3.207 \(\int \frac {c+d x^2}{x^4 (a+b x^2)} \, dx\)

Optimal. Leaf size=59 \[ \frac {\sqrt {b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {b c-a d}{a^2 x}-\frac {c}{3 a x^3} \]

[Out]

-1/3*c/a/x^3+(-a*d+b*c)/a^2/x+(-a*d+b*c)*arctan(x*b^(1/2)/a^(1/2))*b^(1/2)/a^(5/2)

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Rubi [A]  time = 0.04, antiderivative size = 59, 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 = {453, 325, 205} \[ \frac {b c-a d}{a^2 x}+\frac {\sqrt {b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}-\frac {c}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/(x^4*(a + b*x^2)),x]

[Out]

-c/(3*a*x^3) + (b*c - a*d)/(a^2*x) + (Sqrt[b]*(b*c - a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(5/2)

Rule 205

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

Rule 325

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 453

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 {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx &=-\frac {c}{3 a x^3}-\frac {(3 b c-3 a d) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{3 a}\\ &=-\frac {c}{3 a x^3}+\frac {b c-a d}{a^2 x}+\frac {(b (b c-a d)) \int \frac {1}{a+b x^2} \, dx}{a^2}\\ &=-\frac {c}{3 a x^3}+\frac {b c-a d}{a^2 x}+\frac {\sqrt {b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 60, normalized size = 1.02 \[ -\frac {\sqrt {b} (a d-b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}}+\frac {b c-a d}{a^2 x}-\frac {c}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/(x^4*(a + b*x^2)),x]

[Out]

-1/3*c/(a*x^3) + (b*c - a*d)/(a^2*x) - (Sqrt[b]*(-(b*c) + a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(5/2)

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fricas [A]  time = 0.48, size = 136, normalized size = 2.31 \[ \left [-\frac {3 \, {\left (b c - a d\right )} x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 6 \, {\left (b c - a d\right )} x^{2} + 2 \, a c}{6 \, a^{2} x^{3}}, \frac {3 \, {\left (b c - a d\right )} x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 3 \, {\left (b c - a d\right )} x^{2} - a c}{3 \, a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/x^4/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/6*(3*(b*c - a*d)*x^3*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) - 6*(b*c - a*d)*x^2 + 2*a*
c)/(a^2*x^3), 1/3*(3*(b*c - a*d)*x^3*sqrt(b/a)*arctan(x*sqrt(b/a)) + 3*(b*c - a*d)*x^2 - a*c)/(a^2*x^3)]

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giac [A]  time = 0.31, size = 57, normalized size = 0.97 \[ \frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/x^4/(b*x^2+a),x, algorithm="giac")

[Out]

(b^2*c - a*b*d)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) + 1/3*(3*b*c*x^2 - 3*a*d*x^2 - a*c)/(a^2*x^3)

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maple [A]  time = 0.01, size = 72, normalized size = 1.22 \[ -\frac {b d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {b^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {d}{a x}+\frac {b c}{a^{2} x}-\frac {c}{3 a \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)/x^4/(b*x^2+a),x)

[Out]

-b/a/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*d+b^2/a^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)*c-1/3*c/a/x^3-1/a/x
*d+1/a^2/x*b*c

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maxima [A]  time = 2.33, size = 56, normalized size = 0.95 \[ \frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, {\left (b c - a d\right )} x^{2} - a c}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/x^4/(b*x^2+a),x, algorithm="maxima")

[Out]

(b^2*c - a*b*d)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2) + 1/3*(3*(b*c - a*d)*x^2 - a*c)/(a^2*x^3)

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mupad [B]  time = 0.12, size = 53, normalized size = 0.90 \[ -\frac {\frac {c}{3\,a}+\frac {x^2\,\left (a\,d-b\,c\right )}{a^2}}{x^3}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{a^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)/(x^4*(a + b*x^2)),x)

[Out]

- (c/(3*a) + (x^2*(a*d - b*c))/a^2)/x^3 - (b^(1/2)*atan((b^(1/2)*x)/a^(1/2))*(a*d - b*c))/a^(5/2)

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sympy [B]  time = 0.41, size = 129, normalized size = 2.19 \[ \frac {\sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right ) \log {\left (- \frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right )}{a b d - b^{2} c} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right ) \log {\left (\frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right )}{a b d - b^{2} c} + x \right )}}{2} + \frac {- a c + x^{2} \left (- 3 a d + 3 b c\right )}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)/x**4/(b*x**2+a),x)

[Out]

sqrt(-b/a**5)*(a*d - b*c)*log(-a**3*sqrt(-b/a**5)*(a*d - b*c)/(a*b*d - b**2*c) + x)/2 - sqrt(-b/a**5)*(a*d - b
*c)*log(a**3*sqrt(-b/a**5)*(a*d - b*c)/(a*b*d - b**2*c) + x)/2 + (-a*c + x**2*(-3*a*d + 3*b*c))/(3*a**2*x**3)

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