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

Optimal. Leaf size=333 \[ -\frac{15 d^{11/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{3/4} b^{13/4}}+\frac{15 d^{11/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{3/4} b^{13/4}}-\frac{15 d^{11/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{3/4} b^{13/4}}+\frac{15 d^{11/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{3/4} b^{13/4}}-\frac{15 d^5 \sqrt{d x}}{64 b^3 \left (a+b x^2\right )}-\frac{3 d^3 (d x)^{5/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3} \]

[Out]

-(d*(d*x)^(9/2))/(6*b*(a + b*x^2)^3) - (3*d^3*(d*x)^(5/2))/(16*b^2*(a + b*x^2)^2
) - (15*d^5*Sqrt[d*x])/(64*b^3*(a + b*x^2)) - (15*d^(11/2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(3/4)*b^(13/4)) + (15*d^(11
/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(3
/4)*b^(13/4)) - (15*d^(11/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a
^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(3/4)*b^(13/4)) + (15*d^(11/2)*Log[Sqr
t[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt
[2]*a^(3/4)*b^(13/4))

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Rubi [A]  time = 0.715746, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ -\frac{15 d^{11/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{3/4} b^{13/4}}+\frac{15 d^{11/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{3/4} b^{13/4}}-\frac{15 d^{11/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{3/4} b^{13/4}}+\frac{15 d^{11/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{3/4} b^{13/4}}-\frac{15 d^5 \sqrt{d x}}{64 b^3 \left (a+b x^2\right )}-\frac{3 d^3 (d x)^{5/2}}{16 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{9/2}}{6 b \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-(d*(d*x)^(9/2))/(6*b*(a + b*x^2)^3) - (3*d^3*(d*x)^(5/2))/(16*b^2*(a + b*x^2)^2
) - (15*d^5*Sqrt[d*x])/(64*b^3*(a + b*x^2)) - (15*d^(11/2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(3/4)*b^(13/4)) + (15*d^(11
/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(3
/4)*b^(13/4)) - (15*d^(11/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a
^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(3/4)*b^(13/4)) + (15*d^(11/2)*Log[Sqr
t[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt
[2]*a^(3/4)*b^(13/4))

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Rubi in Sympy [A]  time = 152.628, size = 313, normalized size = 0.94 \[ - \frac{d \left (d x\right )^{\frac{9}{2}}}{6 b \left (a + b x^{2}\right )^{3}} - \frac{3 d^{3} \left (d x\right )^{\frac{5}{2}}}{16 b^{2} \left (a + b x^{2}\right )^{2}} - \frac{15 d^{5} \sqrt{d x}}{64 b^{3} \left (a + b x^{2}\right )} - \frac{15 \sqrt{2} d^{\frac{11}{2}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 a^{\frac{3}{4}} b^{\frac{13}{4}}} + \frac{15 \sqrt{2} d^{\frac{11}{2}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 a^{\frac{3}{4}} b^{\frac{13}{4}}} - \frac{15 \sqrt{2} d^{\frac{11}{2}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 a^{\frac{3}{4}} b^{\frac{13}{4}}} + \frac{15 \sqrt{2} d^{\frac{11}{2}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 a^{\frac{3}{4}} b^{\frac{13}{4}}} \]

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)**2,x)

[Out]

-d*(d*x)**(9/2)/(6*b*(a + b*x**2)**3) - 3*d**3*(d*x)**(5/2)/(16*b**2*(a + b*x**2
)**2) - 15*d**5*sqrt(d*x)/(64*b**3*(a + b*x**2)) - 15*sqrt(2)*d**(11/2)*log(-sqr
t(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(512*a**(3/4
)*b**(13/4)) + 15*sqrt(2)*d**(11/2)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqrt(d
*x) + sqrt(a)*d + sqrt(b)*d*x)/(512*a**(3/4)*b**(13/4)) - 15*sqrt(2)*d**(11/2)*a
tan(1 - sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(256*a**(3/4)*b**(13/4))
+ 15*sqrt(2)*d**(11/2)*atan(1 + sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(
256*a**(3/4)*b**(13/4))

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Mathematica [A]  time = 0.267893, size = 260, normalized size = 0.78 \[ \frac{d^5 \sqrt{d x} \left (-\frac{45 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4} \sqrt{x}}+\frac{45 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4} \sqrt{x}}-\frac{90 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4} \sqrt{x}}+\frac{90 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4} \sqrt{x}}-\frac{256 a^2 \sqrt [4]{b}}{\left (a+b x^2\right )^3}+\frac{800 a \sqrt [4]{b}}{\left (a+b x^2\right )^2}-\frac{904 \sqrt [4]{b}}{a+b x^2}\right )}{1536 b^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

(d^5*Sqrt[d*x]*((-256*a^2*b^(1/4))/(a + b*x^2)^3 + (800*a*b^(1/4))/(a + b*x^2)^2
- (904*b^(1/4))/(a + b*x^2) - (90*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/
a^(1/4)])/(a^(3/4)*Sqrt[x]) + (90*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a
^(1/4)])/(a^(3/4)*Sqrt[x]) - (45*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[x] + Sqrt[b]*x])/(a^(3/4)*Sqrt[x]) + (45*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*Sqrt[x])))/(1536*b^(13/4))

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Maple [A]  time = 0.027, size = 280, normalized size = 0.8 \[ -{\frac{113\,{d}^{7}}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}b} \left ( dx \right ) ^{{\frac{9}{2}}}}-{\frac{21\,{d}^{9}a}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{b}^{2}} \left ( dx \right ) ^{{\frac{5}{2}}}}-{\frac{15\,{d}^{11}{a}^{2}}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{b}^{3}}\sqrt{dx}}+{\frac{15\,{d}^{5}\sqrt{2}}{512\,a{b}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({1 \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }+{\frac{15\,{d}^{5}\sqrt{2}}{256\,a{b}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{15\,{d}^{5}\sqrt{2}}{256\,a{b}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-113/192*d^7/(b*d^2*x^2+a*d^2)^3/b*(d*x)^(9/2)-21/32*d^9/(b*d^2*x^2+a*d^2)^3/b^2
*a*(d*x)^(5/2)-15/64*d^11/(b*d^2*x^2+a*d^2)^3/b^3*a^2*(d*x)^(1/2)+15/512*d^5/b^3
*(a*d^2/b)^(1/4)/a*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)))+15/256*d^5/b^
3*(a*d^2/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+15/256
*d^5/b^3*(a*d^2/b)^(1/4)/a*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((d*x)^(11/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288441, size = 470, normalized size = 1.41 \[ -\frac{180 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{22}}{a^{3} b^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{22}}{a^{3} b^{13}}\right )^{\frac{1}{4}} a b^{3}}{\sqrt{d x} d^{5} + \sqrt{d^{11} x + \sqrt{-\frac{d^{22}}{a^{3} b^{13}}} a^{2} b^{6}}}\right ) - 45 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{22}}{a^{3} b^{13}}\right )^{\frac{1}{4}} \log \left (15 \, \sqrt{d x} d^{5} + 15 \, \left (-\frac{d^{22}}{a^{3} b^{13}}\right )^{\frac{1}{4}} a b^{3}\right ) + 45 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{22}}{a^{3} b^{13}}\right )^{\frac{1}{4}} \log \left (15 \, \sqrt{d x} d^{5} - 15 \, \left (-\frac{d^{22}}{a^{3} b^{13}}\right )^{\frac{1}{4}} a b^{3}\right ) + 4 \,{\left (113 \, b^{2} d^{5} x^{4} + 126 \, a b d^{5} x^{2} + 45 \, a^{2} d^{5}\right )} \sqrt{d x}}{768 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

-1/768*(180*(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*b^3)*(-d^22/(a^3*b^13))
^(1/4)*arctan((-d^22/(a^3*b^13))^(1/4)*a*b^3/(sqrt(d*x)*d^5 + sqrt(d^11*x + sqrt
(-d^22/(a^3*b^13))*a^2*b^6))) - 45*(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*
b^3)*(-d^22/(a^3*b^13))^(1/4)*log(15*sqrt(d*x)*d^5 + 15*(-d^22/(a^3*b^13))^(1/4)
*a*b^3) + 45*(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*b^3)*(-d^22/(a^3*b^13)
)^(1/4)*log(15*sqrt(d*x)*d^5 - 15*(-d^22/(a^3*b^13))^(1/4)*a*b^3) + 4*(113*b^2*d
^5*x^4 + 126*a*b*d^5*x^2 + 45*a^2*d^5)*sqrt(d*x))/(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2
*b^4*x^2 + a^3*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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27997, size = 412, normalized size = 1.24 \[ \frac{1}{1536} \, d^{4}{\left (\frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{4}} + \frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{4}} + \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{4}} - \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{4}} - \frac{8 \,{\left (113 \, \sqrt{d x} b^{2} d^{7} x^{4} + 126 \, \sqrt{d x} a b d^{7} x^{2} + 45 \, \sqrt{d x} a^{2} d^{7}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{3}}\right )} \]

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)^2,x, algorithm="giac")

[Out]

1/1536*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))/(a*b^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))/(a*
b^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))/(a*b^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))/(a*b^4) - 8*(113*sqrt(d*x)*b^2*d^7*x^4 +
126*sqrt(d*x)*a*b*d^7*x^2 + 45*sqrt(d*x)*a^2*d^7)/((b*d^2*x^2 + a*d^2)^3*b^3))