32760 \, B a^{4} b x^{2}{\rm sign}\left (b x + a\right ) + 65520 \, A a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 6006 \, B a^{5} x{\rm sign}\left (b x + a\right ) + 30030 \, A a^{4} b x{\rm sign}\left (b x + a\right ) + 5544 \, A a^{5}{\rm sign}\left (b x + a\right )}{72072 \, x^{13}} \]

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^14,x, algorithm="giac")

[Out]

-1/72072*(13*B*a*b^12 - 7*A*b^13)*sign(b*x + a)/a^8 - 1/72072*(10296*B*b^5*x^6*s
ign(b*x + a) + 45045*B*a*b^4*x^5*sign(b*x + a) + 9009*A*b^5*x^5*sign(b*x + a) +
80080*B*a^2*b^3*x^4*sign(b*x + a) + 40040*A*a*b^4*x^4*sign(b*x + a) + 72072*B*a^
3*b^2*x^3*sign(b*x + a) + 72072*A*a^2*b^3*x^3*sign(b*x + a) + 32760*B*a^4*b*x^2*
sign(b*x + a) + 65520*A*a^3*b^2*x^2*sign(b*x + a) + 6006*B*a^5*x*sign(b*x + a) +
30030*A*a^4*b*x*sign(b*x + a) + 5544*A*a^5*sign(b*x + a))/x^13