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

Optimal. Leaf size=169 \[ \frac{3 d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{7/2}}-\frac{d x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{8 a b^3}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) (4 b c-5 a d)}{4 a b^2}+\frac{x \left (c+d x^2\right )^2 (b c-a d)}{a b \sqrt{a+b x^2}} \]

[Out]

-(d*(2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(8*a*b^3) - (d*(4*b*c - 5
*a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(4*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^2)/(
a*b*Sqrt[a + b*x^2]) + (3*d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b
]*x)/Sqrt[a + b*x^2]])/(8*b^(7/2))

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Rubi [A]  time = 0.433848, 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 \[ \frac{3 d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{7/2}}-\frac{d x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{8 a b^3}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) (4 b c-5 a d)}{4 a b^2}+\frac{x \left (c+d x^2\right )^2 (b c-a d)}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*Sqrt[a + b*x^2])/(8*a*b^3) - (d*(4*b*c - 5
*a*d)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(4*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^2)/(
a*b*Sqrt[a + b*x^2]) + (3*d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b
]*x)/Sqrt[a + b*x^2]])/(8*b^(7/2))

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Rubi in Sympy [A]  time = 44.9421, size = 163, normalized size = 0.96 \[ \frac{3 d \left (5 a^{2} d^{2} - 12 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{7}{2}}} - \frac{x \left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{a b \sqrt{a + b x^{2}}} + \frac{d^{2} x \sqrt{a + b x^{2}} \left (a c + x^{2} \left (5 a d - 4 b c\right )\right )}{4 a b^{2}} - \frac{d x \sqrt{a + b x^{2}} \left (3 a d - 2 b c\right ) \left (5 a d - 8 b c\right )}{8 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*d*(5*a**2*d**2 - 12*a*b*c*d + 8*b**2*c**2)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(
8*b**(7/2)) - x*(c + d*x**2)**2*(a*d - b*c)/(a*b*sqrt(a + b*x**2)) + d**2*x*sqrt
(a + b*x**2)*(a*c + x**2*(5*a*d - 4*b*c))/(4*a*b**2) - d*x*sqrt(a + b*x**2)*(3*a
*d - 2*b*c)*(5*a*d - 8*b*c)/(8*a*b**3)

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Mathematica [A]  time = 0.155896, size = 122, normalized size = 0.72 \[ \frac{3 d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{7/2}}+\frac{x \sqrt{a+b x^2} \left (d^2 (12 b c-7 a d)+\frac{8 (b c-a d)^3}{a \left (a+b x^2\right )}+2 b d^3 x^2\right )}{8 b^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + b*x^2]*(d^2*(12*b*c - 7*a*d) + 2*b*d^3*x^2 + (8*(b*c - a*d)^3)/(a*(a
+ b*x^2))))/(8*b^3) + (3*d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*Log[b*x + Sqrt[
b]*Sqrt[a + b*x^2]])/(8*b^(7/2))

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Maple [A]  time = 0.013, size = 219, normalized size = 1.3 \[{\frac{{c}^{3}x}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{{d}^{3}{x}^{5}}{4\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,a{d}^{3}{x}^{3}}{8\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,{a}^{2}{d}^{3}x}{8\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,{a}^{2}{d}^{3}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{3\,c{d}^{2}{x}^{3}}{2\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{9\,ac{d}^{2}x}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{9\,ac{d}^{2}}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}-3\,{\frac{{c}^{2}dx}{b\sqrt{b{x}^{2}+a}}}+3\,{\frac{{c}^{2}d\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^3*x/a/(b*x^2+a)^(1/2)+1/4*d^3*x^5/b/(b*x^2+a)^(1/2)-5/8*d^3*a/b^2*x^3/(b*x^2+a
)^(1/2)-15/8*d^3*a^2/b^3*x/(b*x^2+a)^(1/2)+15/8*d^3*a^2/b^(7/2)*ln(x*b^(1/2)+(b*
x^2+a)^(1/2))+3/2*c*d^2*x^3/b/(b*x^2+a)^(1/2)+9/2*c*d^2*a/b^2*x/(b*x^2+a)^(1/2)-
9/2*c*d^2*a/b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-3*c^2*d*x/b/(b*x^2+a)^(1/2)+3*
c^2*d/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/(b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278341, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, a b^{2} d^{3} x^{5} +{\left (12 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x^{3} +{\left (8 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 36 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} + 3 \,{\left (8 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} +{\left (8 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{16 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{b}}, \frac{{\left (2 \, a b^{2} d^{3} x^{5} +{\left (12 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x^{3} +{\left (8 \, b^{3} c^{3} - 24 \, a b^{2} c^{2} d + 36 \, a^{2} b c d^{2} - 15 \, a^{3} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 3 \,{\left (8 \, a^{2} b^{2} c^{2} d - 12 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} +{\left (8 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{8 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(2*(2*a*b^2*d^3*x^5 + (12*a*b^2*c*d^2 - 5*a^2*b*d^3)*x^3 + (8*b^3*c^3 - 24
*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 15*a^3*d^3)*x)*sqrt(b*x^2 + a)*sqrt(b) + 3*(8*a^
2*b^2*c^2*d - 12*a^3*b*c*d^2 + 5*a^4*d^3 + (8*a*b^3*c^2*d - 12*a^2*b^2*c*d^2 + 5
*a^3*b*d^3)*x^2)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((a*b^4*x^
2 + a^2*b^3)*sqrt(b)), 1/8*((2*a*b^2*d^3*x^5 + (12*a*b^2*c*d^2 - 5*a^2*b*d^3)*x^
3 + (8*b^3*c^3 - 24*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 15*a^3*d^3)*x)*sqrt(b*x^2 + a
)*sqrt(-b) + 3*(8*a^2*b^2*c^2*d - 12*a^3*b*c*d^2 + 5*a^4*d^3 + (8*a*b^3*c^2*d -
12*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^2)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/((a*b^4
*x^2 + a^2*b^3)*sqrt(-b))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.257793, size = 212, normalized size = 1.25 \[ \frac{{\left ({\left (\frac{2 \, d^{3} x^{2}}{b} + \frac{12 \, a b^{4} c d^{2} - 5 \, a^{2} b^{3} d^{3}}{a b^{5}}\right )} x^{2} + \frac{8 \, b^{5} c^{3} - 24 \, a b^{4} c^{2} d + 36 \, a^{2} b^{3} c d^{2} - 15 \, a^{3} b^{2} d^{3}}{a b^{5}}\right )} x}{8 \, \sqrt{b x^{2} + a}} - \frac{3 \,{\left (8 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((2*d^3*x^2/b + (12*a*b^4*c*d^2 - 5*a^2*b^3*d^3)/(a*b^5))*x^2 + (8*b^5*c^3 -
24*a*b^4*c^2*d + 36*a^2*b^3*c*d^2 - 15*a^3*b^2*d^3)/(a*b^5))*x/sqrt(b*x^2 + a)
- 3/8*(8*b^2*c^2*d - 12*a*b*c*d^2 + 5*a^2*d^3)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 +
a)))/b^(7/2)