3.1.49 \(\int \frac {(a+b x^3)^5 (A+B x^3)}{x^{17}} \, dx\)

Optimal. Leaf size=115 \[ -\frac {a^5 A}{16 x^{16}}-\frac {a^4 (a B+5 A b)}{13 x^{13}}-\frac {a^3 b (a B+2 A b)}{2 x^{10}}-\frac {10 a^2 b^2 (a B+A b)}{7 x^7}-\frac {b^4 (5 a B+A b)}{x}-\frac {5 a b^3 (2 a B+A b)}{4 x^4}+\frac {1}{2} b^5 B x^2 \]

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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 {10 a^2 b^2 (a B+A b)}{7 x^7}-\frac {a^4 (a B+5 A b)}{13 x^{13}}-\frac {a^3 b (a B+2 A b)}{2 x^{10}}-\frac {a^5 A}{16 x^{16}}-\frac {5 a b^3 (2 a B+A b)}{4 x^4}-\frac {b^4 (5 a B+A b)}{x}+\frac {1}{2} b^5 B x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^17,x]

[Out]

-(a^5*A)/(16*x^16) - (a^4*(5*A*b + a*B))/(13*x^13) - (a^3*b*(2*A*b + a*B))/(2*x^10) - (10*a^2*b^2*(A*b + a*B))
/(7*x^7) - (5*a*b^3*(A*b + 2*a*B))/(4*x^4) - (b^4*(A*b + 5*a*B))/x + (b^5*B*x^2)/2

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^{17}} \, dx &=\int \left (\frac {a^5 A}{x^{17}}+\frac {a^4 (5 A b+a B)}{x^{14}}+\frac {5 a^3 b (2 A b+a B)}{x^{11}}+\frac {10 a^2 b^2 (A b+a B)}{x^8}+\frac {5 a b^3 (A b+2 a B)}{x^5}+\frac {b^4 (A b+5 a B)}{x^2}+b^5 B x\right ) \, dx\\ &=-\frac {a^5 A}{16 x^{16}}-\frac {a^4 (5 A b+a B)}{13 x^{13}}-\frac {a^3 b (2 A b+a B)}{2 x^{10}}-\frac {10 a^2 b^2 (A b+a B)}{7 x^7}-\frac {5 a b^3 (A b+2 a B)}{4 x^4}-\frac {b^4 (A b+5 a B)}{x}+\frac {1}{2} b^5 B x^2\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 118, normalized size = 1.03 \begin {gather*} -\frac {7 a^5 \left (13 A+16 B x^3\right )+56 a^4 b x^3 \left (10 A+13 B x^3\right )+208 a^3 b^2 x^6 \left (7 A+10 B x^3\right )+520 a^2 b^3 x^9 \left (4 A+7 B x^3\right )+1820 a b^4 x^{12} \left (A+4 B x^3\right )-728 b^5 x^{15} \left (B x^3-2 A\right )}{1456 x^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^17,x]

[Out]

-1/1456*(-728*b^5*x^15*(-2*A + B*x^3) + 1820*a*b^4*x^12*(A + 4*B*x^3) + 520*a^2*b^3*x^9*(4*A + 7*B*x^3) + 208*
a^3*b^2*x^6*(7*A + 10*B*x^3) + 56*a^4*b*x^3*(10*A + 13*B*x^3) + 7*a^5*(13*A + 16*B*x^3))/x^16

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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^{17}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^17,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^17, x]

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fricas [A]  time = 0.56, size = 121, normalized size = 1.05 \begin {gather*} \frac {728 \, B b^{5} x^{18} - 1456 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 1820 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 2080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 728 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 91 \, A a^{5} - 112 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{1456 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^17,x, algorithm="fricas")

[Out]

1/1456*(728*B*b^5*x^18 - 1456*(5*B*a*b^4 + A*b^5)*x^15 - 1820*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 2080*(B*a^3*b^2 +
A*a^2*b^3)*x^9 - 728*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 91*A*a^5 - 112*(B*a^5 + 5*A*a^4*b)*x^3)/x^16

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giac [A]  time = 0.16, size = 128, normalized size = 1.11 \begin {gather*} \frac {1}{2} \, B b^{5} x^{2} - \frac {7280 \, B a b^{4} x^{15} + 1456 \, A b^{5} x^{15} + 3640 \, B a^{2} b^{3} x^{12} + 1820 \, A a b^{4} x^{12} + 2080 \, B a^{3} b^{2} x^{9} + 2080 \, A a^{2} b^{3} x^{9} + 728 \, B a^{4} b x^{6} + 1456 \, A a^{3} b^{2} x^{6} + 112 \, B a^{5} x^{3} + 560 \, A a^{4} b x^{3} + 91 \, A a^{5}}{1456 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^17,x, algorithm="giac")

[Out]

1/2*B*b^5*x^2 - 1/1456*(7280*B*a*b^4*x^15 + 1456*A*b^5*x^15 + 3640*B*a^2*b^3*x^12 + 1820*A*a*b^4*x^12 + 2080*B
*a^3*b^2*x^9 + 2080*A*a^2*b^3*x^9 + 728*B*a^4*b*x^6 + 1456*A*a^3*b^2*x^6 + 112*B*a^5*x^3 + 560*A*a^4*b*x^3 + 9
1*A*a^5)/x^16

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maple [A]  time = 0.04, size = 104, normalized size = 0.90 \begin {gather*} \frac {B \,b^{5} x^{2}}{2}-\frac {\left (A b +5 B a \right ) b^{4}}{x}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{4 x^{4}}-\frac {10 \left (A b +B a \right ) a^{2} b^{2}}{7 x^{7}}-\frac {\left (2 A b +B a \right ) a^{3} b}{2 x^{10}}-\frac {\left (5 A b +B a \right ) a^{4}}{13 x^{13}}-\frac {A \,a^{5}}{16 x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^17,x)

[Out]

-1/16*a^5*A/x^16-1/13*a^4*(5*A*b+B*a)/x^13-1/2*a^3*b*(2*A*b+B*a)/x^10-10/7*a^2*b^2*(A*b+B*a)/x^7-5/4*a*b^3*(A*
b+2*B*a)/x^4-b^4*(A*b+5*B*a)/x+1/2*b^5*B*x^2

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maxima [A]  time = 0.50, size = 122, normalized size = 1.06 \begin {gather*} \frac {1}{2} \, B b^{5} x^{2} - \frac {1456 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 1820 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 728 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 91 \, A a^{5} + 112 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{1456 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^17,x, algorithm="maxima")

[Out]

1/2*B*b^5*x^2 - 1/1456*(1456*(5*B*a*b^4 + A*b^5)*x^15 + 1820*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 2080*(B*a^3*b^2 +
A*a^2*b^3)*x^9 + 728*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 91*A*a^5 + 112*(B*a^5 + 5*A*a^4*b)*x^3)/x^16

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mupad [B]  time = 2.36, size = 121, normalized size = 1.05 \begin {gather*} \frac {B\,b^5\,x^2}{2}-\frac {\frac {A\,a^5}{16}+x^6\,\left (\frac {B\,a^4\,b}{2}+A\,a^3\,b^2\right )+x^{12}\,\left (\frac {5\,B\,a^2\,b^3}{2}+\frac {5\,A\,a\,b^4}{4}\right )+x^3\,\left (\frac {B\,a^5}{13}+\frac {5\,A\,b\,a^4}{13}\right )+x^{15}\,\left (A\,b^5+5\,B\,a\,b^4\right )+x^9\,\left (\frac {10\,B\,a^3\,b^2}{7}+\frac {10\,A\,a^2\,b^3}{7}\right )}{x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^5)/x^17,x)

[Out]

(B*b^5*x^2)/2 - ((A*a^5)/16 + x^6*(A*a^3*b^2 + (B*a^4*b)/2) + x^12*((5*B*a^2*b^3)/2 + (5*A*a*b^4)/4) + x^3*((B
*a^5)/13 + (5*A*a^4*b)/13) + x^15*(A*b^5 + 5*B*a*b^4) + x^9*((10*A*a^2*b^3)/7 + (10*B*a^3*b^2)/7))/x^16

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sympy [A]  time = 66.09, size = 134, normalized size = 1.17 \begin {gather*} \frac {B b^{5} x^{2}}{2} + \frac {- 91 A a^{5} + x^{15} \left (- 1456 A b^{5} - 7280 B a b^{4}\right ) + x^{12} \left (- 1820 A a b^{4} - 3640 B a^{2} b^{3}\right ) + x^{9} \left (- 2080 A a^{2} b^{3} - 2080 B a^{3} b^{2}\right ) + x^{6} \left (- 1456 A a^{3} b^{2} - 728 B a^{4} b\right ) + x^{3} \left (- 560 A a^{4} b - 112 B a^{5}\right )}{1456 x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**17,x)

[Out]

B*b**5*x**2/2 + (-91*A*a**5 + x**15*(-1456*A*b**5 - 7280*B*a*b**4) + x**12*(-1820*A*a*b**4 - 3640*B*a**2*b**3)
+ x**9*(-2080*A*a**2*b**3 - 2080*B*a**3*b**2) + x**6*(-1456*A*a**3*b**2 - 728*B*a**4*b) + x**3*(-560*A*a**4*b
- 112*B*a**5))/(1456*x**16)

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