3.49 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{a^5 A}{16 x^{16}}-\frac{a^4 (a B+5 A b)}{13 x^{13}}-\frac{a^3 b (a B+2 A b)}{2 x^{10}}-\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{b^4 (5 a B+A b)}{x}-\frac{5 a b^3 (2 a B+A b)}{4 x^4}+\frac{1}{2} b^5 B x^2 \]

[Out]

-(a^5*A)/(16*x^16) - (a^4*(5*A*b + a*B))/(13*x^13) - (a^3*b*(2*A*b + a*B))/(2*x^
10) - (10*a^2*b^2*(A*b + a*B))/(7*x^7) - (5*a*b^3*(A*b + 2*a*B))/(4*x^4) - (b^4*
(A*b + 5*a*B))/x + (b^5*B*x^2)/2

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Rubi [A]  time = 0.214096, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{a^5 A}{16 x^{16}}-\frac{a^4 (a B+5 A b)}{13 x^{13}}-\frac{a^3 b (a B+2 A b)}{2 x^{10}}-\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{b^4 (5 a B+A b)}{x}-\frac{5 a b^3 (2 a B+A b)}{4 x^4}+\frac{1}{2} b^5 B x^2 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^17,x]

[Out]

-(a^5*A)/(16*x^16) - (a^4*(5*A*b + a*B))/(13*x^13) - (a^3*b*(2*A*b + a*B))/(2*x^
10) - (10*a^2*b^2*(A*b + a*B))/(7*x^7) - (5*a*b^3*(A*b + 2*a*B))/(4*x^4) - (b^4*
(A*b + 5*a*B))/x + (b^5*B*x^2)/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{16 x^{16}} + B b^{5} \int x\, dx - \frac{a^{4} \left (5 A b + B a\right )}{13 x^{13}} - \frac{a^{3} b \left (2 A b + B a\right )}{2 x^{10}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{7 x^{7}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{4 x^{4}} - \frac{b^{4} \left (A b + 5 B a\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**17,x)

[Out]

-A*a**5/(16*x**16) + B*b**5*Integral(x, x) - a**4*(5*A*b + B*a)/(13*x**13) - a**
3*b*(2*A*b + B*a)/(2*x**10) - 10*a**2*b**2*(A*b + B*a)/(7*x**7) - 5*a*b**3*(A*b
+ 2*B*a)/(4*x**4) - b**4*(A*b + 5*B*a)/x

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Mathematica [A]  time = 0.063221, size = 118, normalized size = 1.03 \[ -\frac{7 a^5 \left (13 A+16 B x^3\right )+56 a^4 b x^3 \left (10 A+13 B x^3\right )+208 a^3 b^2 x^6 \left (7 A+10 B x^3\right )+520 a^2 b^3 x^9 \left (4 A+7 B x^3\right )+1820 a b^4 x^{12} \left (A+4 B x^3\right )-728 b^5 x^{15} \left (B x^3-2 A\right )}{1456 x^{16}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^17,x]

[Out]

-(-728*b^5*x^15*(-2*A + B*x^3) + 1820*a*b^4*x^12*(A + 4*B*x^3) + 520*a^2*b^3*x^9
*(4*A + 7*B*x^3) + 208*a^3*b^2*x^6*(7*A + 10*B*x^3) + 56*a^4*b*x^3*(10*A + 13*B*
x^3) + 7*a^5*(13*A + 16*B*x^3))/(1456*x^16)

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Maple [A]  time = 0.008, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{16\,{x}^{16}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{2\,{x}^{10}}}-{\frac{10\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{7\,{x}^{7}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{4\,{x}^{4}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{x}}+{\frac{{b}^{5}B{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^17,x)

[Out]

-1/16*a^5*A/x^16-1/13*a^4*(5*A*b+B*a)/x^13-1/2*a^3*b*(2*A*b+B*a)/x^10-10/7*a^2*b
^2*(A*b+B*a)/x^7-5/4*a*b^3*(A*b+2*B*a)/x^4-b^4*(A*b+5*B*a)/x+1/2*b^5*B*x^2

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Maxima [A]  time = 1.38613, size = 165, normalized size = 1.43 \[ \frac{1}{2} \, B b^{5} x^{2} - \frac{1456 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 1820 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2080 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 728 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 91 \, A a^{5} + 112 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{1456 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^17,x, algorithm="maxima")

[Out]

1/2*B*b^5*x^2 - 1/1456*(1456*(5*B*a*b^4 + A*b^5)*x^15 + 1820*(2*B*a^2*b^3 + A*a*
b^4)*x^12 + 2080*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 728*(B*a^4*b + 2*A*a^3*b^2)*x^6 +
91*A*a^5 + 112*(B*a^5 + 5*A*a^4*b)*x^3)/x^16

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Fricas [A]  time = 0.21805, size = 163, normalized size = 1.42 \[ \frac{728 \, B b^{5} x^{18} - 1456 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 1820 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 2080 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 728 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 91 \, A a^{5} - 112 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{1456 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^17,x, algorithm="fricas")

[Out]

1/1456*(728*B*b^5*x^18 - 1456*(5*B*a*b^4 + A*b^5)*x^15 - 1820*(2*B*a^2*b^3 + A*a
*b^4)*x^12 - 2080*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 728*(B*a^4*b + 2*A*a^3*b^2)*x^6
- 91*A*a^5 - 112*(B*a^5 + 5*A*a^4*b)*x^3)/x^16

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Sympy [A]  time = 129.937, size = 126, normalized size = 1.1 \[ \frac{B b^{5} x^{2}}{2} - \frac{91 A a^{5} + x^{15} \left (1456 A b^{5} + 7280 B a b^{4}\right ) + x^{12} \left (1820 A a b^{4} + 3640 B a^{2} b^{3}\right ) + x^{9} \left (2080 A a^{2} b^{3} + 2080 B a^{3} b^{2}\right ) + x^{6} \left (1456 A a^{3} b^{2} + 728 B a^{4} b\right ) + x^{3} \left (560 A a^{4} b + 112 B a^{5}\right )}{1456 x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**17,x)

[Out]

B*b**5*x**2/2 - (91*A*a**5 + x**15*(1456*A*b**5 + 7280*B*a*b**4) + x**12*(1820*A
*a*b**4 + 3640*B*a**2*b**3) + x**9*(2080*A*a**2*b**3 + 2080*B*a**3*b**2) + x**6*
(1456*A*a**3*b**2 + 728*B*a**4*b) + x**3*(560*A*a**4*b + 112*B*a**5))/(1456*x**1
6)

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GIAC/XCAS [A]  time = 0.214733, size = 173, normalized size = 1.5 \[ \frac{1}{2} \, B b^{5} x^{2} - \frac{7280 \, B a b^{4} x^{15} + 1456 \, A b^{5} x^{15} + 3640 \, B a^{2} b^{3} x^{12} + 1820 \, A a b^{4} x^{12} + 2080 \, B a^{3} b^{2} x^{9} + 2080 \, A a^{2} b^{3} x^{9} + 728 \, B a^{4} b x^{6} + 1456 \, A a^{3} b^{2} x^{6} + 112 \, B a^{5} x^{3} + 560 \, A a^{4} b x^{3} + 91 \, A a^{5}}{1456 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^17,x, algorithm="giac")

[Out]

1/2*B*b^5*x^2 - 1/1456*(7280*B*a*b^4*x^15 + 1456*A*b^5*x^15 + 3640*B*a^2*b^3*x^1
2 + 1820*A*a*b^4*x^12 + 2080*B*a^3*b^2*x^9 + 2080*A*a^2*b^3*x^9 + 728*B*a^4*b*x^
6 + 1456*A*a^3*b^2*x^6 + 112*B*a^5*x^3 + 560*A*a^4*b*x^3 + 91*A*a^5)/x^16