3.54 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^{22}} \, dx\)

Optimal. Leaf size=48 \[ \frac{\left (a+b x^3\right )^6 (A b-7 a B)}{126 a^2 x^{18}}-\frac{A \left (a+b x^3\right )^6}{21 a x^{21}} \]

[Out]

-(A*(a + b*x^3)^6)/(21*a*x^21) + ((A*b - 7*a*B)*(a + b*x^3)^6)/(126*a^2*x^18)

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Rubi [A]  time = 0.031515, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {446, 78, 37} \[ \frac{\left (a+b x^3\right )^6 (A b-7 a B)}{126 a^2 x^{18}}-\frac{A \left (a+b x^3\right )^6}{21 a x^{21}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^22,x]

[Out]

-(A*(a + b*x^3)^6)/(21*a*x^21) + ((A*b - 7*a*B)*(a + b*x^3)^6)/(126*a^2*x^18)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{22}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^8} \, dx,x,x^3\right )\\ &=-\frac{A \left (a+b x^3\right )^6}{21 a x^{21}}+\frac{(-A b+7 a B) \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^7} \, dx,x,x^3\right )}{21 a}\\ &=-\frac{A \left (a+b x^3\right )^6}{21 a x^{21}}+\frac{(A b-7 a B) \left (a+b x^3\right )^6}{126 a^2 x^{18}}\\ \end{align*}

Mathematica [B]  time = 0.0293528, size = 118, normalized size = 2.46 \[ -\frac{35 a^2 b^3 x^9 \left (3 A+4 B x^3\right )+21 a^3 b^2 x^6 \left (4 A+5 B x^3\right )+7 a^4 b x^3 \left (5 A+6 B x^3\right )+a^5 \left (6 A+7 B x^3\right )+35 a b^4 x^{12} \left (2 A+3 B x^3\right )+21 b^5 x^{15} \left (A+2 B x^3\right )}{126 x^{21}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^22,x]

[Out]

-(21*b^5*x^15*(A + 2*B*x^3) + 35*a*b^4*x^12*(2*A + 3*B*x^3) + 35*a^2*b^3*x^9*(3*A + 4*B*x^3) + 21*a^3*b^2*x^6*
(4*A + 5*B*x^3) + 7*a^4*b*x^3*(5*A + 6*B*x^3) + a^5*(6*A + 7*B*x^3))/(126*x^21)

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Maple [B]  time = 0.009, size = 104, normalized size = 2.2 \begin{align*} -{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{18\,{x}^{18}}}-{\frac{A{a}^{5}}{21\,{x}^{21}}}-{\frac{B{b}^{5}}{3\,{x}^{3}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{6\,{x}^{12}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{6\,{x}^{6}}}-{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{3\,{x}^{15}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^22,x)

[Out]

-1/18*a^4*(5*A*b+B*a)/x^18-1/21*A*a^5/x^21-1/3*B*b^5/x^3-5/6*a^2*b^2*(A*b+B*a)/x^12-1/6*b^4*(A*b+5*B*a)/x^6-1/
3*a^3*b*(2*A*b+B*a)/x^15-5/9*a*b^3*(A*b+2*B*a)/x^9

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Maxima [B]  time = 1.29429, size = 163, normalized size = 3.4 \begin{align*} -\frac{42 \, B b^{5} x^{18} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 70 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 6 \, A a^{5} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{126 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^22,x, algorithm="maxima")

[Out]

-1/126*(42*B*b^5*x^18 + 21*(5*B*a*b^4 + A*b^5)*x^15 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 105*(B*a^3*b^2 + A*a^2
*b^3)*x^9 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 6*A*a^5 + 7*(B*a^5 + 5*A*a^4*b)*x^3)/x^21

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Fricas [B]  time = 1.32716, size = 267, normalized size = 5.56 \begin{align*} -\frac{42 \, B b^{5} x^{18} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 70 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 6 \, A a^{5} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{126 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^22,x, algorithm="fricas")

[Out]

-1/126*(42*B*b^5*x^18 + 21*(5*B*a*b^4 + A*b^5)*x^15 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 105*(B*a^3*b^2 + A*a^2
*b^3)*x^9 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 6*A*a^5 + 7*(B*a^5 + 5*A*a^4*b)*x^3)/x^21

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**22,x)

[Out]

Timed out

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Giac [B]  time = 1.1171, size = 171, normalized size = 3.56 \begin{align*} -\frac{42 \, B b^{5} x^{18} + 105 \, B a b^{4} x^{15} + 21 \, A b^{5} x^{15} + 140 \, B a^{2} b^{3} x^{12} + 70 \, A a b^{4} x^{12} + 105 \, B a^{3} b^{2} x^{9} + 105 \, A a^{2} b^{3} x^{9} + 42 \, B a^{4} b x^{6} + 84 \, A a^{3} b^{2} x^{6} + 7 \, B a^{5} x^{3} + 35 \, A a^{4} b x^{3} + 6 \, A a^{5}}{126 \, x^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^22,x, algorithm="giac")

[Out]

-1/126*(42*B*b^5*x^18 + 105*B*a*b^4*x^15 + 21*A*b^5*x^15 + 140*B*a^2*b^3*x^12 + 70*A*a*b^4*x^12 + 105*B*a^3*b^
2*x^9 + 105*A*a^2*b^3*x^9 + 42*B*a^4*b*x^6 + 84*A*a^3*b^2*x^6 + 7*B*a^5*x^3 + 35*A*a^4*b*x^3 + 6*A*a^5)/x^21