3.1.93 \(\int \frac {A+B x+C x^2+D x^3}{x^3 (a+b x^2)} \, dx\) [93]

Optimal. Leaf size=92 \[ -\frac {A}{2 a x^2}-\frac {B}{a x}-\frac {(b B-a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {(A b-a C) \log (x)}{a^2}+\frac {(A b-a C) \log \left (a+b x^2\right )}{2 a^2} \]

[Out]

-1/2*A/a/x^2-B/a/x-(A*b-C*a)*ln(x)/a^2+1/2*(A*b-C*a)*ln(b*x^2+a)/a^2-(B*b-D*a)*arctan(x*b^(1/2)/a^(1/2))/a^(3/
2)/b^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1816, 649, 211, 266} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b B-a D)}{a^{3/2} \sqrt {b}}+\frac {(A b-a C) \log \left (a+b x^2\right )}{2 a^2}-\frac {\log (x) (A b-a C)}{a^2}-\frac {A}{2 a x^2}-\frac {B}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)),x]

[Out]

-1/2*A/(a*x^2) - B/(a*x) - ((b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) - ((A*b - a*C)*Log[x])/
a^2 + ((A*b - a*C)*Log[a + b*x^2])/(2*a^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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )} \, dx &=\int \left (\frac {A}{a x^3}+\frac {B}{a x^2}+\frac {-A b+a C}{a^2 x}+\frac {-a (b B-a D)+b (A b-a C) x}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {A}{2 a x^2}-\frac {B}{a x}-\frac {(A b-a C) \log (x)}{a^2}+\frac {\int \frac {-a (b B-a D)+b (A b-a C) x}{a+b x^2} \, dx}{a^2}\\ &=-\frac {A}{2 a x^2}-\frac {B}{a x}-\frac {(A b-a C) \log (x)}{a^2}+\frac {(b (A b-a C)) \int \frac {x}{a+b x^2} \, dx}{a^2}-\frac {(b B-a D) \int \frac {1}{a+b x^2} \, dx}{a}\\ &=-\frac {A}{2 a x^2}-\frac {B}{a x}-\frac {(b B-a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {(A b-a C) \log (x)}{a^2}+\frac {(A b-a C) \log \left (a+b x^2\right )}{2 a^2}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 84, normalized size = 0.91 \begin {gather*} \frac {-\frac {a A}{x^2}-\frac {2 a B}{x}+\frac {2 \sqrt {a} (-b B+a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+2 (-A b+a C) \log (x)+(A b-a C) \log \left (a+b x^2\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2 + D*x^3)/(x^3*(a + b*x^2)),x]

[Out]

(-((a*A)/x^2) - (2*a*B)/x + (2*Sqrt[a]*(-(b*B) + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + 2*(-(A*b) + a*C)*
Log[x] + (A*b - a*C)*Log[a + b*x^2])/(2*a^2)

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Maple [A]
time = 0.11, size = 89, normalized size = 0.97

method	result	size
default	\(\frac {\frac {\left (b^{2} A -a b C \right ) \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {\left (-a b B +a^{2} D\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{a^{2}}-\frac {A}{2 a \,x^{2}}-\frac {B}{a x}+\frac {\left (-A b +a C \right ) \ln \left (x \right )}{a^{2}}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

1/a^2*(1/2*(A*b^2-C*a*b)/b*ln(b*x^2+a)+(-B*a*b+D*a^2)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))-1/2*A/a/x^2-B/a/x+1
/a^2*(-A*b+C*a)*ln(x)

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Maxima [A]
time = 0.48, size = 76, normalized size = 0.83 \begin {gather*} \frac {{\left (D a - B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {{\left (C a - A b\right )} \log \left (x\right )}{a^{2}} - \frac {2 \, B x + A}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a),x, algorithm="maxima")

[Out]

(D*a - B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - 1/2*(C*a - A*b)*log(b*x^2 + a)/a^2 + (C*a - A*b)*log(x)/a^2
- 1/2*(2*B*x + A)/(a*x^2)

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Fricas [A]
time = 3.19, size = 205, normalized size = 2.23 \begin {gather*} \left [-\frac {{\left (D a - B b\right )} \sqrt {-a b} x^{2} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, B a b x + {\left (C a b - A b^{2}\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \, {\left (C a b - A b^{2}\right )} x^{2} \log \left (x\right ) + A a b}{2 \, a^{2} b x^{2}}, \frac {2 \, {\left (D a - B b\right )} \sqrt {a b} x^{2} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 2 \, B a b x - {\left (C a b - A b^{2}\right )} x^{2} \log \left (b x^{2} + a\right ) + 2 \, {\left (C a b - A b^{2}\right )} x^{2} \log \left (x\right ) - A a b}{2 \, a^{2} b x^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/2*((D*a - B*b)*sqrt(-a*b)*x^2*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2*B*a*b*x + (C*a*b - A*b^2)*
x^2*log(b*x^2 + a) - 2*(C*a*b - A*b^2)*x^2*log(x) + A*a*b)/(a^2*b*x^2), 1/2*(2*(D*a - B*b)*sqrt(a*b)*x^2*arcta
n(sqrt(a*b)*x/a) - 2*B*a*b*x - (C*a*b - A*b^2)*x^2*log(b*x^2 + a) + 2*(C*a*b - A*b^2)*x^2*log(x) - A*a*b)/(a^2
*b*x^2)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x**3+C*x**2+B*x+A)/x**3/(b*x**2+a),x)

[Out]

Timed out

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Giac [A]
time = 0.92, size = 80, normalized size = 0.87 \begin {gather*} \frac {{\left (D a - B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {{\left (C a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {{\left (C a - A b\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {2 \, B a x + A a}{2 \, a^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^3+C*x^2+B*x+A)/x^3/(b*x^2+a),x, algorithm="giac")

[Out]

(D*a - B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - 1/2*(C*a - A*b)*log(b*x^2 + a)/a^2 + (C*a - A*b)*log(abs(x))
/a^2 - 1/2*(2*B*a*x + A*a)/(a^2*x^2)

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Mupad [B]
time = 1.30, size = 97, normalized size = 1.05 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,D}{\sqrt {a}\,\sqrt {b}}-\frac {B}{a\,x}-\frac {C\,\left (\ln \left (b\,x^2+a\right )-2\,\ln \left (x\right )\right )}{2\,a}-\frac {A}{2\,a\,x^2}+\frac {A\,b\,\ln \left (b\,x^2+a\right )}{2\,a^2}-\frac {A\,b\,\ln \left (x\right )}{a^2}-\frac {B\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{a^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2 + x^3*D)/(x^3*(a + b*x^2)),x)

[Out]

(atan((b^(1/2)*x)/a^(1/2))*D)/(a^(1/2)*b^(1/2)) - B/(a*x) - (C*(log(a + b*x^2) - 2*log(x)))/(2*a) - A/(2*a*x^2
) + (A*b*log(a + b*x^2))/(2*a^2) - (A*b*log(x))/a^2 - (B*b^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/a^(3/2)

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