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]

-(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)

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Rubi [A]  time = 0.0296584, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, 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 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]]

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 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]

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*}

Mathematica [B]  time = 0.0318664, size = 118, normalized size = 2.46 \[ -\frac{35 a^2 b^3 x^6 \left (3 A+4 B x^2\right )+21 a^3 b^2 x^4 \left (4 A+5 B x^2\right )+7 a^4 b x^2 \left (5 A+6 B x^2\right )+a^5 \left (6 A+7 B x^2\right )+35 a b^4 x^8 \left (2 A+3 B x^2\right )+21 b^5 x^{10} \left (A+2 B x^2\right )}{84 x^{14}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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))/(84*x^14)

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Maple [B]  time = 0.006, size = 104, normalized size = 2.2 \begin{align*} -{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{2\,{x}^{10}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{12\,{x}^{12}}}-{\frac{5\,{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{4\,{x}^{8}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{4\,{x}^{4}}}-{\frac{B{b}^{5}}{2\,{x}^{2}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{6\,{x}^{6}}}-{\frac{A{a}^{5}}{14\,{x}^{14}}} \end{align*}

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

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Maxima [B]  time = 1.02963, size = 163, normalized size = 3.4 \begin{align*} -\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}} \end{align*}

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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Fricas [B]  time = 1.40334, size = 265, normalized size = 5.52 \begin{align*} -\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}} \end{align*}

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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Sympy [B]  time = 25.9282, size = 128, normalized size = 2.67 \begin{align*} - \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}} \end{align*}

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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Giac [B]  time = 1.15746, size = 171, normalized size = 3.56 \begin{align*} -\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}} \end{align*}

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