3.384 \(\int \frac{x^3}{a c+b c x^2+d \sqrt{a+b x^2}} \, dx\)
Optimal. Leaf size=69 \[ -\frac{d \sqrt{a+b x^2}}{b^2 c^2}-\frac{\left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^2}+d\right )}{b^2 c^3}+\frac{x^2}{2 b c} \]
[Out]
x^2/(2*b*c) - (d*Sqrt[a + b*x^2])/(b^2*c^2) - ((a*c^2 - d^2)*Log[d + c*Sqrt[a +
b*x^2]])/(b^2*c^3)
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Rubi [A] time = 0.372471, antiderivative size = 69, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069
\[ -\frac{d \sqrt{a+b x^2}}{b^2 c^2}-\frac{\left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^2}+d\right )}{b^2 c^3}+\frac{x^2}{2 b c} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a*c + b*c*x^2 + d*Sqrt[a + b*x^2]),x]
[Out]
x^2/(2*b*c) - (d*Sqrt[a + b*x^2])/(b^2*c^2) - ((a*c^2 - d^2)*Log[d + c*Sqrt[a +
b*x^2]])/(b^2*c^3)
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{\sqrt{a + b x^{2}}} x\, dx}{b^{2} c} - \frac{\int ^{\sqrt{a + b x^{2}}} d\, dx}{b^{2} c^{2}} + \frac{\left (- a c^{2} + d^{2}\right ) \log{\left (c \sqrt{a + b x^{2}} + d \right )}}{b^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(a*c+b*c*x**2+d*(b*x**2+a)**(1/2)),x)
[Out]
Integral(x, (x, sqrt(a + b*x**2)))/(b**2*c) - Integral(d, (x, sqrt(a + b*x**2)))
/(b**2*c**2) + (-a*c**2 + d**2)*log(c*sqrt(a + b*x**2) + d)/(b**2*c**3)
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Mathematica [A] time = 0.10938, size = 95, normalized size = 1.38 \[ \frac{\left (d^2-a c^2\right ) \log \left (a c^2+b c^2 x^2-d^2\right )+\left (2 d^2-2 a c^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{a+b x^2}}{d}\right )+c \left (b c x^2-2 d \sqrt{a+b x^2}\right )}{2 b^2 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(a*c + b*c*x^2 + d*Sqrt[a + b*x^2]),x]
[Out]
(c*(b*c*x^2 - 2*d*Sqrt[a + b*x^2]) + (-2*a*c^2 + 2*d^2)*ArcTanh[(c*Sqrt[a + b*x^
2])/d] + (-(a*c^2) + d^2)*Log[a*c^2 - d^2 + b*c^2*x^2])/(2*b^2*c^3)
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Maple [B] time = 0.03, size = 3426, normalized size = 49.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(a*c+b*c*x^2+d*(b*x^2+a)^(1/2)),x)
[Out]
1/2*d*c^2*a/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^
2*b*(a*c^2-d^2))^(1/2))/b*((x-1/b*(-a*b)^(1/2))^2*b+2*(-a*b)^(1/2)*(x-1/b*(-a*b)
^(1/2)))^(1/2)+1/2*d*c^2*a/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)
^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b^(3/2)*(-a*b)^(1/2)*ln(((x-1/b*(-a*b)^(1
/2))*b+(-a*b)^(1/2))/b^(1/2)+((x-1/b*(-a*b)^(1/2))^2*b+2*(-a*b)^(1/2)*(x-1/b*(-a
*b)^(1/2)))^(1/2))+1/2*d*c^2*a/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-
a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b*((x+1/b*(-a*b)^(1/2))^2*b-2*(-a*b)^
(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2)-1/2*d*c^2*a/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d
^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b^(3/2)*(-a*b)^(1/2)*l
n(((x+1/b*(-a*b)^(1/2))*b-(-a*b)^(1/2))/b^(1/2)+((x+1/b*(-a*b)^(1/2))^2*b-2*(-a*
b)^(1/2)*(x+1/b*(-a*b)^(1/2)))^(1/2))-1/2*d/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2
))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b*((x-(-c^2*b*(a*c^2-d^2
))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/
2)/c^2/b)+d^2/c^2)^(1/2)*a*c^2+1/2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))
/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/
c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)
+d^2/c^2)^(1/2)*d^3-1/2*d/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^
(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b^(3/2)*(-c^2*b*(a*c^2-d^2))^(1/2)*ln(((-c
^2*b*(a*c^2-d^2))^(1/2)/c^2+(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x-
(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*
b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))*a+1/2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c
^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b^(3/2)*(-c^2*b*(a
*c^2-d^2))^(1/2)/c^2*ln(((-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x-(-c^2*b*(a*c^2-d^2))^
(1/2)/c^2/b)*b)/b^(1/2)+((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c
^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))*d^3+1/2/
((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d
^2))^(1/2))/b*d^3/(d^2/c^2)^(1/2)*ln((2*d^2/c^2+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2
*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))
^(1/2)/c^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)
/c^2/b)+d^2/c^2)^(1/2))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))*a-1/2/((-a*b)^(1/2
)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/
b*d^5/c^2/(d^2/c^2)^(1/2)*ln((2*d^2/c^2+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^
2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x-(-c^2*b*(a*c^2-d^2))^(1/2)/c
^2/b)^2*b+2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+
d^2/c^2)^(1/2))/(x-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))-1/2*d/((-a*b)^(1/2)*c^2+(-
c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b*((x+(-
c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*
(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)*a*c^2+1/2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*
c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b*((x+(-c^2*b*(a*
c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^
2))^(1/2)/c^2/b)+d^2/c^2)^(1/2)*d^3+1/2*d/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))
^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b^(3/2)*(-c^2*b*(a*c^2-d^2
))^(1/2)*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b
)*b)/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1
/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))*a-1/2/((-a*b)^(1/2)
*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a*c^2-d^2))^(1/2))/b
^(3/2)*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*ln((-(-c^2*b*(a*c^2-d^2))^(1/2)/c^2+(x+(-c
^2*b*(a*c^2-d^2))^(1/2)/c^2/b)*b)/b^(1/2)+((x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^
2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^
2)^(1/2))*d^3+1/2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^
2-(-c^2*b*(a*c^2-d^2))^(1/2))/b*d^3/(d^2/c^2)^(1/2)*ln((2*d^2/c^2-2*(-c^2*b*(a*c
^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x+(-
c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*
(a*c^2-d^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))*
a-1/2/((-a*b)^(1/2)*c^2+(-c^2*b*(a*c^2-d^2))^(1/2))/((-a*b)^(1/2)*c^2-(-c^2*b*(a
*c^2-d^2))^(1/2))/b*d^5/c^2/(d^2/c^2)^(1/2)*ln((2*d^2/c^2-2*(-c^2*b*(a*c^2-d^2))
^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b)+2*(d^2/c^2)^(1/2)*((x+(-c^2*b*(a
*c^2-d^2))^(1/2)/c^2/b)^2*b-2*(-c^2*b*(a*c^2-d^2))^(1/2)/c^2*(x+(-c^2*b*(a*c^2-d
^2))^(1/2)/c^2/b)+d^2/c^2)^(1/2))/(x+(-c^2*b*(a*c^2-d^2))^(1/2)/c^2/b))-1/2*a/c/
b^2*ln(b*c^2*x^2+a*c^2-d^2)+1/2*x^2/b/c+1/2/b^2/c^3*d^2*ln(b*c^2*x^2+a*c^2-d^2)
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Maxima [A] time = 0.701929, size = 84, normalized size = 1.22 \[ \frac{\frac{{\left (b x^{2} + a\right )} c - 2 \, \sqrt{b x^{2} + a} d}{c^{2}} - \frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{2} + a} c + d\right )}{c^{3}}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b*c*x^2 + a*c + sqrt(b*x^2 + a)*d),x, algorithm="maxima")
[Out]
1/2*(((b*x^2 + a)*c - 2*sqrt(b*x^2 + a)*d)/c^2 - 2*(a*c^2 - d^2)*log(sqrt(b*x^2
+ a)*c + d)/c^3)/b^2
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Fricas [A] time = 0.346862, size = 217, normalized size = 3.14 \[ \frac{2 \, b c^{2} x^{2} - 4 \, \sqrt{b x^{2} + a} c d - 2 \,{\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{2} + a c^{2} - d^{2}\right ) -{\left (a c^{2} - d^{2}\right )} \log \left (-\frac{b c^{2} x^{2} + a c^{2} + 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right ) +{\left (a c^{2} - d^{2}\right )} \log \left (-\frac{b c^{2} x^{2} + a c^{2} - 2 \, \sqrt{b x^{2} + a} c d + d^{2}}{x^{2}}\right )}{4 \, b^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b*c*x^2 + a*c + sqrt(b*x^2 + a)*d),x, algorithm="fricas")
[Out]
1/4*(2*b*c^2*x^2 - 4*sqrt(b*x^2 + a)*c*d - 2*(a*c^2 - d^2)*log(b*c^2*x^2 + a*c^2
- d^2) - (a*c^2 - d^2)*log(-(b*c^2*x^2 + a*c^2 + 2*sqrt(b*x^2 + a)*c*d + d^2)/x
^2) + (a*c^2 - d^2)*log(-(b*c^2*x^2 + a*c^2 - 2*sqrt(b*x^2 + a)*c*d + d^2)/x^2))
/(b^2*c^3)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{a c + b c x^{2} + d \sqrt{a + b x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(a*c+b*c*x**2+d*(b*x**2+a)**(1/2)),x)
[Out]
Integral(x**3/(a*c + b*c*x**2 + d*sqrt(a + b*x**2)), x)
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GIAC/XCAS [A] time = 0.27769, size = 97, normalized size = 1.41 \[ -\frac{\frac{2 \,{\left (a c^{2} - d^{2}\right )}{\rm ln}\left ({\left | \sqrt{b x^{2} + a} c + d \right |}\right )}{b c^{3}} - \frac{{\left (b x^{2} + a\right )} b c - 2 \, \sqrt{b x^{2} + a} b d}{b^{2} c^{2}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b*c*x^2 + a*c + sqrt(b*x^2 + a)*d),x, algorithm="giac")
[Out]
-1/2*(2*(a*c^2 - d^2)*ln(abs(sqrt(b*x^2 + a)*c + d))/(b*c^3) - ((b*x^2 + a)*b*c
- 2*sqrt(b*x^2 + a)*b*d)/(b^2*c^2))/b