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

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

[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)

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Rubi [A]  time = 0.0513359, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {446, 78, 45, 37} \[ -\frac{b \left (a+b x^2\right )^6 (A b-4 a B)}{336 a^3 x^{12}}+\frac{\left (a+b x^2\right )^6 (A b-4 a B)}{56 a^2 x^{14}}-\frac{A \left (a+b x^2\right )^6}{16 a x^{16}} \]

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 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 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 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^{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*}

Mathematica [A]  time = 0.029576, size = 121, normalized size = 1.59 \[ -\frac{84 a^2 b^3 x^6 \left (4 A+5 B x^2\right )+56 a^3 b^2 x^4 \left (5 A+6 B x^2\right )+20 a^4 b x^2 \left (6 A+7 B x^2\right )+3 a^5 \left (7 A+8 B x^2\right )+70 a b^4 x^8 \left (3 A+4 B x^2\right )+28 b^5 x^{10} \left (2 A+3 B x^2\right )}{336 x^{16}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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))/(336*x^16)

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Maple [A]  time = 0.005, size = 104, normalized size = 1.4 \begin{align*} -{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{12\,{x}^{12}}}-{\frac{{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{{x}^{10}}}-{\frac{A{a}^{5}}{16\,{x}^{16}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{B{b}^{5}}{4\,{x}^{4}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{6\,{x}^{6}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{8\,{x}^{8}}} \end{align*}

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]

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

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Maxima [A]  time = 0.997342, size = 163, normalized size = 2.14 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.42996, size = 271, normalized size = 3.57 \begin{align*} -\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{align*}

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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Sympy [A]  time = 59.3195, size = 128, normalized size = 1.68 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.14475, size = 171, normalized size = 2.25 \begin{align*} -\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{align*}

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