3.1.49 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{17}} \, dx\)

Optimal. Leaf size=76 \[ -\frac {b \left (a+b x^2\right )^6 (A b-4 a B)}{336 a^3 x^{12}}+\frac {\left (a+b x^2\right )^6 (A b-4 a B)}{56 a^2 x^{14}}-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}} \]

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Rubi [A]  time = 0.05, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {446, 78, 45, 37} \begin {gather*} -\frac {b \left (a+b x^2\right )^6 (A b-4 a B)}{336 a^3 x^{12}}+\frac {\left (a+b x^2\right )^6 (A b-4 a B)}{56 a^2 x^{14}}-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^17,x]

[Out]

-(A*(a + b*x^2)^6)/(16*a*x^16) + ((A*b - 4*a*B)*(a + b*x^2)^6)/(56*a^2*x^14) - (b*(A*b - 4*a*B)*(a + b*x^2)^6)
/(336*a^3*x^12)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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^2\right )^5 \left (A+B x^2\right )}{x^{17}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}}+\frac {(-2 A b+8 a B) \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^2\right )}{16 a}\\ &=-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}}+\frac {(A b-4 a B) \left (a+b x^2\right )^6}{56 a^2 x^{14}}+\frac {(b (A b-4 a B)) \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^2\right )}{56 a^2}\\ &=-\frac {A \left (a+b x^2\right )^6}{16 a x^{16}}+\frac {(A b-4 a B) \left (a+b x^2\right )^6}{56 a^2 x^{14}}-\frac {b (A b-4 a B) \left (a+b x^2\right )^6}{336 a^3 x^{12}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 121, normalized size = 1.59 \begin {gather*} -\frac {3 a^5 \left (7 A+8 B x^2\right )+20 a^4 b x^2 \left (6 A+7 B x^2\right )+56 a^3 b^2 x^4 \left (5 A+6 B x^2\right )+84 a^2 b^3 x^6 \left (4 A+5 B x^2\right )+70 a b^4 x^8 \left (3 A+4 B x^2\right )+28 b^5 x^{10} \left (2 A+3 B x^2\right )}{336 x^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^17,x]

[Out]

-1/336*(28*b^5*x^10*(2*A + 3*B*x^2) + 70*a*b^4*x^8*(3*A + 4*B*x^2) + 84*a^2*b^3*x^6*(4*A + 5*B*x^2) + 56*a^3*b
^2*x^4*(5*A + 6*B*x^2) + 20*a^4*b*x^2*(6*A + 7*B*x^2) + 3*a^5*(7*A + 8*B*x^2))/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^2\right )^5 \left (A+B x^2\right )}{x^{17}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^17,x]

[Out]

IntegrateAlgebraic[((a + b*x^2)^5*(A + B*x^2))/x^17, x]

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fricas [A]  time = 0.45, size = 121, normalized size = 1.59 \begin {gather*} -\frac {84 \, B b^{5} x^{12} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 21 \, A a^{5} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{336 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^17,x, algorithm="fricas")

[Out]

-1/336*(84*B*b^5*x^12 + 56*(5*B*a*b^4 + A*b^5)*x^10 + 210*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 336*(B*a^3*b^2 + A*a^2
*b^3)*x^6 + 21*A*a^5 + 140*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 24*(B*a^5 + 5*A*a^4*b)*x^2)/x^16

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giac [A]  time = 0.32, size = 127, normalized size = 1.67 \begin {gather*} -\frac {84 \, B b^{5} x^{12} + 280 \, B a b^{4} x^{10} + 56 \, A b^{5} x^{10} + 420 \, B a^{2} b^{3} x^{8} + 210 \, A a b^{4} x^{8} + 336 \, B a^{3} b^{2} x^{6} + 336 \, A a^{2} b^{3} x^{6} + 140 \, B a^{4} b x^{4} + 280 \, A a^{3} b^{2} x^{4} + 24 \, B a^{5} x^{2} + 120 \, A a^{4} b x^{2} + 21 \, A a^{5}}{336 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^17,x, algorithm="giac")

[Out]

-1/336*(84*B*b^5*x^12 + 280*B*a*b^4*x^10 + 56*A*b^5*x^10 + 420*B*a^2*b^3*x^8 + 210*A*a*b^4*x^8 + 336*B*a^3*b^2
*x^6 + 336*A*a^2*b^3*x^6 + 140*B*a^4*b*x^4 + 280*A*a^3*b^2*x^4 + 24*B*a^5*x^2 + 120*A*a^4*b*x^2 + 21*A*a^5)/x^
16

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maple [A]  time = 0.01, size = 104, normalized size = 1.37 \begin {gather*} -\frac {B \,b^{5}}{4 x^{4}}-\frac {\left (A b +5 B a \right ) b^{4}}{6 x^{6}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{8 x^{8}}-\frac {\left (A b +B a \right ) a^{2} b^{2}}{x^{10}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{12 x^{12}}-\frac {A \,a^{5}}{16 x^{16}}-\frac {\left (5 A b +B a \right ) a^{4}}{14 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^17,x)

[Out]

-1/16*A*a^5/x^16-a^2*b^2*(A*b+B*a)/x^10-1/6*b^4*(A*b+5*B*a)/x^6-5/8*a*b^3*(A*b+2*B*a)/x^8-1/4*B*b^5/x^4-1/14*a
^4*(5*A*b+B*a)/x^14-5/12*a^3*b*(2*A*b+B*a)/x^12

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maxima [A]  time = 1.09, size = 121, normalized size = 1.59 \begin {gather*} -\frac {84 \, B b^{5} x^{12} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 21 \, A a^{5} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{336 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^17,x, algorithm="maxima")

[Out]

-1/336*(84*B*b^5*x^12 + 56*(5*B*a*b^4 + A*b^5)*x^10 + 210*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 336*(B*a^3*b^2 + A*a^2
*b^3)*x^6 + 21*A*a^5 + 140*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 24*(B*a^5 + 5*A*a^4*b)*x^2)/x^16

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mupad [B]  time = 0.09, size = 120, normalized size = 1.58 \begin {gather*} -\frac {\frac {A\,a^5}{16}+x^8\,\left (\frac {5\,B\,a^2\,b^3}{4}+\frac {5\,A\,a\,b^4}{8}\right )+x^4\,\left (\frac {5\,B\,a^4\,b}{12}+\frac {5\,A\,a^3\,b^2}{6}\right )+x^2\,\left (\frac {B\,a^5}{14}+\frac {5\,A\,b\,a^4}{14}\right )+x^{10}\,\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )+x^6\,\left (B\,a^3\,b^2+A\,a^2\,b^3\right )+\frac {B\,b^5\,x^{12}}{4}}{x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^17,x)

[Out]

-((A*a^5)/16 + x^8*((5*B*a^2*b^3)/4 + (5*A*a*b^4)/8) + x^4*((5*A*a^3*b^2)/6 + (5*B*a^4*b)/12) + x^2*((B*a^5)/1
4 + (5*A*a^4*b)/14) + x^10*((A*b^5)/6 + (5*B*a*b^4)/6) + x^6*(A*a^2*b^3 + B*a^3*b^2) + (B*b^5*x^12)/4)/x^16

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sympy [A]  time = 32.11, size = 134, normalized size = 1.76 \begin {gather*} \frac {- 21 A a^{5} - 84 B b^{5} x^{12} + x^{10} \left (- 56 A b^{5} - 280 B a b^{4}\right ) + x^{8} \left (- 210 A a b^{4} - 420 B a^{2} b^{3}\right ) + x^{6} \left (- 336 A a^{2} b^{3} - 336 B a^{3} b^{2}\right ) + x^{4} \left (- 280 A a^{3} b^{2} - 140 B a^{4} b\right ) + x^{2} \left (- 120 A a^{4} b - 24 B a^{5}\right )}{336 x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**17,x)

[Out]

(-21*A*a**5 - 84*B*b**5*x**12 + x**10*(-56*A*b**5 - 280*B*a*b**4) + x**8*(-210*A*a*b**4 - 420*B*a**2*b**3) + x
**6*(-336*A*a**2*b**3 - 336*B*a**3*b**2) + x**4*(-280*A*a**3*b**2 - 140*B*a**4*b) + x**2*(-120*A*a**4*b - 24*B
*a**5))/(336*x**16)

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