3.1.46 \(\int \frac {(a+b x^3)^5 (A+B x^3)}{x^{14}} \, dx\)

Optimal. Leaf size=115 \[ -\frac {a^5 A}{13 x^{13}}-\frac {a^4 (a B+5 A b)}{10 x^{10}}-\frac {5 a^3 b (a B+2 A b)}{7 x^7}-\frac {5 a^2 b^2 (a B+A b)}{2 x^4}+\frac {1}{2} b^4 x^2 (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{x}+\frac {1}{5} b^5 B x^5 \]

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Rubi [A]  time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \begin {gather*} -\frac {5 a^2 b^2 (a B+A b)}{2 x^4}-\frac {a^4 (a B+5 A b)}{10 x^{10}}-\frac {5 a^3 b (a B+2 A b)}{7 x^7}-\frac {a^5 A}{13 x^{13}}+\frac {1}{2} b^4 x^2 (5 a B+A b)-\frac {5 a b^3 (2 a B+A b)}{x}+\frac {1}{5} b^5 B x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^14,x]

[Out]

-(a^5*A)/(13*x^13) - (a^4*(5*A*b + a*B))/(10*x^10) - (5*a^3*b*(2*A*b + a*B))/(7*x^7) - (5*a^2*b^2*(A*b + a*B))
/(2*x^4) - (5*a*b^3*(A*b + 2*a*B))/x + (b^4*(A*b + 5*a*B)*x^2)/2 + (b^5*B*x^5)/5

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{14}} \, dx &=\int \left (\frac {a^5 A}{x^{14}}+\frac {a^4 (5 A b+a B)}{x^{11}}+\frac {5 a^3 b (2 A b+a B)}{x^8}+\frac {10 a^2 b^2 (A b+a B)}{x^5}+\frac {5 a b^3 (A b+2 a B)}{x^2}+b^4 (A b+5 a B) x+b^5 B x^4\right ) \, dx\\ &=-\frac {a^5 A}{13 x^{13}}-\frac {a^4 (5 A b+a B)}{10 x^{10}}-\frac {5 a^3 b (2 A b+a B)}{7 x^7}-\frac {5 a^2 b^2 (A b+a B)}{2 x^4}-\frac {5 a b^3 (A b+2 a B)}{x}+\frac {1}{2} b^4 (A b+5 a B) x^2+\frac {1}{5} b^5 B x^5\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 117, normalized size = 1.02 \begin {gather*} -\frac {a^5 \left (70 A+91 B x^3\right )+65 a^4 b x^3 \left (7 A+10 B x^3\right )+325 a^3 b^2 x^6 \left (4 A+7 B x^3\right )+2275 a^2 b^3 x^9 \left (A+4 B x^3\right )-2275 a b^4 x^{12} \left (B x^3-2 A\right )-91 b^5 x^{15} \left (5 A+2 B x^3\right )}{910 x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^14,x]

[Out]

-1/910*(-2275*a*b^4*x^12*(-2*A + B*x^3) - 91*b^5*x^15*(5*A + 2*B*x^3) + 2275*a^2*b^3*x^9*(A + 4*B*x^3) + 325*a
^3*b^2*x^6*(4*A + 7*B*x^3) + 65*a^4*b*x^3*(7*A + 10*B*x^3) + a^5*(70*A + 91*B*x^3))/x^13

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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^{14}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^14,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^14, x]

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fricas [A]  time = 0.64, size = 121, normalized size = 1.05 \begin {gather*} \frac {182 \, B b^{5} x^{18} + 455 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 4550 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 2275 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 650 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 70 \, A a^{5} - 91 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{910 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^14,x, algorithm="fricas")

[Out]

1/910*(182*B*b^5*x^18 + 455*(5*B*a*b^4 + A*b^5)*x^15 - 4550*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 2275*(B*a^3*b^2 + A
*a^2*b^3)*x^9 - 650*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 70*A*a^5 - 91*(B*a^5 + 5*A*a^4*b)*x^3)/x^13

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giac [A]  time = 0.16, size = 128, normalized size = 1.11 \begin {gather*} \frac {1}{5} \, B b^{5} x^{5} + \frac {5}{2} \, B a b^{4} x^{2} + \frac {1}{2} \, A b^{5} x^{2} - \frac {9100 \, B a^{2} b^{3} x^{12} + 4550 \, A a b^{4} x^{12} + 2275 \, B a^{3} b^{2} x^{9} + 2275 \, A a^{2} b^{3} x^{9} + 650 \, B a^{4} b x^{6} + 1300 \, A a^{3} b^{2} x^{6} + 91 \, B a^{5} x^{3} + 455 \, A a^{4} b x^{3} + 70 \, A a^{5}}{910 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^14,x, algorithm="giac")

[Out]

1/5*B*b^5*x^5 + 5/2*B*a*b^4*x^2 + 1/2*A*b^5*x^2 - 1/910*(9100*B*a^2*b^3*x^12 + 4550*A*a*b^4*x^12 + 2275*B*a^3*
b^2*x^9 + 2275*A*a^2*b^3*x^9 + 650*B*a^4*b*x^6 + 1300*A*a^3*b^2*x^6 + 91*B*a^5*x^3 + 455*A*a^4*b*x^3 + 70*A*a^
5)/x^13

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maple [A]  time = 0.06, size = 107, normalized size = 0.93 \begin {gather*} \frac {B \,b^{5} x^{5}}{5}+\frac {A \,b^{5} x^{2}}{2}+\frac {5 B a \,b^{4} x^{2}}{2}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{x}-\frac {5 \left (A b +B a \right ) a^{2} b^{2}}{2 x^{4}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{7 x^{7}}-\frac {\left (5 A b +B a \right ) a^{4}}{10 x^{10}}-\frac {A \,a^{5}}{13 x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^14,x)

[Out]

1/5*b^5*B*x^5+1/2*A*x^2*b^5+5/2*B*x^2*a*b^4-5/2*a^2*b^2*(A*b+B*a)/x^4-1/13*a^5*A/x^13-5/7*a^3*b*(2*A*b+B*a)/x^
7-5*a*b^3*(A*b+2*B*a)/x-1/10*a^4*(5*A*b+B*a)/x^10

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maxima [A]  time = 0.56, size = 122, normalized size = 1.06 \begin {gather*} \frac {1}{5} \, B b^{5} x^{5} + \frac {1}{2} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{2} - \frac {4550 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2275 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 650 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 70 \, A a^{5} + 91 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{910 \, x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^14,x, algorithm="maxima")

[Out]

1/5*B*b^5*x^5 + 1/2*(5*B*a*b^4 + A*b^5)*x^2 - 1/910*(4550*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 2275*(B*a^3*b^2 + A*a
^2*b^3)*x^9 + 650*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 70*A*a^5 + 91*(B*a^5 + 5*A*a^4*b)*x^3)/x^13

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mupad [B]  time = 2.37, size = 123, normalized size = 1.07 \begin {gather*} x^2\,\left (\frac {A\,b^5}{2}+\frac {5\,B\,a\,b^4}{2}\right )-\frac {\frac {A\,a^5}{13}+x^{12}\,\left (10\,B\,a^2\,b^3+5\,A\,a\,b^4\right )+x^6\,\left (\frac {5\,B\,a^4\,b}{7}+\frac {10\,A\,a^3\,b^2}{7}\right )+x^3\,\left (\frac {B\,a^5}{10}+\frac {A\,b\,a^4}{2}\right )+x^9\,\left (\frac {5\,B\,a^3\,b^2}{2}+\frac {5\,A\,a^2\,b^3}{2}\right )}{x^{13}}+\frac {B\,b^5\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^5)/x^14,x)

[Out]

x^2*((A*b^5)/2 + (5*B*a*b^4)/2) - ((A*a^5)/13 + x^12*(10*B*a^2*b^3 + 5*A*a*b^4) + x^6*((10*A*a^3*b^2)/7 + (5*B
*a^4*b)/7) + x^3*((B*a^5)/10 + (A*a^4*b)/2) + x^9*((5*A*a^2*b^3)/2 + (5*B*a^3*b^2)/2))/x^13 + (B*b^5*x^5)/5

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sympy [A]  time = 10.76, size = 134, normalized size = 1.17 \begin {gather*} \frac {B b^{5} x^{5}}{5} + x^{2} \left (\frac {A b^{5}}{2} + \frac {5 B a b^{4}}{2}\right ) + \frac {- 70 A a^{5} + x^{12} \left (- 4550 A a b^{4} - 9100 B a^{2} b^{3}\right ) + x^{9} \left (- 2275 A a^{2} b^{3} - 2275 B a^{3} b^{2}\right ) + x^{6} \left (- 1300 A a^{3} b^{2} - 650 B a^{4} b\right ) + x^{3} \left (- 455 A a^{4} b - 91 B a^{5}\right )}{910 x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**14,x)

[Out]

B*b**5*x**5/5 + x**2*(A*b**5/2 + 5*B*a*b**4/2) + (-70*A*a**5 + x**12*(-4550*A*a*b**4 - 9100*B*a**2*b**3) + x**
9*(-2275*A*a**2*b**3 - 2275*B*a**3*b**2) + x**6*(-1300*A*a**3*b**2 - 650*B*a**4*b) + x**3*(-455*A*a**4*b - 91*
B*a**5))/(910*x**13)

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