3.689 \(\int \frac{(d x)^{5/2}}{a^2+2 a b x^2+b^2 x^4} \, dx\)

Optimal. Leaf size=281 \[ \frac{3 d^{5/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{3 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )} \]

[Out]

-(d*(d*x)^(3/2))/(2*b*(a + b*x^2)) - (3*d^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt
[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*a^(1/4)*b^(7/4)) + (3*d^(5/2)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*a^(1/4)*b^(7/4)) + (3*
d^(5/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d
*x]])/(8*Sqrt[2]*a^(1/4)*b^(7/4)) - (3*d^(5/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqrt[2]*a^(1/4)*b^(7/4))

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Rubi [A]  time = 0.558677, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ \frac{3 d^{5/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{3 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

-(d*(d*x)^(3/2))/(2*b*(a + b*x^2)) - (3*d^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt
[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*a^(1/4)*b^(7/4)) + (3*d^(5/2)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*a^(1/4)*b^(7/4)) + (3*
d^(5/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d
*x]])/(8*Sqrt[2]*a^(1/4)*b^(7/4)) - (3*d^(5/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqrt[2]*a^(1/4)*b^(7/4))

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Rubi in Sympy [A]  time = 117.827, size = 262, normalized size = 0.93 \[ - \frac{d \left (d x\right )^{\frac{3}{2}}}{2 b \left (a + b x^{2}\right )} + \frac{3 \sqrt{2} d^{\frac{5}{2}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{16 \sqrt [4]{a} b^{\frac{7}{4}}} - \frac{3 \sqrt{2} d^{\frac{5}{2}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{16 \sqrt [4]{a} b^{\frac{7}{4}}} - \frac{3 \sqrt{2} d^{\frac{5}{2}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{8 \sqrt [4]{a} b^{\frac{7}{4}}} + \frac{3 \sqrt{2} d^{\frac{5}{2}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{8 \sqrt [4]{a} b^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

-d*(d*x)**(3/2)/(2*b*(a + b*x**2)) + 3*sqrt(2)*d**(5/2)*log(-sqrt(2)*a**(1/4)*b*
*(1/4)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(16*a**(1/4)*b**(7/4)) - 3*s
qrt(2)*d**(5/2)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sq
rt(b)*d*x)/(16*a**(1/4)*b**(7/4)) - 3*sqrt(2)*d**(5/2)*atan(1 - sqrt(2)*b**(1/4)
*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(8*a**(1/4)*b**(7/4)) + 3*sqrt(2)*d**(5/2)*atan(1
+ sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(8*a**(1/4)*b**(7/4))

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Mathematica [A]  time = 0.428697, size = 211, normalized size = 0.75 \[ \frac{(d x)^{5/2} \left (-\frac{8 b^{3/4} x^{3/2}}{a+b x^2}+\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}\right )}{16 b^{7/4} x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

((d*x)^(5/2)*((-8*b^(3/4)*x^(3/2))/(a + b*x^2) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*
b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/a^(1/4) + (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/a^(1/4) - (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x])/a^(1/4)))/(16*b^(7/4)*x^(5/2))

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Maple [A]  time = 0.02, size = 209, normalized size = 0.7 \[ -{\frac{{d}^{3}}{2\,b \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) } \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{3}\sqrt{2}}{16\,{b}^{2}}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{3\,{d}^{3}\sqrt{2}}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{3\,{d}^{3}\sqrt{2}}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

-1/2*d^3/b*(d*x)^(3/2)/(b*d^2*x^2+a*d^2)+3/16*d^3/b^2/(a*d^2/b)^(1/4)*2^(1/2)*ln
((d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x+(a*d^2/b)^(1/4)*
(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+3/8*d^3/b^2/(a*d^2/b)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+3/8*d^3/b^2/(a*d^2/b)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)

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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)^(5/2)/(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285622, size = 308, normalized size = 1.1 \[ -\frac{4 \, \sqrt{d x} d^{2} x - 12 \,{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{3}{4}} a b^{5}}{\sqrt{d x} d^{7} + \sqrt{d^{15} x - \sqrt{-\frac{d^{10}}{a b^{7}}} a b^{3} d^{10}}}\right ) - 3 \,{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{1}{4}} \log \left (27 \, \sqrt{d x} d^{7} + 27 \, \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{3}{4}} a b^{5}\right ) + 3 \,{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{1}{4}} \log \left (27 \, \sqrt{d x} d^{7} - 27 \, \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{3}{4}} a b^{5}\right )}{8 \,{\left (b^{2} x^{2} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(5/2)/(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="fricas")

[Out]

-1/8*(4*sqrt(d*x)*d^2*x - 12*(b^2*x^2 + a*b)*(-d^10/(a*b^7))^(1/4)*arctan((-d^10
/(a*b^7))^(3/4)*a*b^5/(sqrt(d*x)*d^7 + sqrt(d^15*x - sqrt(-d^10/(a*b^7))*a*b^3*d
^10))) - 3*(b^2*x^2 + a*b)*(-d^10/(a*b^7))^(1/4)*log(27*sqrt(d*x)*d^7 + 27*(-d^1
0/(a*b^7))^(3/4)*a*b^5) + 3*(b^2*x^2 + a*b)*(-d^10/(a*b^7))^(1/4)*log(27*sqrt(d*
x)*d^7 - 27*(-d^10/(a*b^7))^(3/4)*a*b^5))/(b^2*x^2 + a*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.275834, size = 355, normalized size = 1.26 \[ -\frac{1}{16} \,{\left (\frac{8 \, \sqrt{d x} d^{3} x}{{\left (b d^{2} x^{2} + a d^{2}\right )} b} - \frac{6 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{4}} - \frac{6 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{4}} + \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{4}} - \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{4}}\right )} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(5/2)/(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="giac")

[Out]

-1/16*(8*sqrt(d*x)*d^3*x/((b*d^2*x^2 + a*d^2)*b) - 6*sqrt(2)*(a*b^3*d^2)^(3/4)*a
rctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a*b^
4) - 6*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) -
2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a*b^4) + 3*sqrt(2)*(a*b^3*d^2)^(3/4)*ln(d*x + sqr
t(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^4) - 3*sqrt(2)*(a*b^3*d^2)^
(3/4)*ln(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^4))*d