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

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

[Out]

(b*x*Sqrt[a + b*x^2])/(2*d) - (Sqrt[b]*(2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[
a + b*x^2]])/(2*d^2) + ((b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*S
qrt[a + b*x^2])])/(Sqrt[c]*d^2)

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Rubi [A]  time = 0.289931, 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 \[ \frac{(b c-a d)^{3/2} \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} d^2}-\frac{\sqrt{b} (2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 d^2}+\frac{b x \sqrt{a+b x^2}}{2 d} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*Sqrt[a + b*x^2])/(2*d) - (Sqrt[b]*(2*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[
a + b*x^2]])/(2*d^2) + ((b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*S
qrt[a + b*x^2])])/(Sqrt[c]*d^2)

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Rubi in Sympy [A]  time = 39.9871, size = 102, normalized size = 0.9 \[ \frac{\sqrt{b} \left (3 a d - 2 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 d^{2}} + \frac{b x \sqrt{a + b x^{2}}}{2 d} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{\sqrt{c} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.318346, size = 110, normalized size = 0.97 \[ \frac{\sqrt{b} (3 a d-2 b c) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\frac{2 (a d-b c)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c}}+b d x \sqrt{a+b x^2}}{2 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.025, size = 1845, normalized size = 16.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.411704, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b x^{2} + a} b d x -{\left (2 \, b c - 3 \, a d\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) -{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{c}} \log \left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \,{\left (a c^{2} x +{\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{\frac{b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d^{2}}, \frac{2 \, \sqrt{b x^{2} + a} b d x - 2 \,{\left (2 \, b c - 3 \, a d\right )} \sqrt{-b} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left (b c - a d\right )} \sqrt{\frac{b c - a d}{c}} \log \left (\frac{{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \,{\left (a c^{2} x +{\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt{b x^{2} + a} \sqrt{\frac{b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d^{2}}, \frac{2 \, \sqrt{b x^{2} + a} b d x - 2 \,{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{c}} \arctan \left (-\frac{{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \, \sqrt{b x^{2} + a} c x \sqrt{-\frac{b c - a d}{c}}}\right ) -{\left (2 \, b c - 3 \, a d\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right )}{4 \, d^{2}}, \frac{\sqrt{b x^{2} + a} b d x -{\left (2 \, b c - 3 \, a d\right )} \sqrt{-b} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left (b c - a d\right )} \sqrt{-\frac{b c - a d}{c}} \arctan \left (-\frac{{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \, \sqrt{b x^{2} + a} c x \sqrt{-\frac{b c - a d}{c}}}\right )}{2 \, d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.283003, size = 205, normalized size = 1.81 \[ \frac{\sqrt{b x^{2} + a} b x}{2 \, d} + \frac{{\left (2 \, b^{\frac{3}{2}} c - 3 \, a \sqrt{b} d\right )}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right )}{4 \, d^{2}} - \frac{{\left (b^{\frac{5}{2}} c^{2} - 2 \, a b^{\frac{3}{2}} c d + a^{2} \sqrt{b} d^{2}\right )} \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{\sqrt{-b^{2} c^{2} + a b c d} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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