3.68 \(\int \frac{\left (a+b x^2\right )^{5/2}}{\left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=194 \[ -\frac{\sqrt{b c-a d} \left (3 a^2 d^2+4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} d^3}+\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d^3}-\frac{x \sqrt{a+b x^2} (b c-a d) (3 a d+4 b c)}{8 c^2 d^2 \left (c+d x^2\right )}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.482747, antiderivative size = 194, 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 \[ -\frac{\sqrt{b c-a d} \left (3 a^2 d^2+4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{8 c^{5/2} d^3}+\frac{b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{d^3}-\frac{x \sqrt{a+b x^2} (b c-a d) (3 a d+4 b c)}{8 c^2 d^2 \left (c+d x^2\right )}-\frac{x \left (a+b x^2\right )^{3/2} (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)/(c + d*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 73.2336, size = 177, normalized size = 0.91 \[ \frac{b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{d^{3}} + \frac{x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{4 c d \left (c + d x^{2}\right )^{2}} + \frac{x \sqrt{a + b x^{2}} \left (a d - b c\right ) \left (3 a d + 4 b c\right )}{8 c^{2} d^{2} \left (c + d x^{2}\right )} + \frac{\sqrt{a d - b c} \left (3 a^{2} d^{2} + 4 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{8 c^{\frac{5}{2}} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)/(d*x**2+c)**3,x)

[Out]

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

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Mathematica [A]  time = 0.343915, size = 184, normalized size = 0.95 \[ \frac{\frac{\left (3 a^3 d^3+a^2 b c d^2+4 a b^2 c^2 d-8 b^3 c^3\right ) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{c^{5/2} \sqrt{a d-b c}}+8 b^{5/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\frac{d x \sqrt{a+b x^2} (a d-b c) \left (a d \left (5 c+3 d x^2\right )+2 b c \left (2 c+3 d x^2\right )\right )}{c^2 \left (c+d x^2\right )^2}}{8 d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)/(c + d*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0.048, size = 14133, normalized size = 72.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)/(d*x^2+c)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}{{\left (d x^{2} + c\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^3,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^3, x)

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Fricas [A]  time = 0.615359, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^3,x, algorithm="fricas")

[Out]

[1/32*(16*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(b)*log(-2*b*x^2 - 2
*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (8*b^2*c^4 + 4*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*
b^2*c^2*d^2 + 4*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d + 4*a*b*c^2*d^2 + 3*
a^2*c*d^3)*x^2)*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)*s
qrt(b*x^2 + a)*sqrt((b*c - a*d)/c))/(d^2*x^4 + 2*c*d*x^2 + c^2)) - 4*(3*(2*b^2*c
^2*d^2 - a*b*c*d^3 - a^2*d^4)*x^3 + (4*b^2*c^3*d + a*b*c^2*d^2 - 5*a^2*c*d^3)*x)
*sqrt(b*x^2 + a))/(c^2*d^5*x^4 + 2*c^3*d^4*x^2 + c^4*d^3), 1/32*(32*(b^2*c^2*d^2
*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(-b)*arctan(b*x/(sqrt(b*x^2 + a)*sqrt(-b))
) + (8*b^2*c^4 + 4*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 + 4*a*b*c*d^3 + 3*
a^2*d^4)*x^4 + 2*(8*b^2*c^3*d + 4*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*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)) - 4*(3*(2*b^2*c^2*d^2 - a*b*c*d^3 - a^2*d^4)*
x^3 + (4*b^2*c^3*d + a*b*c^2*d^2 - 5*a^2*c*d^3)*x)*sqrt(b*x^2 + a))/(c^2*d^5*x^4
+ 2*c^3*d^4*x^2 + c^4*d^3), 1/16*((8*b^2*c^4 + 4*a*b*c^3*d + 3*a^2*c^2*d^2 + (8
*b^2*c^2*d^2 + 4*a*b*c*d^3 + 3*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d + 4*a*b*c^2*d^2 + 3
*a^2*c*d^3)*x^2)*sqrt(-(b*c - a*d)/c)*arctan(-1/2*((2*b*c - a*d)*x^2 + a*c)/(sqr
t(b*x^2 + a)*c*x*sqrt(-(b*c - a*d)/c))) + 8*(b^2*c^2*d^2*x^4 + 2*b^2*c^3*d*x^2 +
b^2*c^4)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(3*(2*b^2*
c^2*d^2 - a*b*c*d^3 - a^2*d^4)*x^3 + (4*b^2*c^3*d + a*b*c^2*d^2 - 5*a^2*c*d^3)*x
)*sqrt(b*x^2 + a))/(c^2*d^5*x^4 + 2*c^3*d^4*x^2 + c^4*d^3), 1/16*(16*(b^2*c^2*d^
2*x^4 + 2*b^2*c^3*d*x^2 + b^2*c^4)*sqrt(-b)*arctan(b*x/(sqrt(b*x^2 + a)*sqrt(-b)
)) + (8*b^2*c^4 + 4*a*b*c^3*d + 3*a^2*c^2*d^2 + (8*b^2*c^2*d^2 + 4*a*b*c*d^3 + 3
*a^2*d^4)*x^4 + 2*(8*b^2*c^3*d + 4*a*b*c^2*d^2 + 3*a^2*c*d^3)*x^2)*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*(3*(2*b^2*c^2*d^2 - a*b*c*d^3 - a^2*d^4)*x^3 + (4*b^2*c^3*d + a*b*
c^2*d^2 - 5*a^2*c*d^3)*x)*sqrt(b*x^2 + a))/(c^2*d^5*x^4 + 2*c^3*d^4*x^2 + c^4*d^
3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)/(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.649774, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)/(d*x^2 + c)^3,x, algorithm="giac")

[Out]

sage0*x