3.694 \(\int \frac{1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx\)

Optimal. Leaf size=300 \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )} \]

[Out]

-7/(6*a^2*d*(d*x)^(3/2)) + 1/(2*a*d*(d*x)^(3/2)*(a + b*x^2)) + (7*b^(3/4)*ArcTan
[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*a^(11/4)*d^(5/2)
) - (7*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqr
t[2]*a^(11/4)*d^(5/2)) + (7*b^(3/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqrt[2]*a^(11/4)*d^(5/2)) - (7*b^(3/4)*Log[
Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqr
t[2]*a^(11/4)*d^(5/2))

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Rubi [A]  time = 0.644911, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]

[Out]

-7/(6*a^2*d*(d*x)^(3/2)) + 1/(2*a*d*(d*x)^(3/2)*(a + b*x^2)) + (7*b^(3/4)*ArcTan
[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*a^(11/4)*d^(5/2)
) - (7*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqr
t[2]*a^(11/4)*d^(5/2)) + (7*b^(3/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqrt[2]*a^(11/4)*d^(5/2)) - (7*b^(3/4)*Log[
Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqr
t[2]*a^(11/4)*d^(5/2))

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Rubi in Sympy [A]  time = 135.371, size = 279, normalized size = 0.93 \[ \frac{1}{2 a d \left (d x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )} - \frac{7}{6 a^{2} d \left (d x\right )^{\frac{3}{2}}} + \frac{7 \sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{16 a^{\frac{11}{4}} d^{\frac{5}{2}}} - \frac{7 \sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{16 a^{\frac{11}{4}} d^{\frac{5}{2}}} + \frac{7 \sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{8 a^{\frac{11}{4}} d^{\frac{5}{2}}} - \frac{7 \sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{8 a^{\frac{11}{4}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

1/(2*a*d*(d*x)**(3/2)*(a + b*x**2)) - 7/(6*a**2*d*(d*x)**(3/2)) + 7*sqrt(2)*b**(
3/4)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)
/(16*a**(11/4)*d**(5/2)) - 7*sqrt(2)*b**(3/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt
(d)*sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(16*a**(11/4)*d**(5/2)) + 7*sqrt(2)*b**
(3/4)*atan(1 - sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(8*a**(11/4)*d**(5
/2)) - 7*sqrt(2)*b**(3/4)*atan(1 + sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d))
)/(8*a**(11/4)*d**(5/2))

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Mathematica [A]  time = 0.312241, size = 233, normalized size = 0.78 \[ \frac{x \left (-\frac{24 a^{3/4} b x^2}{a+b x^2}-32 a^{3/4}+21 \sqrt{2} b^{3/4} x^{3/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-21 \sqrt{2} b^{3/4} x^{3/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+42 \sqrt{2} b^{3/4} x^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-42 \sqrt{2} b^{3/4} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{48 a^{11/4} (d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]

[Out]

(x*(-32*a^(3/4) - (24*a^(3/4)*b*x^2)/(a + b*x^2) + 42*Sqrt[2]*b^(3/4)*x^(3/2)*Ar
cTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 42*Sqrt[2]*b^(3/4)*x^(3/2)*ArcTan[
1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 21*Sqrt[2]*b^(3/4)*x^(3/2)*Log[Sqrt[a]
- Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 21*Sqrt[2]*b^(3/4)*x^(3/2)*Log[
Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(48*a^(11/4)*(d*x)^(5/2
))

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Maple [A]  time = 0.023, size = 226, normalized size = 0.8 \[ -{\frac{2}{3\,{a}^{2}d} \left ( dx \right ) ^{-{\frac{3}{2}}}}-{\frac{b}{2\,{a}^{2}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }\sqrt{dx}}-{\frac{7\,b\sqrt{2}}{16\,{d}^{3}{a}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({1 \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }-{\frac{7\,b\sqrt{2}}{8\,{d}^{3}{a}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }-{\frac{7\,b\sqrt{2}}{8\,{d}^{3}{a}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

-2/3/a^2/d/(d*x)^(3/2)-1/2/d/a^2*b*(d*x)^(1/2)/(b*d^2*x^2+a*d^2)-7/16/d^3/a^3*b*
(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)))-7/8/d^3/a^3*b*(a
*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)-7/8/d^3/a^3*
b*(a*d^2/b)^(1/4)*2^(1/2)*arctan(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(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289106, size = 379, normalized size = 1.26 \[ -\frac{28 \, b x^{2} - 84 \,{\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt{d x} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} d^{3} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}}}{\sqrt{d x} b + \sqrt{a^{6} d^{6} \sqrt{-\frac{b^{3}}{a^{11} d^{10}}} + b^{2} d x}}\right ) + 21 \,{\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt{d x} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \log \left (7 \, a^{3} d^{3} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} + 7 \, \sqrt{d x} b\right ) - 21 \,{\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt{d x} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \log \left (-7 \, a^{3} d^{3} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} + 7 \, \sqrt{d x} b\right ) + 16 \, a}{24 \,{\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(5/2)),x, algorithm="fricas")

[Out]

-1/24*(28*b*x^2 - 84*(a^2*b*d^2*x^3 + a^3*d^2*x)*sqrt(d*x)*(-b^3/(a^11*d^10))^(1
/4)*arctan(a^3*d^3*(-b^3/(a^11*d^10))^(1/4)/(sqrt(d*x)*b + sqrt(a^6*d^6*sqrt(-b^
3/(a^11*d^10)) + b^2*d*x))) + 21*(a^2*b*d^2*x^3 + a^3*d^2*x)*sqrt(d*x)*(-b^3/(a^
11*d^10))^(1/4)*log(7*a^3*d^3*(-b^3/(a^11*d^10))^(1/4) + 7*sqrt(d*x)*b) - 21*(a^
2*b*d^2*x^3 + a^3*d^2*x)*sqrt(d*x)*(-b^3/(a^11*d^10))^(1/4)*log(-7*a^3*d^3*(-b^3
/(a^11*d^10))^(1/4) + 7*sqrt(d*x)*b) + 16*a)/((a^2*b*d^2*x^3 + a^3*d^2*x)*sqrt(d
*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

Integral(1/((d*x)**(5/2)*(a + b*x**2)**2), x)

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GIAC/XCAS [A]  time = 0.271856, size = 373, normalized size = 1.24 \[ -\frac{\sqrt{d x} b}{2 \,{\left (b d^{2} x^{2} + a d^{2}\right )} a^{2} d} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} d^{3}} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} d^{3}} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{16 \, a^{3} d^{3}} + \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{16 \, a^{3} d^{3}} - \frac{2}{3 \, \sqrt{d x} a^{2} d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(5/2)),x, algorithm="giac")

[Out]

-1/2*sqrt(d*x)*b/((b*d^2*x^2 + a*d^2)*a^2*d) - 7/8*sqrt(2)*(a*b^3*d^2)^(1/4)*arc
tan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^3*d^
3) - 7/8*sqrt(2)*(a*b^3*d^2)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4)
- 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^3*d^3) - 7/16*sqrt(2)*(a*b^3*d^2)^(1/4)*ln(d*
x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^3*d^3) + 7/16*sqrt(2)*
(a*b^3*d^2)^(1/4)*ln(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a
^3*d^3) - 2/3/(sqrt(d*x)*a^2*d^2*x)