3.699 \(\int \frac{(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

Optimal. Leaf size=333 \[ \frac{77 d^{13/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3} \]

[Out]

-(d*(d*x)^(11/2))/(6*b*(a + b*x^2)^3) - (11*d^3*(d*x)^(7/2))/(48*b^2*(a + b*x^2)
^2) - (77*d^5*(d*x)^(3/2))/(192*b^3*(a + b*x^2)) - (77*d^(13/2)*ArcTan[1 - (Sqrt
[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(1/4)*b^(15/4)) + (77*
d^(13/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]
*a^(1/4)*b^(15/4)) + (77*d^(13/2)*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^(1/4)*b^(15/4)) - (77*d^(13/2)*Lo
g[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256
*Sqrt[2]*a^(1/4)*b^(15/4))

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Rubi [A]  time = 0.721298, 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 \[ \frac{77 d^{13/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

-(d*(d*x)^(11/2))/(6*b*(a + b*x^2)^3) - (11*d^3*(d*x)^(7/2))/(48*b^2*(a + b*x^2)
^2) - (77*d^5*(d*x)^(3/2))/(192*b^3*(a + b*x^2)) - (77*d^(13/2)*ArcTan[1 - (Sqrt
[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(1/4)*b^(15/4)) + (77*
d^(13/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]
*a^(1/4)*b^(15/4)) + (77*d^(13/2)*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^(1/4)*b^(15/4)) - (77*d^(13/2)*Lo
g[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256
*Sqrt[2]*a^(1/4)*b^(15/4))

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Rubi in Sympy [A]  time = 147.988, size = 313, normalized size = 0.94 \[ - \frac{d \left (d x\right )^{\frac{11}{2}}}{6 b \left (a + b x^{2}\right )^{3}} - \frac{11 d^{3} \left (d x\right )^{\frac{7}{2}}}{48 b^{2} \left (a + b x^{2}\right )^{2}} - \frac{77 d^{5} \left (d x\right )^{\frac{3}{2}}}{192 b^{3} \left (a + b x^{2}\right )} + \frac{77 \sqrt{2} d^{\frac{13}{2}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 \sqrt [4]{a} b^{\frac{15}{4}}} - \frac{77 \sqrt{2} d^{\frac{13}{2}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 \sqrt [4]{a} b^{\frac{15}{4}}} - \frac{77 \sqrt{2} d^{\frac{13}{2}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 \sqrt [4]{a} b^{\frac{15}{4}}} + \frac{77 \sqrt{2} d^{\frac{13}{2}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 \sqrt [4]{a} b^{\frac{15}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

-d*(d*x)**(11/2)/(6*b*(a + b*x**2)**3) - 11*d**3*(d*x)**(7/2)/(48*b**2*(a + b*x*
*2)**2) - 77*d**5*(d*x)**(3/2)/(192*b**3*(a + b*x**2)) + 77*sqrt(2)*d**(13/2)*lo
g(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(512*a
**(1/4)*b**(15/4)) - 77*sqrt(2)*d**(13/2)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*
sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(512*a**(1/4)*b**(15/4)) - 77*sqrt(2)*d**(1
3/2)*atan(1 - sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(256*a**(1/4)*b**(1
5/4)) + 77*sqrt(2)*d**(13/2)*atan(1 + sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(
d)))/(256*a**(1/4)*b**(15/4))

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Mathematica [A]  time = 0.270199, size = 260, normalized size = 0.78 \[ \frac{d^6 \sqrt{d x} \left (-\frac{256 a^2 b^{3/4} x^{3/2}}{\left (a+b x^2\right )^3}+\frac{864 a b^{3/4} x^{3/2}}{\left (a+b x^2\right )^2}-\frac{1224 b^{3/4} x^{3/2}}{a+b x^2}+\frac{231 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{231 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{462 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{462 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}\right )}{1536 b^{15/4} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(d^6*Sqrt[d*x]*((-256*a^2*b^(3/4)*x^(3/2))/(a + b*x^2)^3 + (864*a*b^(3/4)*x^(3/2
))/(a + b*x^2)^2 - (1224*b^(3/4)*x^(3/2))/(a + b*x^2) - (462*Sqrt[2]*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/4) + (462*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/4) + (231*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b
^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4) - (231*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4)))/(1536*b^(15/4)*Sqrt[x])

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Maple [A]  time = 0.027, size = 271, normalized size = 0.8 \[ -{\frac{51\,{d}^{7}}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}b} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{33\,{d}^{9}a}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{b}^{2}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{77\,{d}^{11}{a}^{2}}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{b}^{3}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{77\,{d}^{7}\sqrt{2}}{512\,{b}^{4}}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{77\,{d}^{7}\sqrt{2}}{256\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{77\,{d}^{7}\sqrt{2}}{256\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

-51/64*d^7/(b*d^2*x^2+a*d^2)^3/b*(d*x)^(11/2)-33/32*d^9/(b*d^2*x^2+a*d^2)^3/b^2*
a*(d*x)^(7/2)-77/192*d^11/(b*d^2*x^2+a*d^2)^3/b^3*a^2*(d*x)^(3/2)+77/512*d^7/b^4
/(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)))+77/256*d^7/b^4/
(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+77/256*d^7
/b^4/(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((d*x)^(13/2)/(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 = 0.285922, size = 474, normalized size = 1.42 \[ \frac{924 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{3}{4}} a b^{11}}{\sqrt{d x} d^{19} + \sqrt{d^{39} x - \sqrt{-\frac{d^{26}}{a b^{15}}} a b^{7} d^{26}}}\right ) + 231 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} + 456533 \, \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{3}{4}} a b^{11}\right ) - 231 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} - 456533 \, \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{3}{4}} a b^{11}\right ) - 4 \,{\left (153 \, b^{2} d^{6} x^{5} + 198 \, a b d^{6} x^{3} + 77 \, a^{2} d^{6} x\right )} \sqrt{d x}}{768 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(13/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="fricas")

[Out]

1/768*(924*(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*b^3)*(-d^26/(a*b^15))^(1
/4)*arctan((-d^26/(a*b^15))^(3/4)*a*b^11/(sqrt(d*x)*d^19 + sqrt(d^39*x - sqrt(-d
^26/(a*b^15))*a*b^7*d^26))) + 231*(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*b
^3)*(-d^26/(a*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 + 456533*(-d^26/(a*b^15))^(
3/4)*a*b^11) - 231*(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*b^3)*(-d^26/(a*b
^15))^(1/4)*log(456533*sqrt(d*x)*d^19 - 456533*(-d^26/(a*b^15))^(3/4)*a*b^11) -
4*(153*b^2*d^6*x^5 + 198*a*b*d^6*x^3 + 77*a^2*d^6*x)*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)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278115, size = 408, normalized size = 1.23 \[ \frac{1}{1536} \, d^{5}{\left (\frac{462 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{6}} + \frac{462 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{6}} - \frac{231 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{6}} + \frac{231 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{6}} - \frac{8 \,{\left (153 \, \sqrt{d x} b^{2} d^{7} x^{5} + 198 \, \sqrt{d x} a b d^{7} x^{3} + 77 \, \sqrt{d x} a^{2} d^{7} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(13/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="giac")

[Out]

1/1536*d^5*(462*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^6) + 462*sqrt(2)*(a*b^3*d^2)^(3/4)*ar
ctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a*b^
6) - 231*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^6) + 231*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^6) - 8*(153*sqrt(d*x)*b^2*d^7*x^5 + 198*
sqrt(d*x)*a*b*d^7*x^3 + 77*sqrt(d*x)*a^2*d^7*x)/((b*d^2*x^2 + a*d^2)^3*b^3))