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

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 = 0.203124, 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 \[ \frac{a d \log \left (a+b x^2\right )}{2 b \left (a d^2+b c^2\right )}+\frac{c^2 \log (c+d x)}{d \left (a d^2+b c^2\right )}-\frac{\sqrt{a} c \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} \left (a d^2+b c^2\right )} \]

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))

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Rubi in Sympy [A]  time = 23.6938, size = 82, normalized size = 0.85 \[ - \frac{\sqrt{a} c \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b} \left (a d^{2} + b c^{2}\right )} + \frac{a d \log{\left (a + b x^{2} \right )}}{2 b \left (a d^{2} + b c^{2}\right )} + \frac{c^{2} \log{\left (c + d x \right )}}{d \left (a d^{2} + b c^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(d*x+c)/(b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.0560853, 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 = 0.009, size = 87, normalized size = 0.9 \[{\frac{ad\ln \left ( b{x}^{2}+a \right ) }{2\,b \left ( a{d}^{2}+{c}^{2}b \right ) }}-{\frac{ac}{a{d}^{2}+{c}^{2}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{2}\ln \left ( dx+c \right ) }{d \left ( a{d}^{2}+{c}^{2}b \right ) }} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292032, size = 1, normalized size = 0.01 \[ \left [\frac{b c d \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + a d^{2} \log \left (b x^{2} + a\right ) + 2 \, b c^{2} \log \left (d x + c\right )}{2 \,{\left (b^{2} c^{2} d + a b d^{3}\right )}}, -\frac{2 \, b c d \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - a d^{2} \log \left (b x^{2} + a\right ) - 2 \, b c^{2} \log \left (d x + c\right )}{2 \,{\left (b^{2} c^{2} d + a b d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 + a)*(d*x + c)),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*l
og(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(x/sqrt(a/b)) - 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 [A]  time = 22.4507, size = 1355, normalized size = 14.11 \[ \text{result too large to display} \]

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*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)) +
(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*sqr
t(-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/XCAS [A]  time = 0.264403, size = 115, normalized size = 1.2 \[ \frac{a d{\rm ln}\left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} + a b d^{2}\right )}} + \frac{c^{2}{\rm ln}\left ({\left | d x + c \right |}\right )}{b c^{2} d + a d^{3}} - \frac{a c \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b c^{2} + a d^{2}\right )} \sqrt{a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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