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

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

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Rubi [A]  time = 0.11, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 76} \begin {gather*} \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} \end {gather*}

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 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])

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^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*}

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Mathematica [A]  time = 0.06, size = 106, normalized size = 0.93 \begin {gather*} \frac {1}{36} \left (-\frac {6 a^5 A}{x^6}-\frac {12 a^4 (a B+5 A b)}{x^3}+180 a^3 b \log (x) (a B+2 A b)+120 a^2 b^2 x^3 (a B+A b)+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 ) \end {gather*}

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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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^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 123, normalized size = 1.08 \begin {gather*} \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 \relax (x) - 6 \, A a^{5} - 12 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{36 \, x^{6}} \end {gather*}

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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giac [A]  time = 0.16, size = 148, normalized size = 1.30 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.05, size = 124, normalized size = 1.09 \begin {gather*} \frac {B \,b^{5} x^{12}}{12}+\frac {A \,b^{5} x^{9}}{9}+\frac {5 B a \,b^{4} x^{9}}{9}+\frac {5 A a \,b^{4} x^{6}}{6}+\frac {5 B \,a^{2} b^{3} x^{6}}{3}+\frac {10 A \,a^{2} b^{3} x^{3}}{3}+\frac {10 B \,a^{3} b^{2} x^{3}}{3}+10 A \,a^{3} b^{2} \ln \relax (x )+5 B \,a^{4} b \ln \relax (x )-\frac {5 A \,a^{4} b}{3 x^{3}}-\frac {B \,a^{5}}{3 x^{3}}-\frac {A \,a^{5}}{6 x^{6}} \end {gather*}

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

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maxima [A]  time = 0.65, size = 122, normalized size = 1.07 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.05, size = 113, normalized size = 0.99 \begin {gather*} \ln \relax (x)\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )-\frac {\frac {A\,a^5}{6}+x^3\,\left (\frac {B\,a^5}{3}+\frac {5\,A\,b\,a^4}{3}\right )}{x^6}+x^9\,\left (\frac {A\,b^5}{9}+\frac {5\,B\,a\,b^4}{9}\right )+\frac {B\,b^5\,x^{12}}{12}+\frac {10\,a^2\,b^2\,x^3\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a\,b^3\,x^6\,\left (A\,b+2\,B\,a\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.94, size = 131, normalized size = 1.15 \begin {gather*} \frac {B b^{5} x^{12}}{12} + 5 a^{3} b \left (2 A b + B a\right ) \log {\relax (x )} + 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 {gather*}

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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