3.39 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^7} \, dx\)

Optimal. Leaf size=114 \[ \frac{10}{3} a^2 b^2 x^3 (a B+A b)-\frac{a^4 (a B+5 A b)}{3 x^3}+5 a^3 b \log (x) (a B+2 A b)-\frac{a^5 A}{6 x^6}+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{12} b^5 B x^{12} \]

[Out]

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

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Rubi [A]  time = 0.1058, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ \frac{10}{3} a^2 b^2 x^3 (a B+A b)-\frac{a^4 (a B+5 A b)}{3 x^3}+5 a^3 b \log (x) (a B+2 A b)-\frac{a^5 A}{6 x^6}+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{6} a b^3 x^6 (2 a B+A b)+\frac{1}{12} b^5 B x^{12} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^7,x]

[Out]

-(a^5*A)/(6*x^6) - (a^4*(5*A*b + a*B))/(3*x^3) + (10*a^2*b^2*(A*b + a*B)*x^3)/3 + (5*a*b^3*(A*b + 2*a*B)*x^6)/
6 + (b^4*(A*b + 5*a*B)*x^9)/9 + (b^5*B*x^12)/12 + 5*a^3*b*(2*A*b + a*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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0462325, size = 106, normalized size = 0.93 \[ \frac{1}{36} \left (120 a^2 b^2 x^3 (a B+A b)-\frac{12 a^4 (a B+5 A b)}{x^3}+180 a^3 b \log (x) (a B+2 A b)-\frac{6 a^5 A}{x^6}+4 b^4 x^9 (5 a B+A b)+30 a b^3 x^6 (2 a B+A b)+3 b^5 B x^{12}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^7,x]

[Out]

((-6*a^5*A)/x^6 - (12*a^4*(5*A*b + a*B))/x^3 + 120*a^2*b^2*(A*b + a*B)*x^3 + 30*a*b^3*(A*b + 2*a*B)*x^6 + 4*b^
4*(A*b + 5*a*B)*x^9 + 3*b^5*B*x^12 + 180*a^3*b*(2*A*b + a*B)*Log[x])/36

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Maple [A]  time = 0.009, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{12}}{12}}+{\frac{A{x}^{9}{b}^{5}}{9}}+{\frac{5\,B{x}^{9}a{b}^{4}}{9}}+{\frac{5\,A{x}^{6}a{b}^{4}}{6}}+{\frac{5\,B{x}^{6}{a}^{2}{b}^{3}}{3}}+{\frac{10\,A{x}^{3}{a}^{2}{b}^{3}}{3}}+{\frac{10\,B{x}^{3}{a}^{3}{b}^{2}}{3}}-{\frac{A{a}^{5}}{6\,{x}^{6}}}-{\frac{5\,{a}^{4}bA}{3\,{x}^{3}}}-{\frac{{a}^{5}B}{3\,{x}^{3}}}+10\,A\ln \left ( x \right ){a}^{3}{b}^{2}+5\,B\ln \left ( x \right ){a}^{4}b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.33352, size = 165, normalized size = 1.45 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} + \frac{1}{9} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + \frac{5}{6} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + \frac{10}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + \frac{5}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x^{3}\right ) - \frac{A a^{5} + 2 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47394, size = 273, normalized size = 2.39 \begin{align*} \frac{3 \, B b^{5} x^{18} + 4 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 180 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} \log \left (x\right ) - 6 \, A a^{5} - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{36 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^7,x, algorithm="fricas")

[Out]

1/36*(3*B*b^5*x^18 + 4*(5*B*a*b^4 + A*b^5)*x^15 + 30*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 120*(B*a^3*b^2 + A*a^2*b^3
)*x^9 + 180*(B*a^4*b + 2*A*a^3*b^2)*x^6*log(x) - 6*A*a^5 - 12*(B*a^5 + 5*A*a^4*b)*x^3)/x^6

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Sympy [A]  time = 1.04359, size = 129, normalized size = 1.13 \begin{align*} \frac{B b^{5} x^{12}}{12} + 5 a^{3} b \left (2 A b + B a\right ) \log{\left (x \right )} + x^{9} \left (\frac{A b^{5}}{9} + \frac{5 B a b^{4}}{9}\right ) + x^{6} \left (\frac{5 A a b^{4}}{6} + \frac{5 B a^{2} b^{3}}{3}\right ) + x^{3} \left (\frac{10 A a^{2} b^{3}}{3} + \frac{10 B a^{3} b^{2}}{3}\right ) - \frac{A a^{5} + x^{3} \left (10 A a^{4} b + 2 B a^{5}\right )}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**7,x)

[Out]

B*b**5*x**12/12 + 5*a**3*b*(2*A*b + B*a)*log(x) + x**9*(A*b**5/9 + 5*B*a*b**4/9) + x**6*(5*A*a*b**4/6 + 5*B*a*
*2*b**3/3) + x**3*(10*A*a**2*b**3/3 + 10*B*a**3*b**2/3) - (A*a**5 + x**3*(10*A*a**4*b + 2*B*a**5))/(6*x**6)

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Giac [A]  time = 1.1803, size = 200, normalized size = 1.75 \begin{align*} \frac{1}{12} \, B b^{5} x^{12} + \frac{5}{9} \, B a b^{4} x^{9} + \frac{1}{9} \, A b^{5} x^{9} + \frac{5}{3} \, B a^{2} b^{3} x^{6} + \frac{5}{6} \, A a b^{4} x^{6} + \frac{10}{3} \, B a^{3} b^{2} x^{3} + \frac{10}{3} \, A a^{2} b^{3} x^{3} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{15 \, B a^{4} b x^{6} + 30 \, A a^{3} b^{2} x^{6} + 2 \, B a^{5} x^{3} + 10 \, A a^{4} b x^{3} + A a^{5}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^7,x, algorithm="giac")

[Out]

1/12*B*b^5*x^12 + 5/9*B*a*b^4*x^9 + 1/9*A*b^5*x^9 + 5/3*B*a^2*b^3*x^6 + 5/6*A*a*b^4*x^6 + 10/3*B*a^3*b^2*x^3 +
10/3*A*a^2*b^3*x^3 + 5*(B*a^4*b + 2*A*a^3*b^2)*log(abs(x)) - 1/6*(15*B*a^4*b*x^6 + 30*A*a^3*b^2*x^6 + 2*B*a^5
*x^3 + 10*A*a^4*b*x^3 + A*a^5)/x^6