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

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

[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 = 0.0615403, 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, Rules used = {448} \[ -\frac{10 a^2 b^2 (a B+A b)}{7 x^7}-\frac{a^4 (a B+5 A b)}{13 x^{13}}-\frac{a^3 b (a B+2 A b)}{2 x^{10}}-\frac{a^5 A}{16 x^{16}}-\frac{5 a b^3 (2 a B+A b)}{4 x^4}-\frac{b^4 (5 a B+A b)}{x}+\frac{1}{2} b^5 B x^2 \]

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

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{17}} \, dx &=\int \left (\frac{a^5 A}{x^{17}}+\frac{a^4 (5 A b+a B)}{x^{14}}+\frac{5 a^3 b (2 A b+a B)}{x^{11}}+\frac{10 a^2 b^2 (A b+a B)}{x^8}+\frac{5 a b^3 (A b+2 a B)}{x^5}+\frac{b^4 (A b+5 a B)}{x^2}+b^5 B x\right ) \, dx\\ &=-\frac{a^5 A}{16 x^{16}}-\frac{a^4 (5 A b+a B)}{13 x^{13}}-\frac{a^3 b (2 A b+a B)}{2 x^{10}}-\frac{10 a^2 b^2 (A b+a B)}{7 x^7}-\frac{5 a b^3 (A b+2 a B)}{4 x^4}-\frac{b^4 (A b+5 a B)}{x}+\frac{1}{2} b^5 B x^2\\ \end{align*}

Mathematica [A]  time = 0.0310625, size = 118, normalized size = 1.03 \[ -\frac{520 a^2 b^3 x^9 \left (4 A+7 B x^3\right )+208 a^3 b^2 x^6 \left (7 A+10 B x^3\right )+56 a^4 b x^3 \left (10 A+13 B x^3\right )+7 a^5 \left (13 A+16 B x^3\right )+1820 a b^4 x^{12} \left (A+4 B x^3\right )-728 b^5 x^{15} \left (B x^3-2 A\right )}{1456 x^{16}} \]

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 = 0.006, size = 104, normalized size = 0.9 \begin{align*} -{\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}} \end{align*}

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 = 1.03681, size = 165, normalized size = 1.43 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/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 = 1.43416, size = 281, normalized size = 2.44 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/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 = 52.7244, size = 126, normalized size = 1.1 \begin{align*} \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}} \end{align*}

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**16)

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Giac [A]  time = 1.18053, size = 173, normalized size = 1.5 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/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^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 + 9
1*A*a^5)/x^16