3.1.70 \(\int \frac {A+B x^2}{x^8 (a+b x^2)} \, dx\) [70]

Optimal. Leaf size=99 \[ -\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}-\frac {b (A b-a B)}{3 a^3 x^3}+\frac {b^2 (A b-a B)}{a^4 x}+\frac {b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}} \]

[Out]

-1/7*A/a/x^7+1/5*(A*b-B*a)/a^2/x^5-1/3*b*(A*b-B*a)/a^3/x^3+b^2*(A*b-B*a)/a^4/x+b^(5/2)*(A*b-B*a)*arctan(x*b^(1
/2)/a^(1/2))/a^(9/2)

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Rubi [A]
time = 0.04, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {464, 331, 211} \begin {gather*} \frac {b^{5/2} (A b-a B) \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}+\frac {b^2 (A b-a B)}{a^4 x}-\frac {b (A b-a B)}{3 a^3 x^3}+\frac {A b-a B}{5 a^2 x^5}-\frac {A}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^8*(a + b*x^2)),x]

[Out]

-1/7*A/(a*x^7) + (A*b - a*B)/(5*a^2*x^5) - (b*(A*b - a*B))/(3*a^3*x^3) + (b^2*(A*b - a*B))/(a^4*x) + (b^(5/2)*
(A*b - a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 464

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

Rubi steps

\begin {align*} \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )} \, dx &=-\frac {A}{7 a x^7}-\frac {(7 A b-7 a B) \int \frac {1}{x^6 \left (a+b x^2\right )} \, dx}{7 a}\\ &=-\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}+\frac {(b (A b-a B)) \int \frac {1}{x^4 \left (a+b x^2\right )} \, dx}{a^2}\\ &=-\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}-\frac {b (A b-a B)}{3 a^3 x^3}-\frac {\left (b^2 (A b-a B)\right ) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{a^3}\\ &=-\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}-\frac {b (A b-a B)}{3 a^3 x^3}+\frac {b^2 (A b-a B)}{a^4 x}+\frac {\left (b^3 (A b-a B)\right ) \int \frac {1}{a+b x^2} \, dx}{a^4}\\ &=-\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}-\frac {b (A b-a B)}{3 a^3 x^3}+\frac {b^2 (A b-a B)}{a^4 x}+\frac {b^{5/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 101, normalized size = 1.02 \begin {gather*} -\frac {A}{7 a x^7}+\frac {A b-a B}{5 a^2 x^5}+\frac {b (-A b+a B)}{3 a^3 x^3}-\frac {b^2 (-A b+a B)}{a^4 x}-\frac {b^{5/2} (-A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^8*(a + b*x^2)),x]

[Out]

-1/7*A/(a*x^7) + (A*b - a*B)/(5*a^2*x^5) + (b*(-(A*b) + a*B))/(3*a^3*x^3) - (b^2*(-(A*b) + a*B))/(a^4*x) - (b^
(5/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(9/2)

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Maple [A]
time = 0.10, size = 91, normalized size = 0.92

method	result	size
default	\(\frac {b^{3} \left (A b -B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{4} \sqrt {a b}}-\frac {A}{7 a \,x^{7}}-\frac {-A b +B a}{5 a^{2} x^{5}}-\frac {b \left (A b -B a \right )}{3 a^{3} x^{3}}+\frac {b^{2} \left (A b -B a \right )}{a^{4} x}\)	\(91\)
risch	\(\frac {\frac {b^{2} \left (A b -B a \right ) x^{6}}{a^{4}}-\frac {b \left (A b -B a \right ) x^{4}}{3 a^{3}}+\frac {\left (A b -B a \right ) x^{2}}{5 a^{2}}-\frac {A}{7 a}}{x^{7}}+\frac {\sqrt {-a b}\, b^{3} \ln \left (-b x -\sqrt {-a b}\right ) A}{2 a^{5}}-\frac {\sqrt {-a b}\, b^{2} \ln \left (-b x -\sqrt {-a b}\right ) B}{2 a^{4}}-\frac {\sqrt {-a b}\, b^{3} \ln \left (-b x +\sqrt {-a b}\right ) A}{2 a^{5}}+\frac {\sqrt {-a b}\, b^{2} \ln \left (-b x +\sqrt {-a b}\right ) B}{2 a^{4}}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^8/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 103, normalized size = 1.04 \begin {gather*} -\frac {{\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{6} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x^{2}}{105 \, a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a),x, algorithm="maxima")

[Out]

-(B*a*b^3 - A*b^4)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/105*(105*(B*a*b^2 - A*b^3)*x^6 - 35*(B*a^2*b - A*
a*b^2)*x^4 + 15*A*a^3 + 21*(B*a^3 - A*a^2*b)*x^2)/(a^4*x^7)

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Fricas [A]
time = 0.64, size = 234, normalized size = 2.36 \begin {gather*} \left [-\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{7} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 210 \, {\left (B a b^{2} - A b^{3}\right )} x^{6} - 70 \, {\left (B a^{2} b - A a b^{2}\right )} x^{4} + 30 \, A a^{3} + 42 \, {\left (B a^{3} - A a^{2} b\right )} x^{2}}{210 \, a^{4} x^{7}}, -\frac {105 \, {\left (B a b^{2} - A b^{3}\right )} x^{7} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 105 \, {\left (B a b^{2} - A b^{3}\right )} x^{6} - 35 \, {\left (B a^{2} b - A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 21 \, {\left (B a^{3} - A a^{2} b\right )} x^{2}}{105 \, a^{4} x^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/210*(105*(B*a*b^2 - A*b^3)*x^7*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 210*(B*a*b^2 -
A*b^3)*x^6 - 70*(B*a^2*b - A*a*b^2)*x^4 + 30*A*a^3 + 42*(B*a^3 - A*a^2*b)*x^2)/(a^4*x^7), -1/105*(105*(B*a*b^
2 - A*b^3)*x^7*sqrt(b/a)*arctan(x*sqrt(b/a)) + 105*(B*a*b^2 - A*b^3)*x^6 - 35*(B*a^2*b - A*a*b^2)*x^4 + 15*A*a
^3 + 21*(B*a^3 - A*a^2*b)*x^2)/(a^4*x^7)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (87) = 174\).
time = 0.29, size = 187, normalized size = 1.89 \begin {gather*} \frac {\sqrt {- \frac {b^{5}}{a^{9}}} \left (- A b + B a\right ) \log {\left (- \frac {a^{5} \sqrt {- \frac {b^{5}}{a^{9}}} \left (- A b + B a\right )}{- A b^{4} + B a b^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {b^{5}}{a^{9}}} \left (- A b + B a\right ) \log {\left (\frac {a^{5} \sqrt {- \frac {b^{5}}{a^{9}}} \left (- A b + B a\right )}{- A b^{4} + B a b^{3}} + x \right )}}{2} + \frac {- 15 A a^{3} + x^{6} \cdot \left (105 A b^{3} - 105 B a b^{2}\right ) + x^{4} \left (- 35 A a b^{2} + 35 B a^{2} b\right ) + x^{2} \cdot \left (21 A a^{2} b - 21 B a^{3}\right )}{105 a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**8/(b*x**2+a),x)

[Out]

sqrt(-b**5/a**9)*(-A*b + B*a)*log(-a**5*sqrt(-b**5/a**9)*(-A*b + B*a)/(-A*b**4 + B*a*b**3) + x)/2 - sqrt(-b**5
/a**9)*(-A*b + B*a)*log(a**5*sqrt(-b**5/a**9)*(-A*b + B*a)/(-A*b**4 + B*a*b**3) + x)/2 + (-15*A*a**3 + x**6*(1
05*A*b**3 - 105*B*a*b**2) + x**4*(-35*A*a*b**2 + 35*B*a**2*b) + x**2*(21*A*a**2*b - 21*B*a**3))/(105*a**4*x**7
)

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Giac [A]
time = 1.06, size = 106, normalized size = 1.07 \begin {gather*} -\frac {{\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} - \frac {105 \, B a b^{2} x^{6} - 105 \, A b^{3} x^{6} - 35 \, B a^{2} b x^{4} + 35 \, A a b^{2} x^{4} + 21 \, B a^{3} x^{2} - 21 \, A a^{2} b x^{2} + 15 \, A a^{3}}{105 \, a^{4} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a),x, algorithm="giac")

[Out]

-(B*a*b^3 - A*b^4)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4) - 1/105*(105*B*a*b^2*x^6 - 105*A*b^3*x^6 - 35*B*a^2*b
*x^4 + 35*A*a*b^2*x^4 + 21*B*a^3*x^2 - 21*A*a^2*b*x^2 + 15*A*a^3)/(a^4*x^7)

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Mupad [B]
time = 0.07, size = 89, normalized size = 0.90 \begin {gather*} \frac {b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{9/2}}-\frac {\frac {A}{7\,a}-\frac {x^2\,\left (A\,b-B\,a\right )}{5\,a^2}-\frac {b^2\,x^6\,\left (A\,b-B\,a\right )}{a^4}+\frac {b\,x^4\,\left (A\,b-B\,a\right )}{3\,a^3}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^8*(a + b*x^2)),x)

[Out]

(b^(5/2)*atan((b^(1/2)*x)/a^(1/2))*(A*b - B*a))/a^(9/2) - (A/(7*a) - (x^2*(A*b - B*a))/(5*a^2) - (b^2*x^6*(A*b
- B*a))/a^4 + (b*x^4*(A*b - B*a))/(3*a^3))/x^7

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