3.753 \(\int \frac{\left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=174 \[ \frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac{d^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^3}-\frac{d x \sqrt{c+d x^2} (b c-2 a d)}{2 a b^2}+\frac{x \left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]

[Out]

-(d*(b*c - 2*a*d)*x*Sqrt[c + d*x^2])/(2*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^(3/2
))/(2*a*b*(a + b*x^2)) + ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*ArcTan[(Sqrt[b*c - a*d
]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*b^3) + (d^(3/2)*(5*b*c - 4*a*d)*ArcT
anh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^3)

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Rubi [A]  time = 0.527996, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 a^{3/2} b^3}+\frac{d^{3/2} (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^3}-\frac{d x \sqrt{c+d x^2} (b c-2 a d)}{2 a b^2}+\frac{x \left (c+d x^2\right )^{3/2} (b c-a d)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^(5/2)/(a + b*x^2)^2,x]

[Out]

-(d*(b*c - 2*a*d)*x*Sqrt[c + d*x^2])/(2*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^(3/2
))/(2*a*b*(a + b*x^2)) + ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*ArcTan[(Sqrt[b*c - a*d
]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*a^(3/2)*b^3) + (d^(3/2)*(5*b*c - 4*a*d)*ArcT
anh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*b^3)

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Rubi in Sympy [A]  time = 74.451, size = 155, normalized size = 0.89 \[ - \frac{d^{\frac{3}{2}} \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{2 b^{3}} - \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{2 a b \left (a + b x^{2}\right )} + \frac{d x \sqrt{c + d x^{2}} \left (2 a d - b c\right )}{2 a b^{2}} + \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 a^{\frac{3}{2}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**(5/2)/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.325655, size = 143, normalized size = 0.82 \[ \frac{\frac{(b c-a d)^{3/2} (4 a d+b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{a^{3/2}}+d^{3/2} (5 b c-4 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+b x \sqrt{c+d x^2} \left (\frac{(b c-a d)^2}{a \left (a+b x^2\right )}+d^2\right )}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^(5/2)/(a + b*x^2)^2,x]

[Out]

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

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Maple [B]  time = 0.025, size = 7451, normalized size = 42.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^(5/2)/(b*x^2+a)^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(5/2)/(b*x^2 + a)^2, x)

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Fricas [A]  time = 0.985829, 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)^(5/2)/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c + d x^{2}\right )^{\frac{5}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**(5/2)/(b*x**2+a)**2,x)

[Out]

Integral((c + d*x**2)**(5/2)/(a + b*x**2)**2, x)

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GIAC/XCAS [A]  time = 0.558972, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

sage0*x