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

Optimal. Leaf size=96 \[ \frac{a d \log \left (a+b x^2\right )}{2 b \left (a d^2+b c^2\right )}+\frac{c^2 \log (c+d x)}{d \left (a d^2+b c^2\right )}-\frac{\sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \left (a d^2+b c^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.108043, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1629, 635, 205, 260} \[ \frac{a d \log \left (a+b x^2\right )}{2 b \left (a d^2+b c^2\right )}+\frac{c^2 \log (c+d x)}{d \left (a d^2+b c^2\right )}-\frac{\sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \left (a d^2+b c^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((c + d*x)*(a + b*x^2)),x]

[Out]

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

Rule 1629

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

Rule 635

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 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 260

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]

Rubi steps

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

Mathematica [A]  time = 0.0418023, size = 73, normalized size = 0.76 \[ \frac{-2 \sqrt{a} \sqrt{b} c d \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+a d^2 \log \left (a+b x^2\right )+2 b c^2 \log (c+d x)}{2 a b d^3+2 b^2 c^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 87, normalized size = 0.9 \begin{align*}{\frac{ad\ln \left ( b{x}^{2}+a \right ) }{2\,b \left ( a{d}^{2}+{c}^{2}b \right ) }}-{\frac{ac}{a{d}^{2}+{c}^{2}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{2}\ln \left ( dx+c \right ) }{d \left ( a{d}^{2}+{c}^{2}b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77094, size = 354, normalized size = 3.69 \begin{align*} \left [\frac{b c d \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + a d^{2} \log \left (b x^{2} + a\right ) + 2 \, b c^{2} \log \left (d x + c\right )}{2 \,{\left (b^{2} c^{2} d + a b d^{3}\right )}}, -\frac{2 \, b c d \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - a d^{2} \log \left (b x^{2} + a\right ) - 2 \, b c^{2} \log \left (d x + c\right )}{2 \,{\left (b^{2} c^{2} d + a b d^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 7.01037, size = 1355, normalized size = 14.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*log(x + (-4*a**3*b*c**4*d**5/(a*d**2 + b*c**2)**2 + 2*a**3*c**2*d**5/(a*d**2 + b*c**2) + 4*a**2*b**2*c**6
*d**3/(a*d**2 + b*c**2)**2 - 4*a**2*b*c**4*d**3/(a*d**2 + b*c**2) - a**2*c**2*d**3 + 20*a*b**3*c**8*d/(a*d**2
+ b*c**2)**2 - 14*a*b**2*c**6*d/(a*d**2 + b*c**2) + 7*a*b*c**4*d + 12*b**4*c**10/(d*(a*d**2 + b*c**2)**2) - 8*
b**3*c**8/(d*(a*d**2 + b*c**2)))/(a**2*c*d**4 - 3*a*b*c**3*d**2 + 4*b**2*c**5))/(d*(a*d**2 + b*c**2)) + (a*d/(
2*b*(a*d**2 + b*c**2)) - c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2)))*log(x + (-4*a**3*b*d**7*(a*d/(2*b*(a*d**2
+ b*c**2)) - c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2)))**2 + 2*a**3*d**6*(a*d/(2*b*(a*d**2 + b*c**2)) - c*sq
rt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2))) + 4*a**2*b**2*c**2*d**5*(a*d/(2*b*(a*d**2 + b*c**2)) - c*sqrt(-a*b**3)
/(2*b**2*(a*d**2 + b*c**2)))**2 - 4*a**2*b*c**2*d**4*(a*d/(2*b*(a*d**2 + b*c**2)) - c*sqrt(-a*b**3)/(2*b**2*(a
*d**2 + b*c**2))) - a**2*c**2*d**3 + 20*a*b**3*c**4*d**3*(a*d/(2*b*(a*d**2 + b*c**2)) - c*sqrt(-a*b**3)/(2*b**
2*(a*d**2 + b*c**2)))**2 - 14*a*b**2*c**4*d**2*(a*d/(2*b*(a*d**2 + b*c**2)) - c*sqrt(-a*b**3)/(2*b**2*(a*d**2
+ b*c**2))) + 7*a*b*c**4*d + 12*b**4*c**6*d*(a*d/(2*b*(a*d**2 + b*c**2)) - c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b
*c**2)))**2 - 8*b**3*c**6*(a*d/(2*b*(a*d**2 + b*c**2)) - c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2))))/(a**2*c*
d**4 - 3*a*b*c**3*d**2 + 4*b**2*c**5)) + (a*d/(2*b*(a*d**2 + b*c**2)) + c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c*
*2)))*log(x + (-4*a**3*b*d**7*(a*d/(2*b*(a*d**2 + b*c**2)) + c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2)))**2 +
2*a**3*d**6*(a*d/(2*b*(a*d**2 + b*c**2)) + c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2))) + 4*a**2*b**2*c**2*d**5
*(a*d/(2*b*(a*d**2 + b*c**2)) + c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2)))**2 - 4*a**2*b*c**2*d**4*(a*d/(2*b*
(a*d**2 + b*c**2)) + c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2))) - a**2*c**2*d**3 + 20*a*b**3*c**4*d**3*(a*d/(
2*b*(a*d**2 + b*c**2)) + c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2)))**2 - 14*a*b**2*c**4*d**2*(a*d/(2*b*(a*d**
2 + b*c**2)) + c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2))) + 7*a*b*c**4*d + 12*b**4*c**6*d*(a*d/(2*b*(a*d**2 +
b*c**2)) + c*sqrt(-a*b**3)/(2*b**2*(a*d**2 + b*c**2)))**2 - 8*b**3*c**6*(a*d/(2*b*(a*d**2 + b*c**2)) + c*sqrt
(-a*b**3)/(2*b**2*(a*d**2 + b*c**2))))/(a**2*c*d**4 - 3*a*b*c**3*d**2 + 4*b**2*c**5))

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Giac [A]  time = 1.18663, size = 115, normalized size = 1.2 \begin{align*} \frac{a d \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} + a b d^{2}\right )}} + \frac{c^{2} \log \left ({\left | d x + c \right |}\right )}{b c^{2} d + a d^{3}} - \frac{a c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b c^{2} + a d^{2}\right )} \sqrt{a b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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