3.36 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^4} \, dx\)

Optimal. Leaf size=113 \[ \frac{5}{3} a^2 b^2 x^6 (a B+A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+a^4 \log (x) (a B+5 A b)-\frac{a^5 A}{3 x^3}+\frac{1}{12} b^4 x^{12} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]

[Out]

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

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Rubi [A]  time = 0.117054, antiderivative size = 113, 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} \[ \frac{5}{3} a^2 b^2 x^6 (a B+A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+a^4 \log (x) (a B+5 A b)-\frac{a^5 A}{3 x^3}+\frac{1}{12} b^4 x^{12} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0394044, size = 115, normalized size = 1.02 \[ \frac{5}{3} a^2 b^2 x^6 (a B+A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+\log (x) \left (5 a^4 A b+a^5 B\right )-\frac{a^5 A}{3 x^3}+\frac{1}{12} b^4 x^{12} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^4,x]

[Out]

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

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Maple [A]  time = 0.007, size = 123, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{15}}{15}}+{\frac{A{x}^{12}{b}^{5}}{12}}+{\frac{5\,B{x}^{12}a{b}^{4}}{12}}+{\frac{5\,A{x}^{9}a{b}^{4}}{9}}+{\frac{10\,B{x}^{9}{a}^{2}{b}^{3}}{9}}+{\frac{5\,A{x}^{6}{a}^{2}{b}^{3}}{3}}+{\frac{5\,B{x}^{6}{a}^{3}{b}^{2}}{3}}+{\frac{10\,A{x}^{3}{a}^{3}{b}^{2}}{3}}+{\frac{5\,B{x}^{3}{a}^{4}b}{3}}-{\frac{A{a}^{5}}{3\,{x}^{3}}}+5\,A\ln \left ( x \right ){a}^{4}b+B\ln \left ( x \right ){a}^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.48549, size = 162, normalized size = 1.43 \begin{align*} \frac{1}{15} \, B b^{5} x^{15} + \frac{1}{12} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{12} + \frac{5}{9} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac{5}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + \frac{5}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} - \frac{A a^{5}}{3 \, x^{3}} + \frac{1}{3} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} \log \left (x^{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47553, size = 281, normalized size = 2.49 \begin{align*} \frac{12 \, B b^{5} x^{18} + 15 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 300 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 60 \, A a^{5} + 180 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \log \left (x\right )}{180 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^4,x, algorithm="fricas")

[Out]

1/180*(12*B*b^5*x^18 + 15*(5*B*a*b^4 + A*b^5)*x^15 + 100*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 300*(B*a^3*b^2 + A*a^2
*b^3)*x^9 + 300*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 60*A*a^5 + 180*(B*a^5 + 5*A*a^4*b)*x^3*log(x))/x^3

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Sympy [A]  time = 0.506031, size = 133, normalized size = 1.18 \begin{align*} - \frac{A a^{5}}{3 x^{3}} + \frac{B b^{5} x^{15}}{15} + a^{4} \left (5 A b + B a\right ) \log{\left (x \right )} + x^{12} \left (\frac{A b^{5}}{12} + \frac{5 B a b^{4}}{12}\right ) + x^{9} \left (\frac{5 A a b^{4}}{9} + \frac{10 B a^{2} b^{3}}{9}\right ) + x^{6} \left (\frac{5 A a^{2} b^{3}}{3} + \frac{5 B a^{3} b^{2}}{3}\right ) + x^{3} \left (\frac{10 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**4,x)

[Out]

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

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Giac [A]  time = 1.22758, size = 193, normalized size = 1.71 \begin{align*} \frac{1}{15} \, B b^{5} x^{15} + \frac{5}{12} \, B a b^{4} x^{12} + \frac{1}{12} \, A b^{5} x^{12} + \frac{10}{9} \, B a^{2} b^{3} x^{9} + \frac{5}{9} \, A a b^{4} x^{9} + \frac{5}{3} \, B a^{3} b^{2} x^{6} + \frac{5}{3} \, A a^{2} b^{3} x^{6} + \frac{5}{3} \, B a^{4} b x^{3} + \frac{10}{3} \, A a^{3} b^{2} x^{3} +{\left (B a^{5} + 5 \, A a^{4} b\right )} \log \left ({\left | x \right |}\right ) - \frac{B a^{5} x^{3} + 5 \, A a^{4} b x^{3} + A a^{5}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^4,x, algorithm="giac")

[Out]

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