3.759 \(\int \frac{(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=504 \[ -\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-9*d^3*(d*x)^(5/2))/(16*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(9/2))/
(4*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*d^5*Sqrt[d*x]*(a + b*x^2
))/(16*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*a^(1/4)*d^(11/2)*(a + b*x^2)*A
rcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*b^(13/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*a^(1/4)*d^(11/2)*(a + b*x^2)*ArcTan[1 + (S
qrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*b^(13/4)*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4]) + (45*a^(1/4)*d^(11/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqr
t[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*b^(13/4)*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4]) - (45*a^(1/4)*d^(11/2)*(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]*b^(13/4)
*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.819311, antiderivative size = 504, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(-9*d^3*(d*x)^(5/2))/(16*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(9/2))/
(4*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*d^5*Sqrt[d*x]*(a + b*x^2
))/(16*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*a^(1/4)*d^(11/2)*(a + b*x^2)*A
rcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*b^(13/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*a^(1/4)*d^(11/2)*(a + b*x^2)*ArcTan[1 + (S
qrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*b^(13/4)*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4]) + (45*a^(1/4)*d^(11/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqr
t[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*b^(13/4)*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4]) - (45*a^(1/4)*d^(11/2)*(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]*b^(13/4)
*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)**(11/2)/(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 = 0.350338, size = 474, normalized size = 0.94 \[ -\frac{a^2 (d x)^{11/2} \left (a+b x^2\right )}{4 b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{2 \sqrt [4]{b} \sqrt{x}-\sqrt{2} \sqrt [4]{a}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt{x}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{2 (d x)^{11/2} \left (a+b x^2\right )^3}{b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{17 a (d x)^{11/2} \left (a+b x^2\right )^2}{16 b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

-(a^2*(d*x)^(11/2)*(a + b*x^2))/(4*b^3*x^5*((a + b*x^2)^2)^(3/2)) + (17*a*(d*x)^
(11/2)*(a + b*x^2)^2)/(16*b^3*x^5*((a + b*x^2)^2)^(3/2)) + (2*(d*x)^(11/2)*(a +
b*x^2)^3)/(b^3*x^5*((a + b*x^2)^2)^(3/2)) - (45*a^(1/4)*(d*x)^(11/2)*(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]*b^(13/4)*x^(11/2)*((a + b*x^2)^2)^(3/2)) - (45*a^(1/4)*(d*x)^(11/2)*(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
]*b^(13/4)*x^(11/2)*((a + b*x^2)^2)^(3/2)) + (45*a^(1/4)*(d*x)^(11/2)*(a + b*x^2
)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*b^(1
3/4)*x^(11/2)*((a + b*x^2)^2)^(3/2)) - (45*a^(1/4)*(d*x)^(11/2)*(a + b*x^2)^3*Lo
g[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*b^(13/4)*x
^(11/2)*((a + b*x^2)^2)^(3/2))

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Maple [B]  time = 0.027, size = 702, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

-1/128*(45*(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+90*(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^2-90*(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+90*(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+180*(a*d^2/b)^(1/4)*2^(1/2)*a
rctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2*a*b*d^2-180*(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^2-256*(d*x)^(1/2)*x^4*b^2*d^2+45*(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+90*(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^2-90*(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^2-136*(
d*x)^(5/2)*a*b-512*(d*x)^(1/2)*x^2*a*b*d^2-360*(d*x)^(1/2)*a^2*d^2)*d^3*(b*x^2+a
)/b^3/((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)^(11/2)/(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 = 0.295295, size = 383, normalized size = 0.76 \[ \frac{180 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \arctan \left (\frac{\left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}} b^{3}}{\sqrt{d x} d^{5} + \sqrt{d^{11} x + \sqrt{-\frac{a d^{22}}{b^{13}}} b^{6}}}\right ) - 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (45 \, \sqrt{d x} d^{5} + 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}} b^{3}\right ) + 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (45 \, \sqrt{d x} d^{5} - 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}} b^{3}\right ) + 4 \,{\left (32 \, b^{2} d^{5} x^{4} + 81 \, a b d^{5} x^{2} + 45 \, a^{2} d^{5}\right )} \sqrt{d x}}{64 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(11/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="fricas")

[Out]

1/64*(180*(-a*d^22/b^13)^(1/4)*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*arctan((-a*d^22
/b^13)^(1/4)*b^3/(sqrt(d*x)*d^5 + sqrt(d^11*x + sqrt(-a*d^22/b^13)*b^6))) - 45*(
-a*d^22/b^13)^(1/4)*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*log(45*sqrt(d*x)*d^5 + 45*
(-a*d^22/b^13)^(1/4)*b^3) + 45*(-a*d^22/b^13)^(1/4)*(b^5*x^4 + 2*a*b^4*x^2 + a^2
*b^3)*log(45*sqrt(d*x)*d^5 - 45*(-a*d^22/b^13)^(1/4)*b^3) + 4*(32*b^2*d^5*x^4 +
81*a*b*d^5*x^2 + 45*a^2*d^5)*sqrt(d*x))/(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^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((d*x)**(11/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287295, size = 527, normalized size = 1.05 \[ -\frac{1}{128} \, d^{4}{\left (\frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{256 \, \sqrt{d x} d}{b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (17 \, \sqrt{d x} a b d^{5} x^{2} + 13 \, \sqrt{d x} a^{2} d^{5}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(11/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="giac")

[Out]

-1/128*d^4*(90*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))/(b^4*sign(b*d^4*x^2 + a*d^4)) + 90*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))/(b^4*sign(b*d^4*x^2 + a*d^4)) + 45*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))/(b^4*sign(b*d^4*x^
2 + a*d^4)) - 45*sqrt(2)*(a*b^3*d^2)^(1/4)*d*ln(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sq
rt(d*x) + sqrt(a*d^2/b))/(b^4*sign(b*d^4*x^2 + a*d^4)) - 256*sqrt(d*x)*d/(b^3*si
gn(b*d^4*x^2 + a*d^4)) - 8*(17*sqrt(d*x)*a*b*d^5*x^2 + 13*sqrt(d*x)*a^2*d^5)/((b
*d^2*x^2 + a*d^2)^2*b^3*sign(b*d^4*x^2 + a*d^4)))