3.763 \(\int \frac{(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=459 \[ \frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(d*Sqrt[d*x])/(16*a*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*Sqrt[d*x])/(4*b*(a +
b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*d^(3/2)*(a + b*x^2)*ArcTan[1 - (Sq
rt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(7/4)*b^(5/4)*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4]) + (3*d^(3/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*S
qrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(7/4)*b^(5/4)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4]) - (3*d^(3/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(7/4)*b^(5/4)*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4]) + (3*d^(3/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(7/4)*b^(5/4)*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.756855, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{d \sqrt{d x}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d \sqrt{d x}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{7/4} b^{5/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(d*Sqrt[d*x])/(16*a*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*Sqrt[d*x])/(4*b*(a +
b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*d^(3/2)*(a + b*x^2)*ArcTan[1 - (Sq
rt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(7/4)*b^(5/4)*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4]) + (3*d^(3/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*S
qrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(7/4)*b^(5/4)*Sqrt[a^2 + 2*a*b*x^2 +
b^2*x^4]) - (3*d^(3/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(7/4)*b^(5/4)*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4]) + (3*d^(3/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(7/4)*b^(5/4)*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.406402, size = 436, normalized size = 0.95 \[ -\frac{3 (d x)^{3/2} \left (a+b x^2\right )^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} x^{3/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{3 (d x)^{3/2} \left (a+b x^2\right )^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4} x^{3/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{3 (d x)^{3/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{2 \sqrt [4]{b} \sqrt{x}-\sqrt{2} \sqrt [4]{a}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4} x^{3/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{3 (d x)^{3/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt{x}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4} x^{3/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{(d x)^{3/2} \left (a+b x^2\right )^2}{16 a b x \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{(d x)^{3/2} \left (a+b x^2\right )}{4 b x \left (\left (a+b x^2\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(3/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

-((d*x)^(3/2)*(a + b*x^2))/(4*b*x*((a + b*x^2)^2)^(3/2)) + ((d*x)^(3/2)*(a + b*x
^2)^2)/(16*a*b*x*((a + b*x^2)^2)^(3/2)) + (3*(d*x)^(3/2)*(a + b*x^2)^3*ArcTan[(-
(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(32*Sqrt[2]*a^(7/4)*b
^(5/4)*x^(3/2)*((a + b*x^2)^2)^(3/2)) + (3*(d*x)^(3/2)*(a + b*x^2)^3*ArcTan[(Sqr
t[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(32*Sqrt[2]*a^(7/4)*b^(5/4
)*x^(3/2)*((a + b*x^2)^2)^(3/2)) - (3*(d*x)^(3/2)*(a + b*x^2)^3*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(7/4)*b^(5/4)*x^(3/2)*
((a + b*x^2)^2)^(3/2)) + (3*(d*x)^(3/2)*(a + b*x^2)^3*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(7/4)*b^(5/4)*x^(3/2)*((a + b*x^
2)^2)^(3/2))

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Maple [B]  time = 0.024, size = 674, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

1/128*(3*(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))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*x^4*b^
2*d^2+6*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*
d^2/b)^(1/4))*x^4*b^2*d^2-6*(a*d^2/b)^(1/4)*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)
+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*b^2*d^2+6*(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))/((a*d^2/b)^(1/4)*(d*x)^(1
/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*x^2*a*b*d^2+12*(a*d^2/b)^(1/4)*2^(1/2)*arctan(
(2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2*a*b*d^2-12*(a*d^2/b)^
(1/4)*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2
*a*b*d^2+3*(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))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*a^2*
d^2+6*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^
2/b)^(1/4))*a^2*d^2-6*(a*d^2/b)^(1/4)*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^
2/b)^(1/4))/(a*d^2/b)^(1/4))*a^2*d^2+8*(d*x)^(5/2)*a*b-24*(d*x)^(1/2)*a^2*d^2)/d
*(b*x^2+a)/b/a^2/((b*x^2+a)^2)^(3/2)

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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)^(3/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294974, size = 382, normalized size = 0.83 \[ -\frac{12 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} b \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}}}{\sqrt{d x} d + \sqrt{a^{4} b^{2} \sqrt{-\frac{d^{6}}{a^{7} b^{5}}} + d^{3} x}}\right ) - 3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (3 \, a^{2} b \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} + 3 \, \sqrt{d x} d\right ) + 3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (-3 \, a^{2} b \left (-\frac{d^{6}}{a^{7} b^{5}}\right )^{\frac{1}{4}} + 3 \, \sqrt{d x} d\right ) - 4 \,{\left (b d x^{2} - 3 \, a d\right )} \sqrt{d x}}{64 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(3/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="fricas")

[Out]

-1/64*(12*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-d^6/(a^7*b^5))^(1/4)*arctan(a^2*
b*(-d^6/(a^7*b^5))^(1/4)/(sqrt(d*x)*d + sqrt(a^4*b^2*sqrt(-d^6/(a^7*b^5)) + d^3*
x))) - 3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-d^6/(a^7*b^5))^(1/4)*log(3*a^2*b*
(-d^6/(a^7*b^5))^(1/4) + 3*sqrt(d*x)*d) + 3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*
(-d^6/(a^7*b^5))^(1/4)*log(-3*a^2*b*(-d^6/(a^7*b^5))^(1/4) + 3*sqrt(d*x)*d) - 4*
(b*d*x^2 - 3*a*d)*sqrt(d*x))/(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.286759, size = 497, normalized size = 1.08 \[ \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} b^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} b^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{128 \, a^{2} b^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{128 \, a^{2} b^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{\sqrt{d x} b d^{5} x^{2} - 3 \, \sqrt{d x} a d^{5}}{16 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a b{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(3/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="giac")

[Out]

3/64*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2
*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^2*b^2*sign(b*d^4*x^2 + a*d^4)) + 3/64*sqrt(2)*(a
*b^3*d^2)^(1/4)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a
*d^2/b)^(1/4))/(a^2*b^2*sign(b*d^4*x^2 + a*d^4)) + 3/128*sqrt(2)*(a*b^3*d^2)^(1/
4)*d*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^2*b^2*sign(b
*d^4*x^2 + a*d^4)) - 3/128*sqrt(2)*(a*b^3*d^2)^(1/4)*d*ln(d*x - sqrt(2)*(a*d^2/b
)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^2*b^2*sign(b*d^4*x^2 + a*d^4)) + 1/16*(sqr
t(d*x)*b*d^5*x^2 - 3*sqrt(d*x)*a*d^5)/((b*d^2*x^2 + a*d^2)^2*a*b*sign(b*d^4*x^2
+ a*d^4))