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

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

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {446, 78, 37} \begin {gather*} \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}} \end {gather*}

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 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]

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 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]]

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*}

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Mathematica [B]  time = 0.04, size = 118, normalized size = 2.46 \begin {gather*} -\frac {a^5 \left (6 A+7 B x^3\right )+7 a^4 b x^3 \left (5 A+6 B x^3\right )+21 a^3 b^2 x^6 \left (4 A+5 B x^3\right )+35 a^2 b^3 x^9 \left (3 A+4 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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/126*(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))/x^21

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{22}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.61, size = 121, normalized size = 2.52 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 0.18, size = 127, normalized size = 2.65 \begin {gather*} -\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 {gather*}

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

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maple [B]  time = 0.05, size = 104, normalized size = 2.17 \begin {gather*} -\frac {B \,b^{5}}{3 x^{3}}-\frac {\left (A b +5 B a \right ) b^{4}}{6 x^{6}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{9 x^{9}}-\frac {5 \left (A b +B a \right ) a^{2} b^{2}}{6 x^{12}}-\frac {\left (2 A b +B a \right ) a^{3} b}{3 x^{15}}-\frac {\left (5 A b +B a \right ) a^{4}}{18 x^{18}}-\frac {A \,a^{5}}{21 x^{21}} \end {gather*}

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]

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

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maxima [B]  time = 0.61, size = 121, normalized size = 2.52 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 2.37, size = 122, normalized size = 2.54 \begin {gather*} -\frac {\frac {A\,a^5}{21}+x^6\,\left (\frac {B\,a^4\,b}{3}+\frac {2\,A\,a^3\,b^2}{3}\right )+x^{12}\,\left (\frac {10\,B\,a^2\,b^3}{9}+\frac {5\,A\,a\,b^4}{9}\right )+x^3\,\left (\frac {B\,a^5}{18}+\frac {5\,A\,b\,a^4}{18}\right )+x^{15}\,\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )+x^9\,\left (\frac {5\,B\,a^3\,b^2}{6}+\frac {5\,A\,a^2\,b^3}{6}\right )+\frac {B\,b^5\,x^{18}}{3}}{x^{21}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((A*a^5)/21 + x^6*((2*A*a^3*b^2)/3 + (B*a^4*b)/3) + x^12*((10*B*a^2*b^3)/9 + (5*A*a*b^4)/9) + x^3*((B*a^5)/18
+ (5*A*a^4*b)/18) + x^15*((A*b^5)/6 + (5*B*a*b^4)/6) + x^9*((5*A*a^2*b^3)/6 + (5*B*a^3*b^2)/6) + (B*b^5*x^18)
/3)/x^21

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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