3.689 \(\int \frac{\left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx\)

Optimal. Leaf size=113 \[ \frac{(b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} b^2}+\frac{\sqrt{d} (3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^2}+\frac{d x \sqrt{c+d x^2}}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.251784, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{(b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a} b^2}+\frac{\sqrt{d} (3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^2}+\frac{d x \sqrt{c+d x^2}}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.3367, size = 102, normalized size = 0.9 \[ \frac{d x \sqrt{c + d x^{2}}}{2 b} - \frac{\sqrt{d} \left (2 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 b^{2}} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{\sqrt{a} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.284517, size = 110, normalized size = 0.97 \[ \frac{\sqrt{d} (3 b c-2 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\frac{2 (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{a}}+b d x \sqrt{c+d x^2}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.016, size = 1875, normalized size = 16.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6/(-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)^(3/2)+1/4*d/b*((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+3/4/b*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))*c-1/2/(-a*b)^(1/2)/b*((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)*a*d+1/2/(-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
)*c-1/2/b^2*d^(3/2)*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))*a
-1/2/(-a*b)^(1/2)/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)))*a^2*d^2
+1/(-a*b)^(1/2)/b/(-(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)))*a*d*c-1/2/(
-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^2-1/6/(-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)^(3/2)+1/4*d/b*((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+3/4/b*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))*c+1/2/(-a*b)^(1/2)/b*((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)*a*d-1/2/(-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)*c-1/2/b^2*
d^(3/2)*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))*a+1/2/(-a*b)
^(1/2)/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)))*a^2*d^2-1/(-a*b)^(
1/2)/b/(-(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)))*a*d*c+1/2/(-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^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((d*x^2 + c)^(3/2)/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.424702, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{d x^{2} + c} b d x -{\left (3 \, b c - 2 \, a d\right )} \sqrt{d} \log \left (-2 \, d x^{2} + 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) -{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \,{\left (a^{2} c x -{\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, b^{2}}, \frac{2 \, \sqrt{d x^{2} + c} b d x + 2 \,{\left (3 \, b c - 2 \, a d\right )} \sqrt{-d} \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) -{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{a}} \log \left (\frac{{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \,{\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \,{\left (a^{2} c x -{\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt{d x^{2} + c} \sqrt{-\frac{b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, b^{2}}, \frac{2 \, \sqrt{d x^{2} + c} b d x - 2 \,{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c} a x \sqrt{\frac{b c - a d}{a}}}\right ) -{\left (3 \, b c - 2 \, a d\right )} \sqrt{d} \log \left (-2 \, d x^{2} + 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right )}{4 \, b^{2}}, \frac{\sqrt{d x^{2} + c} b d x +{\left (3 \, b c - 2 \, a d\right )} \sqrt{-d} \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) -{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{a}} \arctan \left (-\frac{{\left (b c - 2 \, a d\right )} x^{2} - a c}{2 \, \sqrt{d x^{2} + c} a x \sqrt{\frac{b c - a d}{a}}}\right )}{2 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.251766, size = 205, normalized size = 1.81 \[ \frac{\sqrt{d x^{2} + c} d x}{2 \, b} - \frac{{\left (3 \, b c \sqrt{d} - 2 \, a d^{\frac{3}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, b^{2}} - \frac{{\left (b^{2} c^{2} \sqrt{d} - 2 \, a b c d^{\frac{3}{2}} + a^{2} d^{\frac{5}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(d*x^2 + c)*d*x/b - 1/4*(3*b*c*sqrt(d) - 2*a*d^(3/2))*ln((sqrt(d)*x - sq
rt(d*x^2 + c))^2)/b^2 - (b^2*c^2*sqrt(d) - 2*a*b*c*d^(3/2) + a^2*d^(5/2))*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^2)