3.1.40 \(\int \frac {(a+b x^3)^5 (A+B x^3)}{x^8} \, dx\)

Optimal. Leaf size=110 \[ -\frac {a^5 A}{7 x^7}-\frac {a^4 (a B+5 A b)}{4 x^4}-\frac {5 a^3 b (a B+2 A b)}{x}+5 a^2 b^2 x^2 (a B+A b)+\frac {1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac {1}{11} b^5 B x^{11} \]

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Rubi [A]  time = 0.07, antiderivative size = 110, 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*} 5 a^2 b^2 x^2 (a B+A b)-\frac {a^4 (a B+5 A b)}{4 x^4}-\frac {5 a^3 b (a B+2 A b)}{x}-\frac {a^5 A}{7 x^7}+\frac {1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac {1}{11} b^5 B x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^8,x]

[Out]

-(a^5*A)/(7*x^7) - (a^4*(5*A*b + a*B))/(4*x^4) - (5*a^3*b*(2*A*b + a*B))/x + 5*a^2*b^2*(A*b + a*B)*x^2 + a*b^3
*(A*b + 2*a*B)*x^5 + (b^4*(A*b + 5*a*B)*x^8)/8 + (b^5*B*x^11)/11

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

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Mathematica [A]  time = 0.04, size = 110, normalized size = 1.00 \begin {gather*} -\frac {a^5 A}{7 x^7}-\frac {a^4 (a B+5 A b)}{4 x^4}-\frac {5 a^3 b (a B+2 A b)}{x}+5 a^2 b^2 x^2 (a B+A b)+\frac {1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac {1}{11} b^5 B x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^8,x]

[Out]

-1/7*(a^5*A)/x^7 - (a^4*(5*A*b + a*B))/(4*x^4) - (5*a^3*b*(2*A*b + a*B))/x + 5*a^2*b^2*(A*b + a*B)*x^2 + a*b^3
*(A*b + 2*a*B)*x^5 + (b^4*(A*b + 5*a*B)*x^8)/8 + (b^5*B*x^11)/11

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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^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^8,x]

[Out]

IntegrateAlgebraic[((a + b*x^3)^5*(A + B*x^3))/x^8, x]

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fricas [A]  time = 1.11, size = 121, normalized size = 1.10 \begin {gather*} \frac {56 \, B b^{5} x^{18} + 77 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 616 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 3080 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 3080 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 88 \, A a^{5} - 154 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^8,x, algorithm="fricas")

[Out]

1/616*(56*B*b^5*x^18 + 77*(5*B*a*b^4 + A*b^5)*x^15 + 616*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 3080*(B*a^3*b^2 + A*a^
2*b^3)*x^9 - 3080*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 88*A*a^5 - 154*(B*a^5 + 5*A*a^4*b)*x^3)/x^7

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giac [A]  time = 0.17, size = 127, normalized size = 1.15 \begin {gather*} \frac {1}{11} \, B b^{5} x^{11} + \frac {5}{8} \, B a b^{4} x^{8} + \frac {1}{8} \, A b^{5} x^{8} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + 5 \, B a^{3} b^{2} x^{2} + 5 \, A a^{2} b^{3} x^{2} - \frac {140 \, B a^{4} b x^{6} + 280 \, A a^{3} b^{2} x^{6} + 7 \, B a^{5} x^{3} + 35 \, A a^{4} b x^{3} + 4 \, A a^{5}}{28 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^8,x, algorithm="giac")

[Out]

1/11*B*b^5*x^11 + 5/8*B*a*b^4*x^8 + 1/8*A*b^5*x^8 + 2*B*a^2*b^3*x^5 + A*a*b^4*x^5 + 5*B*a^3*b^2*x^2 + 5*A*a^2*
b^3*x^2 - 1/28*(140*B*a^4*b*x^6 + 280*A*a^3*b^2*x^6 + 7*B*a^5*x^3 + 35*A*a^4*b*x^3 + 4*A*a^5)/x^7

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maple [A]  time = 0.05, size = 117, normalized size = 1.06 \begin {gather*} \frac {B \,b^{5} x^{11}}{11}+\frac {A \,b^{5} x^{8}}{8}+\frac {5 B a \,b^{4} x^{8}}{8}+A a \,b^{4} x^{5}+2 B \,a^{2} b^{3} x^{5}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{x}-\frac {\left (5 A b +B a \right ) a^{4}}{4 x^{4}}-\frac {A \,a^{5}}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^8,x)

[Out]

1/11*b^5*B*x^11+1/8*A*x^8*b^5+5/8*B*x^8*a*b^4+A*a*b^4*x^5+2*B*a^2*b^3*x^5+5*A*x^2*a^2*b^3+5*B*x^2*a^3*b^2-1/4*
a^4*(5*A*b+B*a)/x^4-1/7*a^5*A/x^7-5*a^3*b*(2*A*b+B*a)/x

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maxima [A]  time = 0.51, size = 121, normalized size = 1.10 \begin {gather*} \frac {1}{11} \, B b^{5} x^{11} + \frac {1}{8} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + 5 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} - \frac {140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 4 \, A a^{5} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{28 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^8,x, algorithm="maxima")

[Out]

1/11*B*b^5*x^11 + 1/8*(5*B*a*b^4 + A*b^5)*x^8 + (2*B*a^2*b^3 + A*a*b^4)*x^5 + 5*(B*a^3*b^2 + A*a^2*b^3)*x^2 -
1/28*(140*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 4*A*a^5 + 7*(B*a^5 + 5*A*a^4*b)*x^3)/x^7

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mupad [B]  time = 2.34, size = 113, normalized size = 1.03 \begin {gather*} x^8\,\left (\frac {A\,b^5}{8}+\frac {5\,B\,a\,b^4}{8}\right )-\frac {\frac {A\,a^5}{7}+x^6\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )+x^3\,\left (\frac {B\,a^5}{4}+\frac {5\,A\,b\,a^4}{4}\right )}{x^7}+\frac {B\,b^5\,x^{11}}{11}+5\,a^2\,b^2\,x^2\,\left (A\,b+B\,a\right )+a\,b^3\,x^5\,\left (A\,b+2\,B\,a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^3)*(a + b*x^3)^5)/x^8,x)

[Out]

x^8*((A*b^5)/8 + (5*B*a*b^4)/8) - ((A*a^5)/7 + x^6*(10*A*a^3*b^2 + 5*B*a^4*b) + x^3*((B*a^5)/4 + (5*A*a^4*b)/4
))/x^7 + (B*b^5*x^11)/11 + 5*a^2*b^2*x^2*(A*b + B*a) + a*b^3*x^5*(A*b + 2*B*a)

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sympy [A]  time = 1.12, size = 129, normalized size = 1.17 \begin {gather*} \frac {B b^{5} x^{11}}{11} + x^{8} \left (\frac {A b^{5}}{8} + \frac {5 B a b^{4}}{8}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{2} \left (5 A a^{2} b^{3} + 5 B a^{3} b^{2}\right ) + \frac {- 4 A a^{5} + x^{6} \left (- 280 A a^{3} b^{2} - 140 B a^{4} b\right ) + x^{3} \left (- 35 A a^{4} b - 7 B a^{5}\right )}{28 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**8,x)

[Out]

B*b**5*x**11/11 + x**8*(A*b**5/8 + 5*B*a*b**4/8) + x**5*(A*a*b**4 + 2*B*a**2*b**3) + x**2*(5*A*a**2*b**3 + 5*B
*a**3*b**2) + (-4*A*a**5 + x**6*(-280*A*a**3*b**2 - 140*B*a**4*b) + x**3*(-35*A*a**4*b - 7*B*a**5))/(28*x**7)

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