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

Optimal. Leaf size=169 \[ \frac{(2 b c-a d) \left (5 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}+\frac{d x \sqrt{a+b x^2} \left (15 a^2 d^2-44 a b c d+44 b^2 c^2\right )}{48 b^3}+\frac{5 d x \sqrt{a+b x^2} \left (c+d x^2\right ) (2 b c-a d)}{24 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 b} \]

[Out]

(d*(44*b^2*c^2 - 44*a*b*c*d + 15*a^2*d^2)*x*Sqrt[a + b*x^2])/(48*b^3) + (5*d*(2*
b*c - a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(24*b^2) + (d*x*Sqrt[a + b*x^2]*(c + d
*x^2)^2)/(6*b) + ((2*b*c - a*d)*(8*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqr
t[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))

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Rubi [A]  time = 0.334012, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{(2 b c-a d) \left (5 a^2 d^2-8 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{7/2}}+\frac{d x \sqrt{a+b x^2} \left (15 a^2 d^2-44 a b c d+44 b^2 c^2\right )}{48 b^3}+\frac{5 d x \sqrt{a+b x^2} \left (c+d x^2\right ) (2 b c-a d)}{24 b^2}+\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

(d*(44*b^2*c^2 - 44*a*b*c*d + 15*a^2*d^2)*x*Sqrt[a + b*x^2])/(48*b^3) + (5*d*(2*
b*c - a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(24*b^2) + (d*x*Sqrt[a + b*x^2]*(c + d
*x^2)^2)/(6*b) + ((2*b*c - a*d)*(8*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqr
t[b]*x)/Sqrt[a + b*x^2]])/(16*b^(7/2))

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Rubi in Sympy [A]  time = 42.7041, size = 163, normalized size = 0.96 \[ \frac{d x \sqrt{a + b x^{2}} \left (c + d x^{2}\right )^{2}}{6 b} - \frac{5 d x \sqrt{a + b x^{2}} \left (c + d x^{2}\right ) \left (a d - 2 b c\right )}{24 b^{2}} + \frac{d x \sqrt{a + b x^{2}} \left (15 a^{2} d^{2} - 44 a b c d + 44 b^{2} c^{2}\right )}{48 b^{3}} - \frac{\left (a d - 2 b c\right ) \left (5 a^{2} d^{2} - 8 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x*sqrt(a + b*x**2)*(c + d*x**2)**2/(6*b) - 5*d*x*sqrt(a + b*x**2)*(c + d*x**2)
*(a*d - 2*b*c)/(24*b**2) + d*x*sqrt(a + b*x**2)*(15*a**2*d**2 - 44*a*b*c*d + 44*
b**2*c**2)/(48*b**3) - (a*d - 2*b*c)*(5*a**2*d**2 - 8*a*b*c*d + 8*b**2*c**2)*ata
nh(sqrt(b)*x/sqrt(a + b*x**2))/(16*b**(7/2))

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Mathematica [A]  time = 0.156407, size = 140, normalized size = 0.83 \[ \frac{\sqrt{b} d x \sqrt{a+b x^2} \left (15 a^2 d^2-2 a b d \left (27 c+5 d x^2\right )+4 b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+3 \left (-5 a^3 d^3+18 a^2 b c d^2-24 a b^2 c^2 d+16 b^3 c^3\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{48 b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*d*x*Sqrt[a + b*x^2]*(15*a^2*d^2 - 2*a*b*d*(27*c + 5*d*x^2) + 4*b^2*(18*
c^2 + 9*c*d*x^2 + 2*d^2*x^4)) + 3*(16*b^3*c^3 - 24*a*b^2*c^2*d + 18*a^2*b*c*d^2
- 5*a^3*d^3)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(48*b^(7/2))

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Maple [A]  time = 0.017, size = 228, normalized size = 1.4 \[{{c}^{3}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{d}^{3}{x}^{5}}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,a{d}^{3}{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{d}^{3}{a}^{2}x}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{a}^{3}{d}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{3\,c{d}^{2}{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{9\,c{d}^{2}ax}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{9\,{a}^{2}c{d}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{3\,{c}^{2}dx}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,a{c}^{2}d}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^3*ln(x*b^(1/2)+(b*x^2+a)^(1/2))/b^(1/2)+1/6*d^3*x^5/b*(b*x^2+a)^(1/2)-5/24*d^3
*a/b^2*x^3*(b*x^2+a)^(1/2)+5/16*d^3*a^2/b^3*x*(b*x^2+a)^(1/2)-5/16*d^3*a^3/b^(7/
2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+3/4*c*d^2*x^3/b*(b*x^2+a)^(1/2)-9/8*c*d^2*a/b^2
*x*(b*x^2+a)^(1/2)+9/8*c*d^2*a^2/b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+3/2*c^2*d
*x/b*(b*x^2+a)^(1/2)-3/2*c^2*d*a/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270199, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, b^{2} d^{3} x^{5} + 2 \,{\left (18 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )} x^{3} + 3 \,{\left (24 \, b^{2} c^{2} d - 18 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 3 \,{\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{96 \, b^{\frac{7}{2}}}, \frac{{\left (8 \, b^{2} d^{3} x^{5} + 2 \,{\left (18 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )} x^{3} + 3 \,{\left (24 \, b^{2} c^{2} d - 18 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{48 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(2*(8*b^2*d^3*x^5 + 2*(18*b^2*c*d^2 - 5*a*b*d^3)*x^3 + 3*(24*b^2*c^2*d - 1
8*a*b*c*d^2 + 5*a^2*d^3)*x)*sqrt(b*x^2 + a)*sqrt(b) - 3*(16*b^3*c^3 - 24*a*b^2*c
^2*d + 18*a^2*b*c*d^2 - 5*a^3*d^3)*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqr
t(b)))/b^(7/2), 1/48*((8*b^2*d^3*x^5 + 2*(18*b^2*c*d^2 - 5*a*b*d^3)*x^3 + 3*(24*
b^2*c^2*d - 18*a*b*c*d^2 + 5*a^2*d^3)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 3*(16*b^3*c^
3 - 24*a*b^2*c^2*d + 18*a^2*b*c*d^2 - 5*a^3*d^3)*arctan(sqrt(-b)*x/sqrt(b*x^2 +
a)))/(sqrt(-b)*b^3)]

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Sympy [A]  time = 39.8273, size = 400, normalized size = 2.37 \[ \frac{5 a^{\frac{5}{2}} d^{3} x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{9 a^{\frac{3}{2}} c d^{2} x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{3}{2}} d^{3} x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 \sqrt{a} c^{2} d x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{3 \sqrt{a} c d^{2} x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} d^{3} x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{5 a^{3} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{9 a^{2} c d^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} - \frac{3 a c^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} + c^{3} \left (\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (x \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (x \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}\right ) + \frac{3 c d^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{d^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(5/2)*d**3*x/(16*b**3*sqrt(1 + b*x**2/a)) - 9*a**(3/2)*c*d**2*x/(8*b**2*sqr
t(1 + b*x**2/a)) + 5*a**(3/2)*d**3*x**3/(48*b**2*sqrt(1 + b*x**2/a)) + 3*sqrt(a)
*c**2*d*x*sqrt(1 + b*x**2/a)/(2*b) - 3*sqrt(a)*c*d**2*x**3/(8*b*sqrt(1 + b*x**2/
a)) - sqrt(a)*d**3*x**5/(24*b*sqrt(1 + b*x**2/a)) - 5*a**3*d**3*asinh(sqrt(b)*x/
sqrt(a))/(16*b**(7/2)) + 9*a**2*c*d**2*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) - 3
*a*c**2*d*asinh(sqrt(b)*x/sqrt(a))/(2*b**(3/2)) + c**3*Piecewise((sqrt(-a/b)*asi
n(x*sqrt(-b/a))/sqrt(a), (a > 0) & (b < 0)), (sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(
a), (a > 0) & (b > 0)), (sqrt(-a/b)*acosh(x*sqrt(-b/a))/sqrt(-a), (b > 0) & (a <
0))) + 3*c*d**2*x**5/(4*sqrt(a)*sqrt(1 + b*x**2/a)) + d**3*x**7/(6*sqrt(a)*sqrt
(1 + b*x**2/a))

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GIAC/XCAS [A]  time = 0.242198, size = 203, normalized size = 1.2 \[ \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, d^{3} x^{2}}{b} + \frac{18 \, b^{4} c d^{2} - 5 \, a b^{3} d^{3}}{b^{5}}\right )} x^{2} + \frac{3 \,{\left (24 \, b^{4} c^{2} d - 18 \, a b^{3} c d^{2} + 5 \, a^{2} b^{2} d^{3}\right )}}{b^{5}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (16 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*d^3*x^2/b + (18*b^4*c*d^2 - 5*a*b^3*d^3)/b^5)*x^2 + 3*(24*b^4*c^2*d -
18*a*b^3*c*d^2 + 5*a^2*b^2*d^3)/b^5)*sqrt(b*x^2 + a)*x - 1/16*(16*b^3*c^3 - 24*
a*b^2*c^2*d + 18*a^2*b*c*d^2 - 5*a^3*d^3)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/
b^(7/2)