3.47 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^{15}} \, dx\)

Optimal. Leaf size=48 \[ \frac {\left (a+b x^2\right )^6 (A b-7 a B)}{84 a^2 x^{12}}-\frac {A \left (a+b x^2\right )^6}{14 a x^{14}} \]

[Out]

-1/14*A*(b*x^2+a)^6/a/x^14+1/84*(A*b-7*B*a)*(b*x^2+a)^6/a^2/x^12

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {446, 78, 37} \[ \frac {\left (a+b x^2\right )^6 (A b-7 a B)}{84 a^2 x^{12}}-\frac {A \left (a+b x^2\right )^6}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^15,x]

[Out]

-(A*(a + b*x^2)^6)/(14*a*x^14) + ((A*b - 7*a*B)*(a + b*x^2)^6)/(84*a^2*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 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^{15}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^8} \, dx,x,x^2\right )\\ &=-\frac {A \left (a+b x^2\right )^6}{14 a x^{14}}+\frac {(-A b+7 a B) \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^2\right )}{14 a}\\ &=-\frac {A \left (a+b x^2\right )^6}{14 a x^{14}}+\frac {(A b-7 a B) \left (a+b x^2\right )^6}{84 a^2 x^{12}}\\ \end {align*}

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Mathematica [B]  time = 0.03, size = 118, normalized size = 2.46 \[ -\frac {a^5 \left (6 A+7 B x^2\right )+7 a^4 b x^2 \left (5 A+6 B x^2\right )+21 a^3 b^2 x^4 \left (4 A+5 B x^2\right )+35 a^2 b^3 x^6 \left (3 A+4 B x^2\right )+35 a b^4 x^8 \left (2 A+3 B x^2\right )+21 b^5 x^{10} \left (A+2 B x^2\right )}{84 x^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^15,x]

[Out]

-1/84*(21*b^5*x^10*(A + 2*B*x^2) + 35*a*b^4*x^8*(2*A + 3*B*x^2) + 35*a^2*b^3*x^6*(3*A + 4*B*x^2) + 21*a^3*b^2*
x^4*(4*A + 5*B*x^2) + 7*a^4*b*x^2*(5*A + 6*B*x^2) + a^5*(6*A + 7*B*x^2))/x^14

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fricas [B]  time = 0.42, size = 121, normalized size = 2.52 \[ -\frac {42 \, B b^{5} x^{12} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 6 \, A a^{5} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{84 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^15,x, algorithm="fricas")

[Out]

-1/84*(42*B*b^5*x^12 + 21*(5*B*a*b^4 + A*b^5)*x^10 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 105*(B*a^3*b^2 + A*a^2*b
^3)*x^6 + 6*A*a^5 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 7*(B*a^5 + 5*A*a^4*b)*x^2)/x^14

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giac [B]  time = 0.40, size = 127, normalized size = 2.65 \[ -\frac {42 \, B b^{5} x^{12} + 105 \, B a b^{4} x^{10} + 21 \, A b^{5} x^{10} + 140 \, B a^{2} b^{3} x^{8} + 70 \, A a b^{4} x^{8} + 105 \, B a^{3} b^{2} x^{6} + 105 \, A a^{2} b^{3} x^{6} + 42 \, B a^{4} b x^{4} + 84 \, A a^{3} b^{2} x^{4} + 7 \, B a^{5} x^{2} + 35 \, A a^{4} b x^{2} + 6 \, A a^{5}}{84 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^15,x, algorithm="giac")

[Out]

-1/84*(42*B*b^5*x^12 + 105*B*a*b^4*x^10 + 21*A*b^5*x^10 + 140*B*a^2*b^3*x^8 + 70*A*a*b^4*x^8 + 105*B*a^3*b^2*x
^6 + 105*A*a^2*b^3*x^6 + 42*B*a^4*b*x^4 + 84*A*a^3*b^2*x^4 + 7*B*a^5*x^2 + 35*A*a^4*b*x^2 + 6*A*a^5)/x^14

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maple [B]  time = 0.01, size = 104, normalized size = 2.17 \[ -\frac {B \,b^{5}}{2 x^{2}}-\frac {\left (A b +5 B a \right ) b^{4}}{4 x^{4}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{6 x^{6}}-\frac {5 \left (A b +B a \right ) a^{2} b^{2}}{4 x^{8}}-\frac {\left (2 A b +B a \right ) a^{3} b}{2 x^{10}}-\frac {A \,a^{5}}{14 x^{14}}-\frac {\left (5 A b +B a \right ) a^{4}}{12 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^15,x)

[Out]

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

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maxima [B]  time = 1.06, size = 121, normalized size = 2.52 \[ -\frac {42 \, B b^{5} x^{12} + 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 70 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 6 \, A a^{5} + 42 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 7 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{84 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^15,x, algorithm="maxima")

[Out]

-1/84*(42*B*b^5*x^12 + 21*(5*B*a*b^4 + A*b^5)*x^10 + 70*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 105*(B*a^3*b^2 + A*a^2*b
^3)*x^6 + 6*A*a^5 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 7*(B*a^5 + 5*A*a^4*b)*x^2)/x^14

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mupad [B]  time = 0.06, size = 121, normalized size = 2.52 \[ -\frac {\frac {A\,a^5}{14}+x^4\,\left (\frac {B\,a^4\,b}{2}+A\,a^3\,b^2\right )+x^8\,\left (\frac {5\,B\,a^2\,b^3}{3}+\frac {5\,A\,a\,b^4}{6}\right )+x^2\,\left (\frac {B\,a^5}{12}+\frac {5\,A\,b\,a^4}{12}\right )+x^{10}\,\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )+x^6\,\left (\frac {5\,B\,a^3\,b^2}{4}+\frac {5\,A\,a^2\,b^3}{4}\right )+\frac {B\,b^5\,x^{12}}{2}}{x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^5)/x^15,x)

[Out]

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

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sympy [B]  time = 15.80, size = 134, normalized size = 2.79 \[ \frac {- 6 A a^{5} - 42 B b^{5} x^{12} + x^{10} \left (- 21 A b^{5} - 105 B a b^{4}\right ) + x^{8} \left (- 70 A a b^{4} - 140 B a^{2} b^{3}\right ) + x^{6} \left (- 105 A a^{2} b^{3} - 105 B a^{3} b^{2}\right ) + x^{4} \left (- 84 A a^{3} b^{2} - 42 B a^{4} b\right ) + x^{2} \left (- 35 A a^{4} b - 7 B a^{5}\right )}{84 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**15,x)

[Out]

(-6*A*a**5 - 42*B*b**5*x**12 + x**10*(-21*A*b**5 - 105*B*a*b**4) + x**8*(-70*A*a*b**4 - 140*B*a**2*b**3) + x**
6*(-105*A*a**2*b**3 - 105*B*a**3*b**2) + x**4*(-84*A*a**3*b**2 - 42*B*a**4*b) + x**2*(-35*A*a**4*b - 7*B*a**5)
)/(84*x**14)

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