3.51 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^{19}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{5 a^2 b^3 B}{3 x^6}-\frac{10 a^3 b^2 B}{9 x^9}-\frac{5 a^4 b B}{12 x^{12}}-\frac{a^5 B}{15 x^{15}}-\frac{A \left (a+b x^3\right )^6}{18 a x^{18}}-\frac{5 a b^4 B}{3 x^3}+b^5 B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0538979, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {446, 78, 43} \[ -\frac{5 a^2 b^3 B}{3 x^6}-\frac{10 a^3 b^2 B}{9 x^9}-\frac{5 a^4 b B}{12 x^{12}}-\frac{a^5 B}{15 x^{15}}-\frac{A \left (a+b x^3\right )^6}{18 a x^{18}}-\frac{5 a b^4 B}{3 x^3}+b^5 B \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^19,x]

[Out]

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

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{19}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^7} \, dx,x,x^3\right )\\ &=-\frac{A \left (a+b x^3\right )^6}{18 a x^{18}}+\frac{1}{3} B \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^6} \, dx,x,x^3\right )\\ &=-\frac{A \left (a+b x^3\right )^6}{18 a x^{18}}+\frac{1}{3} B \operatorname{Subst}\left (\int \left (\frac{a^5}{x^6}+\frac{5 a^4 b}{x^5}+\frac{10 a^3 b^2}{x^4}+\frac{10 a^2 b^3}{x^3}+\frac{5 a b^4}{x^2}+\frac{b^5}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^5 B}{15 x^{15}}-\frac{5 a^4 b B}{12 x^{12}}-\frac{10 a^3 b^2 B}{9 x^9}-\frac{5 a^2 b^3 B}{3 x^6}-\frac{5 a b^4 B}{3 x^3}-\frac{A \left (a+b x^3\right )^6}{18 a x^{18}}+b^5 B \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0376973, size = 121, normalized size = 1.33 \[ -\frac{100 a^2 b^3 x^9 \left (2 A+3 B x^3\right )+50 a^3 b^2 x^6 \left (3 A+4 B x^3\right )+15 a^4 b x^3 \left (4 A+5 B x^3\right )+2 a^5 \left (5 A+6 B x^3\right )+150 a b^4 x^{12} \left (A+2 B x^3\right )+60 A b^5 x^{15}-180 b^5 B x^{18} \log (x)}{180 x^{18}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^19,x]

[Out]

-(60*A*b^5*x^15 + 150*a*b^4*x^12*(A + 2*B*x^3) + 100*a^2*b^3*x^9*(2*A + 3*B*x^3) + 50*a^3*b^2*x^6*(3*A + 4*B*x
^3) + 15*a^4*b*x^3*(4*A + 5*B*x^3) + 2*a^5*(5*A + 6*B*x^3) - 180*b^5*B*x^18*Log[x])/(180*x^18)

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Maple [A]  time = 0.007, size = 124, normalized size = 1.4 \begin{align*} -{\frac{5\,a{b}^{4}A}{6\,{x}^{6}}}-{\frac{5\,{a}^{2}{b}^{3}B}{3\,{x}^{6}}}-{\frac{{b}^{5}A}{3\,{x}^{3}}}-{\frac{5\,a{b}^{4}B}{3\,{x}^{3}}}-{\frac{{a}^{4}bA}{3\,{x}^{15}}}-{\frac{{a}^{5}B}{15\,{x}^{15}}}-{\frac{A{a}^{5}}{18\,{x}^{18}}}-{\frac{10\,A{b}^{3}{a}^{2}}{9\,{x}^{9}}}-{\frac{10\,{a}^{3}{b}^{2}B}{9\,{x}^{9}}}-{\frac{5\,{a}^{3}{b}^{2}A}{6\,{x}^{12}}}-{\frac{5\,{a}^{4}bB}{12\,{x}^{12}}}+{b}^{5}B\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^19,x)

[Out]

-5/6*a*b^4/x^6*A-5/3*a^2*b^3*B/x^6-1/3*b^5/x^3*A-5/3*a*b^4*B/x^3-1/3*a^4/x^15*A*b-1/15*a^5*B/x^15-1/18*A*a^5/x
^18-10/9*a^2*b^3/x^9*A-10/9*a^3*b^2*B/x^9-5/6*a^3*b^2/x^12*A-5/12*a^4*b*B/x^12+b^5*B*ln(x)

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Maxima [A]  time = 1.10264, size = 166, normalized size = 1.82 \begin{align*} \frac{1}{3} \, B b^{5} \log \left (x^{3}\right ) - \frac{60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 10 \, A a^{5} + 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{180 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^19,x, algorithm="maxima")

[Out]

1/3*B*b^5*log(x^3) - 1/180*(60*(5*B*a*b^4 + A*b^5)*x^15 + 150*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 200*(B*a^3*b^2 +
A*a^2*b^3)*x^9 + 75*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 10*A*a^5 + 12*(B*a^5 + 5*A*a^4*b)*x^3)/x^18

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Fricas [A]  time = 1.45278, size = 281, normalized size = 3.09 \begin{align*} \frac{180 \, B b^{5} x^{18} \log \left (x\right ) - 60 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 150 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 200 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 10 \, A a^{5} - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{180 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^19,x, algorithm="fricas")

[Out]

1/180*(180*B*b^5*x^18*log(x) - 60*(5*B*a*b^4 + A*b^5)*x^15 - 150*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 200*(B*a^3*b^2
+ A*a^2*b^3)*x^9 - 75*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 10*A*a^5 - 12*(B*a^5 + 5*A*a^4*b)*x^3)/x^18

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Sympy [A]  time = 153.16, size = 124, normalized size = 1.36 \begin{align*} B b^{5} \log{\left (x \right )} - \frac{10 A a^{5} + x^{15} \left (60 A b^{5} + 300 B a b^{4}\right ) + x^{12} \left (150 A a b^{4} + 300 B a^{2} b^{3}\right ) + x^{9} \left (200 A a^{2} b^{3} + 200 B a^{3} b^{2}\right ) + x^{6} \left (150 A a^{3} b^{2} + 75 B a^{4} b\right ) + x^{3} \left (60 A a^{4} b + 12 B a^{5}\right )}{180 x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**19,x)

[Out]

B*b**5*log(x) - (10*A*a**5 + x**15*(60*A*b**5 + 300*B*a*b**4) + x**12*(150*A*a*b**4 + 300*B*a**2*b**3) + x**9*
(200*A*a**2*b**3 + 200*B*a**3*b**2) + x**6*(150*A*a**3*b**2 + 75*B*a**4*b) + x**3*(60*A*a**4*b + 12*B*a**5))/(
180*x**18)

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Giac [A]  time = 1.18607, size = 184, normalized size = 2.02 \begin{align*} B b^{5} \log \left ({\left | x \right |}\right ) - \frac{147 \, B b^{5} x^{18} + 300 \, B a b^{4} x^{15} + 60 \, A b^{5} x^{15} + 300 \, B a^{2} b^{3} x^{12} + 150 \, A a b^{4} x^{12} + 200 \, B a^{3} b^{2} x^{9} + 200 \, A a^{2} b^{3} x^{9} + 75 \, B a^{4} b x^{6} + 150 \, A a^{3} b^{2} x^{6} + 12 \, B a^{5} x^{3} + 60 \, A a^{4} b x^{3} + 10 \, A a^{5}}{180 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^19,x, algorithm="giac")

[Out]

B*b^5*log(abs(x)) - 1/180*(147*B*b^5*x^18 + 300*B*a*b^4*x^15 + 60*A*b^5*x^15 + 300*B*a^2*b^3*x^12 + 150*A*a*b^
4*x^12 + 200*B*a^3*b^2*x^9 + 200*A*a^2*b^3*x^9 + 75*B*a^4*b*x^6 + 150*A*a^3*b^2*x^6 + 12*B*a^5*x^3 + 60*A*a^4*
b*x^3 + 10*A*a^5)/x^18