3.713 \(\int \frac{(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=385 \[ \frac{4389 d^{21/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}+\frac{4389 d^{21/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5} \]

[Out]

-(d*(d*x)^(19/2))/(10*b*(a + b*x^2)^5) - (19*d^3*(d*x)^(15/2))/(160*b^2*(a + b*x
^2)^4) - (19*d^5*(d*x)^(11/2))/(128*b^3*(a + b*x^2)^3) - (209*d^7*(d*x)^(7/2))/(
1024*b^4*(a + b*x^2)^2) - (1463*d^9*(d*x)^(3/2))/(4096*b^5*(a + b*x^2)) - (4389*
d^(21/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2
]*a^(1/4)*b^(23/4)) + (4389*d^(21/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(
1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(1/4)*b^(23/4)) + (4389*d^(21/2)*Log[Sqrt[a]*Sqr
t[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^
(1/4)*b^(23/4)) - (4389*d^(21/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(1/4)*b^(23/4))

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Rubi [A]  time = 0.923893, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ \frac{4389 d^{21/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}+\frac{4389 d^{21/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(19/2))/(10*b*(a + b*x^2)^5) - (19*d^3*(d*x)^(15/2))/(160*b^2*(a + b*x
^2)^4) - (19*d^5*(d*x)^(11/2))/(128*b^3*(a + b*x^2)^3) - (209*d^7*(d*x)^(7/2))/(
1024*b^4*(a + b*x^2)^2) - (1463*d^9*(d*x)^(3/2))/(4096*b^5*(a + b*x^2)) - (4389*
d^(21/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2
]*a^(1/4)*b^(23/4)) + (4389*d^(21/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(
1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(1/4)*b^(23/4)) + (4389*d^(21/2)*Log[Sqrt[a]*Sqr
t[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^
(1/4)*b^(23/4)) - (4389*d^(21/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(1/4)*b^(23/4))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**(21/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.420735, size = 298, normalized size = 0.77 \[ \frac{d^9 (d x)^{3/2} \left (-\frac{16384 a^4 b^{3/4}}{\left (a+b x^2\right )^5}+\frac{84992 a^3 b^{3/4}}{\left (a+b x^2\right )^4}-\frac{180992 a^2 b^{3/4}}{\left (a+b x^2\right )^3}+\frac{205984 a b^{3/4}}{\left (a+b x^2\right )^2}-\frac{152120 b^{3/4}}{a+b x^2}+\frac{21945 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a} x^{3/2}}-\frac{21945 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a} x^{3/2}}-\frac{43890 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} x^{3/2}}+\frac{43890 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a} x^{3/2}}\right )}{163840 b^{23/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^9*(d*x)^(3/2)*((-16384*a^4*b^(3/4))/(a + b*x^2)^5 + (84992*a^3*b^(3/4))/(a +
b*x^2)^4 - (180992*a^2*b^(3/4))/(a + b*x^2)^3 + (205984*a*b^(3/4))/(a + b*x^2)^2
- (152120*b^(3/4))/(a + b*x^2) - (43890*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)])/(a^(1/4)*x^(3/2)) + (43890*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)])/(a^(1/4)*x^(3/2)) + (21945*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*x^(3/2)) - (21945*Sqrt[2]*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(1/4)*x^(3/2))))/(163840*b^(2
3/4))

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Maple [A]  time = 0.034, size = 335, normalized size = 0.9 \[ -{\frac{1463\,{d}^{19}{a}^{4}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{5}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{209\,{d}^{17}{a}^{3}}{128\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{4}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{5947\,{d}^{15}{a}^{2}}{2048\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{6289\,{d}^{13}a}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}} \left ( dx \right ) ^{{\frac{15}{2}}}}-{\frac{3803\,{d}^{11}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{4389\,{d}^{11}\sqrt{2}}{32768\,{b}^{6}}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{4389\,{d}^{11}\sqrt{2}}{16384\,{b}^{6}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{4389\,{d}^{11}\sqrt{2}}{16384\,{b}^{6}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-1463/4096*d^19/(b*d^2*x^2+a*d^2)^5/b^5*a^4*(d*x)^(3/2)-209/128*d^17/(b*d^2*x^2+
a*d^2)^5/b^4*a^3*(d*x)^(7/2)-5947/2048*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(1
1/2)-6289/2560*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(15/2)-3803/4096*d^11/(b*d^2
*x^2+a*d^2)^5/b*(d*x)^(19/2)+4389/32768*d^11/b^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))/(d*x+(a*d^2/b)^(1/4)*(d*x)
^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+4389/16384*d^11/b^6/(a*d^2/b)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+4389/16384*d^11/b^6/(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)^(21/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29012, size = 630, normalized size = 1.64 \[ \frac{87780 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{3}{4}} a b^{17}}{\sqrt{d x} d^{31} + \sqrt{d^{63} x - \sqrt{-\frac{d^{42}}{a b^{23}}} a b^{11} d^{42}}}\right ) + 21945 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{1}{4}} \log \left (84546715869 \, \sqrt{d x} d^{31} + 84546715869 \, \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{3}{4}} a b^{17}\right ) - 21945 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{1}{4}} \log \left (84546715869 \, \sqrt{d x} d^{31} - 84546715869 \, \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{3}{4}} a b^{17}\right ) - 4 \,{\left (19015 \, b^{4} d^{10} x^{9} + 50312 \, a b^{3} d^{10} x^{7} + 59470 \, a^{2} b^{2} d^{10} x^{5} + 33440 \, a^{3} b d^{10} x^{3} + 7315 \, a^{4} d^{10} x\right )} \sqrt{d x}}{81920 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(21/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(87780*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^
4*b^6*x^2 + a^5*b^5)*(-d^42/(a*b^23))^(1/4)*arctan((-d^42/(a*b^23))^(3/4)*a*b^17
/(sqrt(d*x)*d^31 + sqrt(d^63*x - sqrt(-d^42/(a*b^23))*a*b^11*d^42))) + 21945*(b^
10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^
5)*(-d^42/(a*b^23))^(1/4)*log(84546715869*sqrt(d*x)*d^31 + 84546715869*(-d^42/(a
*b^23))^(3/4)*a*b^17) - 21945*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3
*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(-d^42/(a*b^23))^(1/4)*log(84546715869*sqrt(
d*x)*d^31 - 84546715869*(-d^42/(a*b^23))^(3/4)*a*b^17) - 4*(19015*b^4*d^10*x^9 +
50312*a*b^3*d^10*x^7 + 59470*a^2*b^2*d^10*x^5 + 33440*a^3*b*d^10*x^3 + 7315*a^4
*d^10*x)*sqrt(d*x))/(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 +
5*a^4*b^6*x^2 + a^5*b^5)

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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)**(21/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283535, size = 459, normalized size = 1.19 \[ \frac{1}{163840} \, d^{9}{\left (\frac{43890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{8}} + \frac{43890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{8}} - \frac{21945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{8}} + \frac{21945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{8}} - \frac{8 \,{\left (19015 \, \sqrt{d x} b^{4} d^{11} x^{9} + 50312 \, \sqrt{d x} a b^{3} d^{11} x^{7} + 59470 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{5} + 33440 \, \sqrt{d x} a^{3} b d^{11} x^{3} + 7315 \, \sqrt{d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(21/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^9*(43890*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^8) + 43890*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^8) - 21945*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^8) + 21945*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^8) - 8*(19015*sqrt(d*x)*b^4*d^
11*x^9 + 50312*sqrt(d*x)*a*b^3*d^11*x^7 + 59470*sqrt(d*x)*a^2*b^2*d^11*x^5 + 334
40*sqrt(d*x)*a^3*b*d^11*x^3 + 7315*sqrt(d*x)*a^4*d^11*x)/((b*d^2*x^2 + a*d^2)^5*
b^5))