3.677 \(\int \frac{x^3 \sqrt{c+d x^2}}{a+b x^2} \, dx\)

Optimal. Leaf size=88 \[ \frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-\frac{a \sqrt{c+d x^2}}{b^2}+\frac{\left (c+d x^2\right )^{3/2}}{3 b d} \]

[Out]

-((a*Sqrt[c + d*x^2])/b^2) + (c + d*x^2)^(3/2)/(3*b*d) + (a*Sqrt[b*c - a*d]*ArcT
anh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(5/2)

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Rubi [A]  time = 0.235172, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2}}-\frac{a \sqrt{c+d x^2}}{b^2}+\frac{\left (c+d x^2\right )^{3/2}}{3 b d} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*Sqrt[c + d*x^2])/(a + b*x^2),x]

[Out]

-((a*Sqrt[c + d*x^2])/b^2) + (c + d*x^2)^(3/2)/(3*b*d) + (a*Sqrt[b*c - a*d]*ArcT
anh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(5/2)

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Rubi in Sympy [A]  time = 26.7021, size = 73, normalized size = 0.83 \[ - \frac{a \sqrt{c + d x^{2}}}{b^{2}} + \frac{a \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{5}{2}}} + \frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(d*x**2+c)**(1/2)/(b*x**2+a),x)

[Out]

-a*sqrt(c + d*x**2)/b**2 + a*sqrt(a*d - b*c)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(
a*d - b*c))/b**(5/2) + (c + d*x**2)**(3/2)/(3*b*d)

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Mathematica [A]  time = 0.188574, size = 85, normalized size = 0.97 \[ \frac{a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{5/2}}+\frac{\sqrt{c+d x^2} \left (b \left (c+d x^2\right )-3 a d\right )}{3 b^2 d} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*Sqrt[c + d*x^2])/(a + b*x^2),x]

[Out]

(Sqrt[c + d*x^2]*(-3*a*d + b*(c + d*x^2)))/(3*b^2*d) + (a*Sqrt[b*c - a*d]*ArcTan
h[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(5/2)

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Maple [B]  time = 0.02, size = 963, normalized size = 10.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(d*x^2+c)^(1/2)/(b*x^2+a),x)

[Out]

1/3*(d*x^2+c)^(3/2)/b/d-1/2*a/b^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*a/b^3*d^(1/2)*(-a*b)^(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*a^2/b^3/(-(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*a/b^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*a/b^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*a/b^3*d^(1/2)*(-a*b)^(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*a^2/b^3/(-(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*a/b^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^3/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262815, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a d \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left (b d x^{2} + b c - 3 \, a d\right )} \sqrt{d x^{2} + c}}{12 \, b^{2} d}, \frac{3 \, a d \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) + 2 \,{\left (b d x^{2} + b c - 3 \, a d\right )} \sqrt{d x^{2} + c}}{6 \, b^{2} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)*x^3/(b*x^2 + a),x, algorithm="fricas")

[Out]

[1/12*(3*a*d*sqrt((b*c - a*d)/b)*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*
d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2
+ c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(b*d*x^2 + b*c - 3*a
*d)*sqrt(d*x^2 + c))/(b^2*d), 1/6*(3*a*d*sqrt(-(b*c - a*d)/b)*arctan(1/2*(b*d*x^
2 + 2*b*c - a*d)/(sqrt(d*x^2 + c)*b*sqrt(-(b*c - a*d)/b))) + 2*(b*d*x^2 + b*c -
3*a*d)*sqrt(d*x^2 + c))/(b^2*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \sqrt{c + d x^{2}}}{a + b x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(d*x**2+c)**(1/2)/(b*x**2+a),x)

[Out]

Integral(x**3*sqrt(c + d*x**2)/(a + b*x**2), x)

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GIAC/XCAS [A]  time = 0.227211, size = 130, normalized size = 1.48 \[ -\frac{\frac{3 \,{\left (a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} - \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} - 3 \, \sqrt{d x^{2} + c} a b d}{b^{3}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)*x^3/(b*x^2 + a),x, algorithm="giac")

[Out]

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