3.715 \(\int \frac{(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=388 \[ \frac{231 d^{17/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{5/4} b^{19/4}}-\frac{231 d^{17/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{5/4} b^{19/4}}-\frac{231 d^{17/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{5/4} b^{19/4}}+\frac{231 d^{17/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{5/4} b^{19/4}}+\frac{231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}-\frac{77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5} \]

[Out]

-(d*(d*x)^(15/2))/(10*b*(a + b*x^2)^5) - (3*d^3*(d*x)^(11/2))/(32*b^2*(a + b*x^2
)^4) - (11*d^5*(d*x)^(7/2))/(128*b^3*(a + b*x^2)^3) - (77*d^7*(d*x)^(3/2))/(1024
*b^4*(a + b*x^2)^2) + (231*d^7*(d*x)^(3/2))/(4096*a*b^4*(a + b*x^2)) - (231*d^(1
7/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^
(5/4)*b^(19/4)) + (231*d^(17/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*
Sqrt[d])])/(8192*Sqrt[2]*a^(5/4)*b^(19/4)) + (231*d^(17/2)*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^(5/4)*
b^(19/4)) - (231*d^(17/2)*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^(5/4)*b^(19/4))

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Rubi [A]  time = 0.922398, antiderivative size = 388, 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 \[ \frac{231 d^{17/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{5/4} b^{19/4}}-\frac{231 d^{17/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{5/4} b^{19/4}}-\frac{231 d^{17/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{5/4} b^{19/4}}+\frac{231 d^{17/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{5/4} b^{19/4}}+\frac{231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}-\frac{77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(15/2))/(10*b*(a + b*x^2)^5) - (3*d^3*(d*x)^(11/2))/(32*b^2*(a + b*x^2
)^4) - (11*d^5*(d*x)^(7/2))/(128*b^3*(a + b*x^2)^3) - (77*d^7*(d*x)^(3/2))/(1024
*b^4*(a + b*x^2)^2) + (231*d^7*(d*x)^(3/2))/(4096*a*b^4*(a + b*x^2)) - (231*d^(1
7/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^
(5/4)*b^(19/4)) + (231*d^(17/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*
Sqrt[d])])/(8192*Sqrt[2]*a^(5/4)*b^(19/4)) + (231*d^(17/2)*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^(5/4)*
b^(19/4)) - (231*d^(17/2)*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^(5/4)*b^(19/4))

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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)**(17/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.42868, size = 308, normalized size = 0.79 \[ \frac{d^8 \sqrt{d x} \left (\frac{1155 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4}}-\frac{1155 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4}}-\frac{2310 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac{2310 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+\frac{16384 a^3 b^{3/4} x^{3/2}}{\left (a+b x^2\right )^5}-\frac{64512 a^2 b^{3/4} x^{3/2}}{\left (a+b x^2\right )^4}+\frac{9240 b^{3/4} x^{3/2}}{a^2+a b x^2}+\frac{93952 a b^{3/4} x^{3/2}}{\left (a+b x^2\right )^3}-\frac{58144 b^{3/4} x^{3/2}}{\left (a+b x^2\right )^2}\right )}{163840 b^{19/4} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^8*Sqrt[d*x]*((16384*a^3*b^(3/4)*x^(3/2))/(a + b*x^2)^5 - (64512*a^2*b^(3/4)*x
^(3/2))/(a + b*x^2)^4 + (93952*a*b^(3/4)*x^(3/2))/(a + b*x^2)^3 - (58144*b^(3/4)
*x^(3/2))/(a + b*x^2)^2 + (9240*b^(3/4)*x^(3/2))/(a^2 + a*b*x^2) - (2310*Sqrt[2]
*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(5/4) + (2310*Sqrt[2]*ArcTan[1
+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(5/4) + (1155*Sqrt[2]*Log[Sqrt[a] - Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4) - (1155*Sqrt[2]*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4)))/(163840*b^(19/4)*Sqrt[x
])

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Maple [A]  time = 0.033, size = 341, normalized size = 0.9 \[ -{\frac{77\,{d}^{17}{a}^{3}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{4}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{11\,{d}^{15}{a}^{2}}{128\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{3}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{313\,{d}^{13}a}{2048\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{331\,{d}^{11}}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{15}{2}}}}+{\frac{231\,{d}^{9}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{231\,{d}^{9}\sqrt{2}}{32768\,a{b}^{5}}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{231\,{d}^{9}\sqrt{2}}{16384\,a{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{231\,{d}^{9}\sqrt{2}}{16384\,a{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-77/4096*d^17/(b*d^2*x^2+a*d^2)^5/b^4*a^3*(d*x)^(3/2)-11/128*d^15/(b*d^2*x^2+a*d
^2)^5/b^3*a^2*(d*x)^(7/2)-313/2048*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(11/2)-3
31/2560*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(15/2)+231/4096*d^9/(b*d^2*x^2+a*d^2)^5
/a*(d*x)^(19/2)+231/32768*d^9/a/b^5/(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)))+231/16384*d^9/a/b^5/(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+231/16384*d^9/a/b^5/(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)^(17/2)/(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 = 0.292578, size = 657, normalized size = 1.69 \[ \frac{4620 \,{\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac{d^{34}}{a^{5} b^{19}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{34}}{a^{5} b^{19}}\right )^{\frac{3}{4}} a^{4} b^{14}}{\sqrt{d x} d^{25} + \sqrt{d^{51} x - \sqrt{-\frac{d^{34}}{a^{5} b^{19}}} a^{3} b^{9} d^{34}}}\right ) + 1155 \,{\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac{d^{34}}{a^{5} b^{19}}\right )^{\frac{1}{4}} \log \left (12326391 \, \sqrt{d x} d^{25} + 12326391 \, \left (-\frac{d^{34}}{a^{5} b^{19}}\right )^{\frac{3}{4}} a^{4} b^{14}\right ) - 1155 \,{\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac{d^{34}}{a^{5} b^{19}}\right )^{\frac{1}{4}} \log \left (12326391 \, \sqrt{d x} d^{25} - 12326391 \, \left (-\frac{d^{34}}{a^{5} b^{19}}\right )^{\frac{3}{4}} a^{4} b^{14}\right ) + 4 \,{\left (1155 \, b^{4} d^{8} x^{9} - 2648 \, a b^{3} d^{8} x^{7} - 3130 \, a^{2} b^{2} d^{8} x^{5} - 1760 \, a^{3} b d^{8} x^{3} - 385 \, a^{4} d^{8} x\right )} \sqrt{d x}}{81920 \,{\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(17/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(4620*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*
a^5*b^5*x^2 + a^6*b^4)*(-d^34/(a^5*b^19))^(1/4)*arctan((-d^34/(a^5*b^19))^(3/4)*
a^4*b^14/(sqrt(d*x)*d^25 + sqrt(d^51*x - sqrt(-d^34/(a^5*b^19))*a^3*b^9*d^34)))
+ 1155*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5
*x^2 + a^6*b^4)*(-d^34/(a^5*b^19))^(1/4)*log(12326391*sqrt(d*x)*d^25 + 12326391*
(-d^34/(a^5*b^19))^(3/4)*a^4*b^14) - 1155*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b
^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^34/(a^5*b^19))^(1/4)*log(
12326391*sqrt(d*x)*d^25 - 12326391*(-d^34/(a^5*b^19))^(3/4)*a^4*b^14) + 4*(1155*
b^4*d^8*x^9 - 2648*a*b^3*d^8*x^7 - 3130*a^2*b^2*d^8*x^5 - 1760*a^3*b*d^8*x^3 - 3
85*a^4*d^8*x)*sqrt(d*x))/(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b
^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)

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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)**(17/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280274, size = 463, normalized size = 1.19 \[ \frac{1}{163840} \, d^{7}{\left (\frac{2310 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{2} b^{7}} + \frac{2310 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{2} b^{7}} - \frac{1155 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{2} b^{7}} + \frac{1155 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{2} b^{7}} + \frac{8 \,{\left (1155 \, \sqrt{d x} b^{4} d^{11} x^{9} - 2648 \, \sqrt{d x} a b^{3} d^{11} x^{7} - 3130 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{5} - 1760 \, \sqrt{d x} a^{3} b d^{11} x^{3} - 385 \, \sqrt{d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a b^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(17/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^7*(2310*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^2*b^7) + 2310*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^2*b^7) - 1155*sqrt(2)*(a*b^3*d^2)^(3/4)*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqr
t(d*x) + sqrt(a*d^2/b))/(a^2*b^7) + 1155*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^2*b^7) + 8*(1155*sqrt(d*x)*b^4
*d^11*x^9 - 2648*sqrt(d*x)*a*b^3*d^11*x^7 - 3130*sqrt(d*x)*a^2*b^2*d^11*x^5 - 17
60*sqrt(d*x)*a^3*b*d^11*x^3 - 385*sqrt(d*x)*a^4*d^11*x)/((b*d^2*x^2 + a*d^2)^5*a
*b^4))