3.719 \(\int \frac{(d x)^{9/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=394 \[ \frac{63 d^{9/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5} \]

[Out]

-(d*(d*x)^(7/2))/(10*b*(a + b*x^2)^5) - (7*d^3*(d*x)^(3/2))/(160*b^2*(a + b*x^2)
^4) + (7*d^3*(d*x)^(3/2))/(640*a*b^2*(a + b*x^2)^3) + (63*d^3*(d*x)^(3/2))/(5120
*a^2*b^2*(a + b*x^2)^2) + (63*d^3*(d*x)^(3/2))/(4096*a^3*b^2*(a + b*x^2)) - (63*
d^(9/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]
*a^(13/4)*b^(11/4)) + (63*d^(9/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4
)*Sqrt[d])])/(8192*Sqrt[2]*a^(13/4)*b^(11/4)) + (63*d^(9/2)*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^(13/4
)*b^(11/4)) - (63*d^(9/2)*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^(13/4)*b^(11/4))

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Rubi [A]  time = 0.94411, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{63 d^{9/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{13/4} b^{11/4}}-\frac{63 d^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{13/4} b^{11/4}}+\frac{63 d^3 (d x)^{3/2}}{4096 a^3 b^2 \left (a+b x^2\right )}+\frac{63 d^3 (d x)^{3/2}}{5120 a^2 b^2 \left (a+b x^2\right )^2}+\frac{7 d^3 (d x)^{3/2}}{640 a b^2 \left (a+b x^2\right )^3}-\frac{7 d^3 (d x)^{3/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{7/2}}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(7/2))/(10*b*(a + b*x^2)^5) - (7*d^3*(d*x)^(3/2))/(160*b^2*(a + b*x^2)
^4) + (7*d^3*(d*x)^(3/2))/(640*a*b^2*(a + b*x^2)^3) + (63*d^3*(d*x)^(3/2))/(5120
*a^2*b^2*(a + b*x^2)^2) + (63*d^3*(d*x)^(3/2))/(4096*a^3*b^2*(a + b*x^2)) - (63*
d^(9/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]
*a^(13/4)*b^(11/4)) + (63*d^(9/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4
)*Sqrt[d])])/(8192*Sqrt[2]*a^(13/4)*b^(11/4)) + (63*d^(9/2)*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^(13/4
)*b^(11/4)) - (63*d^(9/2)*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^(13/4)*b^(11/4))

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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)**(9/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.441838, size = 308, normalized size = 0.78 \[ \frac{d^4 \sqrt{d x} \left (\frac{315 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4}}-\frac{315 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4}}-\frac{630 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{13/4}}+\frac{630 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{13/4}}+\frac{2520 b^{3/4} x^{3/2}}{a^3 \left (a+b x^2\right )}+\frac{2016 b^{3/4} x^{3/2}}{a^2 \left (a+b x^2\right )^2}+\frac{1792 b^{3/4} x^{3/2}}{a \left (a+b x^2\right )^3}-\frac{23552 b^{3/4} x^{3/2}}{\left (a+b x^2\right )^4}+\frac{16384 a b^{3/4} x^{3/2}}{\left (a+b x^2\right )^5}\right )}{163840 b^{11/4} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(9/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^4*Sqrt[d*x]*((16384*a*b^(3/4)*x^(3/2))/(a + b*x^2)^5 - (23552*b^(3/4)*x^(3/2)
)/(a + b*x^2)^4 + (1792*b^(3/4)*x^(3/2))/(a*(a + b*x^2)^3) + (2016*b^(3/4)*x^(3/
2))/(a^2*(a + b*x^2)^2) + (2520*b^(3/4)*x^(3/2))/(a^3*(a + b*x^2)) - (630*Sqrt[2
]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(13/4) + (630*Sqrt[2]*ArcTan[
1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(13/4) + (315*Sqrt[2]*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(13/4) - (315*Sqrt[2]*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(13/4)))/(163840*b^(11/4)*Sqrt
[x])

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Maple [A]  time = 0.033, size = 339, normalized size = 0.9 \[ -{\frac{21\,{d}^{13}a}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{d}^{11}}{128\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{287\,{d}^{9}}{2048\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{189\,{d}^{7}b}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{15}{2}}}}+{\frac{63\,{d}^{5}{b}^{2}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{3}} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{63\,{d}^{5}\sqrt{2}}{32768\,{a}^{3}{b}^{3}}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{63\,{d}^{5}\sqrt{2}}{16384\,{a}^{3}{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{63\,{d}^{5}\sqrt{2}}{16384\,{a}^{3}{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(9/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-21/4096*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(3/2)-3/128*d^11/(b*d^2*x^2+a*d^2)
^5/b*(d*x)^(7/2)+287/2048*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(11/2)+189/2560*d^7/(b
*d^2*x^2+a*d^2)^5/a^2*b*(d*x)^(15/2)+63/4096*d^5/(b*d^2*x^2+a*d^2)^5/a^3*b^2*(d*
x)^(19/2)+63/32768*d^5/a^3/b^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))/(d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*
d^2/b)^(1/2)))+63/16384*d^5/a^3/b^3/(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^
2/b)^(1/4)*(d*x)^(1/2)+1)+63/16384*d^5/a^3/b^3/(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)^(9/2)/(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 = 0.287692, size = 672, normalized size = 1.71 \[ \frac{1260 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{250047 \, a^{10} b^{8} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{3}{4}}}{250047 \, \sqrt{d x} d^{13} + \sqrt{-62523502209 \, a^{7} b^{5} d^{18} \sqrt{-\frac{d^{18}}{a^{13} b^{11}}} + 62523502209 \, d^{27} x}}\right ) + 315 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{1}{4}} \log \left (250047 \, a^{10} b^{8} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{3}{4}} + 250047 \, \sqrt{d x} d^{13}\right ) - 315 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{1}{4}} \log \left (-250047 \, a^{10} b^{8} \left (-\frac{d^{18}}{a^{13} b^{11}}\right )^{\frac{3}{4}} + 250047 \, \sqrt{d x} d^{13}\right ) + 4 \,{\left (315 \, b^{4} d^{4} x^{9} + 1512 \, a b^{3} d^{4} x^{7} + 2870 \, a^{2} b^{2} d^{4} x^{5} - 480 \, a^{3} b d^{4} x^{3} - 105 \, a^{4} d^{4} x\right )} \sqrt{d x}}{81920 \,{\left (a^{3} b^{7} x^{10} + 5 \, a^{4} b^{6} x^{8} + 10 \, a^{5} b^{5} x^{6} + 10 \, a^{6} b^{4} x^{4} + 5 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(9/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(1260*(a^3*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4 +
5*a^7*b^3*x^2 + a^8*b^2)*(-d^18/(a^13*b^11))^(1/4)*arctan(250047*a^10*b^8*(-d^18
/(a^13*b^11))^(3/4)/(250047*sqrt(d*x)*d^13 + sqrt(-62523502209*a^7*b^5*d^18*sqrt
(-d^18/(a^13*b^11)) + 62523502209*d^27*x))) + 315*(a^3*b^7*x^10 + 5*a^4*b^6*x^8
+ 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4 + 5*a^7*b^3*x^2 + a^8*b^2)*(-d^18/(a^13*b^11))
^(1/4)*log(250047*a^10*b^8*(-d^18/(a^13*b^11))^(3/4) + 250047*sqrt(d*x)*d^13) -
315*(a^3*b^7*x^10 + 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4 + 5*a^7*b^3*
x^2 + a^8*b^2)*(-d^18/(a^13*b^11))^(1/4)*log(-250047*a^10*b^8*(-d^18/(a^13*b^11)
)^(3/4) + 250047*sqrt(d*x)*d^13) + 4*(315*b^4*d^4*x^9 + 1512*a*b^3*d^4*x^7 + 287
0*a^2*b^2*d^4*x^5 - 480*a^3*b*d^4*x^3 - 105*a^4*d^4*x)*sqrt(d*x))/(a^3*b^7*x^10
+ 5*a^4*b^6*x^8 + 10*a^5*b^5*x^6 + 10*a^6*b^4*x^4 + 5*a^7*b^3*x^2 + a^8*b^2)

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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)**(9/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282487, size = 463, normalized size = 1.18 \[ \frac{1}{163840} \, d^{3}{\left (\frac{630 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{4} b^{5}} + \frac{630 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{4} b^{5}} - \frac{315 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{4} b^{5}} + \frac{315 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{4} b^{5}} + \frac{8 \,{\left (315 \, \sqrt{d x} b^{4} d^{11} x^{9} + 1512 \, \sqrt{d x} a b^{3} d^{11} x^{7} + 2870 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{5} - 480 \, \sqrt{d x} a^{3} b d^{11} x^{3} - 105 \, \sqrt{d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{3} b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(9/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^3*(630*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^4*b^5) + 630*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^4*b^5) - 315*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^4*b^5) + 315*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^4*b^5) + 8*(315*sqrt(d*x)*b^4*d^11
*x^9 + 1512*sqrt(d*x)*a*b^3*d^11*x^7 + 2870*sqrt(d*x)*a^2*b^2*d^11*x^5 - 480*sqr
t(d*x)*a^3*b*d^11*x^3 - 105*sqrt(d*x)*a^4*d^11*x)/((b*d^2*x^2 + a*d^2)^5*a^3*b^2
))