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

Optimal. Leaf size=157 \[ \frac{a^{3/2} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^3}-\frac{\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^3 d^{3/2}}+\frac{x \sqrt{c+d x^2} (b c-4 a d)}{8 b^2 d}+\frac{x^3 \sqrt{c+d x^2}}{4 b} \]

[Out]

((b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(8*b^2*d) + (x^3*Sqrt[c + d*x^2])/(4*b) + (a^(
3/2)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^3
- ((b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^
3*d^(3/2))

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Rubi [A]  time = 0.607539, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{a^{3/2} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^3}-\frac{\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^3 d^{3/2}}+\frac{x \sqrt{c+d x^2} (b c-4 a d)}{8 b^2 d}+\frac{x^3 \sqrt{c+d x^2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(8*b^2*d) + (x^3*Sqrt[c + d*x^2])/(4*b) + (a^(
3/2)*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^3
- ((b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(8*b^
3*d^(3/2))

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Rubi in Sympy [A]  time = 66.7809, size = 143, normalized size = 0.91 \[ - \frac{a^{\frac{3}{2}} \sqrt{a d - b c} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{b^{3}} + \frac{x^{3} \sqrt{c + d x^{2}}}{4 b} - \frac{x \sqrt{c + d x^{2}} \left (4 a d - b c\right )}{8 b^{2} d} + \frac{\left (8 a^{2} d^{2} - 4 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 b^{3} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(d*x**2+c)**(1/2)/(b*x**2+a),x)

[Out]

-a**(3/2)*sqrt(a*d - b*c)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/b*
*3 + x**3*sqrt(c + d*x**2)/(4*b) - x*sqrt(c + d*x**2)*(4*a*d - b*c)/(8*b**2*d) +
(8*a**2*d**2 - 4*a*b*c*d - b**2*c**2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/(8*b**3
*d**(3/2))

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Mathematica [A]  time = 0.359307, size = 148, normalized size = 0.94 \[ \frac{8 a^{3/2} d^{3/2} \sqrt{b c-a d} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )-\left (-8 a^2 d^2+4 a b c d+b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+b \sqrt{d} x \sqrt{c+d x^2} \left (-4 a d+b c+2 b d x^2\right )}{8 b^3 d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[d]*x*Sqrt[c + d*x^2]*(b*c - 4*a*d + 2*b*d*x^2) + 8*a^(3/2)*d^(3/2)*Sqrt[
b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] - (b^2*c^2 + 4*
a*b*c*d - 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(8*b^3*d^(3/2))

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Maple [B]  time = 0.022, size = 1088, normalized size = 6.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(d*x^2+c)^(1/2)/(b*x^2+a),x)

[Out]

1/4/b*x*(d*x^2+c)^(3/2)/d-1/8/b*c/d*x*(d*x^2+c)^(1/2)-1/8/b*c^2/d^(3/2)*ln(x*d^(
1/2)+(d*x^2+c)^(1/2))-1/2/b^2*a*x*(d*x^2+c)^(1/2)-1/2/b^2*a*c/d^(1/2)*ln(x*d^(1/
2)+(d*x^2+c)^(1/2))+1/2/b^2*a^2/(-a*b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b
)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/2/b^3*a^2*d^(1/2)*ln((d*(-a*
b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^
(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+1/2/b^3*a^3/(-a*b)^(1/2)/(-(a*d
-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a
*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2
))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d-1/2/b^2*a^2/(-a*b)^(1/2)/(-(a*d-b
*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d
-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))
-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*c-1/2/b^2*a^2/(-a*b)^(1/2)*((x+1/b*(-
a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/2/b
^3*a^2*d^(1/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a
*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/2/b
^3*a^3/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(
x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^
(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*d+1/2/b^2
*a^2/(-a*b)^(1/2)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+
1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1
/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*c

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.412373, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*sqrt(-a*b*c + a^2*d)*a*d^(3/2)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x
^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqr
t(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*(2*b^2*d*x^3
+ (b^2*c - 4*a*b*d)*x)*sqrt(d*x^2 + c)*sqrt(d) - (b^2*c^2 + 4*a*b*c*d - 8*a^2*d
^2)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/(b^3*d^(3/2)), 1/8*(2*s
qrt(-a*b*c + a^2*d)*a*sqrt(-d)*d*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^4 + a^
2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*
c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + (2*b^2*d*x^3 + (b^2*c
- 4*a*b*d)*x)*sqrt(d*x^2 + c)*sqrt(-d) - (b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*arct
an(sqrt(-d)*x/sqrt(d*x^2 + c)))/(b^3*sqrt(-d)*d), -1/16*(8*sqrt(a*b*c - a^2*d)*a
*d^(3/2)*arctan(-1/2*((b*c - 2*a*d)*x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 +
c)*x)) - 2*(2*b^2*d*x^3 + (b^2*c - 4*a*b*d)*x)*sqrt(d*x^2 + c)*sqrt(d) + (b^2*c
^2 + 4*a*b*c*d - 8*a^2*d^2)*log(-2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))
/(b^3*d^(3/2)), -1/8*(4*sqrt(a*b*c - a^2*d)*a*sqrt(-d)*d*arctan(-1/2*((b*c - 2*a
*d)*x^2 - a*c)/(sqrt(a*b*c - a^2*d)*sqrt(d*x^2 + c)*x)) - (2*b^2*d*x^3 + (b^2*c
- 4*a*b*d)*x)*sqrt(d*x^2 + c)*sqrt(-d) + (b^2*c^2 + 4*a*b*c*d - 8*a^2*d^2)*arcta
n(sqrt(-d)*x/sqrt(d*x^2 + c)))/(b^3*sqrt(-d)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \sqrt{c + d x^{2}}}{a + b x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(d*x**2+c)**(1/2)/(b*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.242496, size = 254, normalized size = 1.62 \[ \frac{1}{8} \, \sqrt{d x^{2} + c} x{\left (\frac{2 \, x^{2}}{b} + \frac{b^{5} c d - 4 \, a b^{4} d^{2}}{b^{6} d^{2}}\right )} - \frac{{\left (a^{2} b c \sqrt{d} - a^{3} d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b^{3}} + \frac{{\left (b^{2} c^{2} \sqrt{d} + 4 \, a b c d^{\frac{3}{2}} - 8 \, a^{2} d^{\frac{5}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{16 \, b^{3} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(d*x^2 + c)*x*(2*x^2/b + (b^5*c*d - 4*a*b^4*d^2)/(b^6*d^2)) - (a^2*b*c*s
qrt(d) - a^3*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*
d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*b^3) + 1/16*(b^2*c^2*sqrt(d
) + 4*a*b*c*d^(3/2) - 8*a^2*d^(5/2))*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/(b^3*d^
2)