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

Optimal. Leaf size=158 \[ \frac{\left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^3 \sqrt{d}}-\frac{\sqrt{a} (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^3}+\frac{x \sqrt{c+d x^2} (5 b c-4 a d)}{8 b^2}+\frac{d x^3 \sqrt{c+d x^2}}{4 b} \]

[Out]

((5*b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(8*b^2) + (d*x^3*Sqrt[c + d*x^2])/(4*b) - (S
qrt[a]*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/
b^3 + ((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]]
)/(8*b^3*Sqrt[d])

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Rubi [A]  time = 0.652622, antiderivative size = 158, 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 \[ \frac{\left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{8 b^3 \sqrt{d}}-\frac{\sqrt{a} (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^3}+\frac{x \sqrt{c+d x^2} (5 b c-4 a d)}{8 b^2}+\frac{d x^3 \sqrt{c+d x^2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

((5*b*c - 4*a*d)*x*Sqrt[c + d*x^2])/(8*b^2) + (d*x^3*Sqrt[c + d*x^2])/(4*b) - (S
qrt[a]*(b*c - a*d)^(3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/
b^3 + ((3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]]
)/(8*b^3*Sqrt[d])

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Rubi in Sympy [A]  time = 84.2162, size = 146, normalized size = 0.92 \[ - \frac{\sqrt{a} \left (a d - b c\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{b^{3}} + \frac{d x^{3} \sqrt{c + d x^{2}}}{4 b} - \frac{x \sqrt{c + d x^{2}} \left (4 a d - 5 b c\right )}{8 b^{2}} + \frac{\left (8 a^{2} d^{2} - 12 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8 b^{3} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a)*(a*d - b*c)**(3/2)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/
b**3 + d*x**3*sqrt(c + d*x**2)/(4*b) - x*sqrt(c + d*x**2)*(4*a*d - 5*b*c)/(8*b**
2) + (8*a**2*d**2 - 12*a*b*c*d + 3*b**2*c**2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/
(8*b**3*sqrt(d))

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Mathematica [A]  time = 0.28518, size = 139, normalized size = 0.88 \[ \frac{\frac{\left (8 a^2 d^2-12 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}+b x \sqrt{c+d x^2} \left (-4 a d+5 b c+2 b d x^2\right )-8 \sqrt{a} (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*Sqrt[c + d*x^2]*(5*b*c - 4*a*d + 2*b*d*x^2) - 8*Sqrt[a]*(b*c - a*d)^(3/2)*A
rcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] + ((3*b^2*c^2 - 12*a*b*c*d
+ 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(8*b^3)

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Maple [B]  time = 0.018, size = 1973, normalized size = 12.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/b*x*(d*x^2+c)^(3/2)+3/8/b*c*x*(d*x^2+c)^(1/2)+3/8/b*c^2/d^(1/2)*ln(x*d^(1/2)
+(d*x^2+c)^(1/2))-1/6*a/(-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)^(3/2)-1/4*a/b^2*d*((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*a/b^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))*c+1/2*a^2/(-a*b)^(1/
2)/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)*d-1/2*a/(-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)*c+1/2*a^2/b^3*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))+1/2*a^3/(-a*b)^(1/2)/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^2-a^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)))*d*c+1/2*a/(-a*b)^(1/2)/b/(-(a*d-b*c)/b)^(1/2)*l
n((-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/(-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)^(3/2)-1/4*a/b^2*d*((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*a/b^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))*c-1/
2*a^2/(-a*b)^(1/2)/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)*d+1/2*a/(-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)*c+1/2*a^2/b^3*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))-1/2*a^3/(-a*b)^(1/2)/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^2+a^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)))*d*c-1/2*a/(-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)))*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)*x^2/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.786378, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*sqrt(-a*b*c + a^2*d)*(b*c - a*d)*sqrt(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*(
2*b^2*d*x^3 + (5*b^2*c - 4*a*b*d)*x)*sqrt(d*x^2 + c)*sqrt(d) - (3*b^2*c^2 - 12*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*sq
rt(d)), -1/8*(2*sqrt(-a*b*c + a^2*d)*(b*c - a*d)*sqrt(-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 + (5*b^2*c - 4*a*b*d)*x)*sqrt(d*x^2 + c)*sqrt(-d) - (3*b^2*c^2
- 12*a*b*c*d + 8*a^2*d^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)))/(b^3*sqrt(-d)), 1/
16*(8*sqrt(a*b*c - a^2*d)*(b*c - a*d)*sqrt(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*(2*b^2*d*x^3 + (5*b^2*c - 4*a*b
*d)*x)*sqrt(d*x^2 + c)*sqrt(d) + (3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*log(-2*sqr
t(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/(b^3*sqrt(d)), 1/8*(4*sqrt(a*b*c - a^
2*d)*(b*c - a*d)*sqrt(-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 + (5*b^2*c - 4*a*b*d)*x)*sqrt(d*x^2 + c)
*sqrt(-d) + (3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*arctan(sqrt(-d)*x/sqrt(d*x^2 +
c)))/(b^3*sqrt(-d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.243154, size = 267, normalized size = 1.69 \[ \frac{1}{8} \, \sqrt{d x^{2} + c}{\left (\frac{2 \, d x^{2}}{b} + \frac{5 \, b^{5} c d^{2} - 4 \, a b^{4} d^{3}}{b^{6} d^{2}}\right )} x + \frac{{\left (a b^{2} c^{2} \sqrt{d} - 2 \, a^{2} b c d^{\frac{3}{2}} + a^{3} d^{\frac{5}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b^{3}} - \frac{{\left (3 \, b^{2} c^{2} - 12 \, a b c d + 8 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{16 \, b^{3} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(d*x^2 + c)*(2*d*x^2/b + (5*b^5*c*d^2 - 4*a*b^4*d^3)/(b^6*d^2))*x + (a*b
^2*c^2*sqrt(d) - 2*a^2*b*c*d^(3/2) + a^3*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^3) - 1/16*(3*b^2*c^2 - 12*a*b*c*d + 8*a^2*d^2)*ln((sqrt(d)*x - sqrt(d*x^2 + c
))^2)/(b^3*sqrt(d))