3.696 \(\int \frac{(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

Optimal. Leaf size=368 \[ -\frac{663 a^{5/4} d^{19/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{21/4}}-\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} b^{21/4}}-\frac{663 a d^9 \sqrt{d x}}{64 b^5}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac{663 d^7 (d x)^{5/2}}{320 b^4} \]

[Out]

(-663*a*d^9*Sqrt[d*x])/(64*b^5) + (663*d^7*(d*x)^(5/2))/(320*b^4) - (d*(d*x)^(17
/2))/(6*b*(a + b*x^2)^3) - (17*d^3*(d*x)^(13/2))/(48*b^2*(a + b*x^2)^2) - (221*d
^5*(d*x)^(9/2))/(192*b^3*(a + b*x^2)) - (663*a^(5/4)*d^(19/2)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*d
^(19/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*
b^(21/4)) - (663*a^(5/4)*d^(19/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*d^(19/2)*L
og[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(25
6*Sqrt[2]*b^(21/4))

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Rubi [A]  time = 0.878508, antiderivative size = 368, 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 \[ -\frac{663 a^{5/4} d^{19/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} b^{21/4}}-\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} b^{21/4}}+\frac{663 a^{5/4} d^{19/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} b^{21/4}}-\frac{663 a d^9 \sqrt{d x}}{64 b^5}-\frac{221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac{17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac{663 d^7 (d x)^{5/2}}{320 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(-663*a*d^9*Sqrt[d*x])/(64*b^5) + (663*d^7*(d*x)^(5/2))/(320*b^4) - (d*(d*x)^(17
/2))/(6*b*(a + b*x^2)^3) - (17*d^3*(d*x)^(13/2))/(48*b^2*(a + b*x^2)^2) - (221*d
^5*(d*x)^(9/2))/(192*b^3*(a + b*x^2)) - (663*a^(5/4)*d^(19/2)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*d
^(19/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*
b^(21/4)) - (663*a^(5/4)*d^(19/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*d^(19/2)*L
og[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(25
6*Sqrt[2]*b^(21/4))

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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)**(19/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.355021, size = 283, normalized size = 0.77 \[ \frac{d^9 \sqrt{d x} \left (-\frac{9945 \sqrt{2} a^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt{x}}+\frac{9945 \sqrt{2} a^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt{x}}-\frac{19890 \sqrt{2} a^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{x}}+\frac{19890 \sqrt{2} a^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{x}}-\frac{1280 a^4 \sqrt [4]{b}}{\left (a+b x^2\right )^3}+\frac{7840 a^3 \sqrt [4]{b}}{\left (a+b x^2\right )^2}-\frac{24680 a^2 \sqrt [4]{b}}{a+b x^2}-61440 a \sqrt [4]{b}+3072 b^{5/4} x^2\right )}{7680 b^{21/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(d^9*Sqrt[d*x]*(-61440*a*b^(1/4) + 3072*b^(5/4)*x^2 - (1280*a^4*b^(1/4))/(a + b*
x^2)^3 + (7840*a^3*b^(1/4))/(a + b*x^2)^2 - (24680*a^2*b^(1/4))/(a + b*x^2) - (1
9890*Sqrt[2]*a^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/Sqrt[x] + (1
9890*Sqrt[2]*a^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/Sqrt[x] - (9
945*Sqrt[2]*a^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/
Sqrt[x] + (9945*Sqrt[2]*a^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/Sqrt[x]))/(7680*b^(21/4))

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Maple [A]  time = 0.031, size = 306, normalized size = 0.8 \[{\frac{2\,{d}^{7}}{5\,{b}^{4}} \left ( dx \right ) ^{{\frac{5}{2}}}}-8\,{\frac{a{d}^{9}\sqrt{dx}}{{b}^{5}}}-{\frac{617\,{d}^{11}{a}^{2}}{192\,{b}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{9}{2}}}}-{\frac{173\,{d}^{13}{a}^{3}}{32\,{b}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{5}{2}}}}-{\frac{151\,{d}^{15}{a}^{4}}{64\,{b}^{5} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}}\sqrt{dx}}+{\frac{663\,a{d}^{9}\sqrt{2}}{512\,{b}^{5}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({1 \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }+{\frac{663\,a{d}^{9}\sqrt{2}}{256\,{b}^{5}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{663\,a{d}^{9}\sqrt{2}}{256\,{b}^{5}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

2/5*d^7*(d*x)^(5/2)/b^4-8*a*d^9*(d*x)^(1/2)/b^5-617/192*d^11/b^3*a^2/(b*d^2*x^2+
a*d^2)^3*(d*x)^(9/2)-173/32*d^13/b^4*a^3/(b*d^2*x^2+a*d^2)^3*(d*x)^(5/2)-151/64*
d^15/b^5*a^4/(b*d^2*x^2+a*d^2)^3*(d*x)^(1/2)+663/512*d^9/b^5*a*(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)))+663/256*d^9/b^5*a*(a*d^2/b)^(1/4)
*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+663/256*d^9/b^5*a*(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)^(19/2)/(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 = 0.293018, size = 508, normalized size = 1.38 \[ -\frac{39780 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \arctan \left (\frac{\left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}}{\sqrt{d x} a d^{9} + \sqrt{a^{2} d^{19} x + \sqrt{-\frac{a^{5} d^{38}}{b^{21}}} b^{10}}}\right ) - 9945 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \log \left (663 \, \sqrt{d x} a d^{9} + 663 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}\right ) + 9945 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \log \left (663 \, \sqrt{d x} a d^{9} - 663 \, \left (-\frac{a^{5} d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}\right ) - 4 \,{\left (384 \, b^{4} d^{9} x^{8} - 6528 \, a b^{3} d^{9} x^{6} - 24973 \, a^{2} b^{2} d^{9} x^{4} - 27846 \, a^{3} b d^{9} x^{2} - 9945 \, a^{4} d^{9}\right )} \sqrt{d x}}{3840 \,{\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(19/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="fricas")

[Out]

-1/3840*(39780*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a
^3*b^5)*arctan((-a^5*d^38/b^21)^(1/4)*b^5/(sqrt(d*x)*a*d^9 + sqrt(a^2*d^19*x + s
qrt(-a^5*d^38/b^21)*b^10))) - 9945*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4
+ 3*a^2*b^6*x^2 + a^3*b^5)*log(663*sqrt(d*x)*a*d^9 + 663*(-a^5*d^38/b^21)^(1/4)
*b^5) + 9945*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3
*b^5)*log(663*sqrt(d*x)*a*d^9 - 663*(-a^5*d^38/b^21)^(1/4)*b^5) - 4*(384*b^4*d^9
*x^8 - 6528*a*b^3*d^9*x^6 - 24973*a^2*b^2*d^9*x^4 - 27846*a^3*b*d^9*x^2 - 9945*a
^4*d^9)*sqrt(d*x))/(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3*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)**(19/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27964, size = 459, normalized size = 1.25 \[ \frac{1}{7680} \, d^{8}{\left (\frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{6}} + \frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{6}} + \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{6}} - \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{6}} - \frac{40 \,{\left (617 \, \sqrt{d x} a^{2} b^{2} d^{7} x^{4} + 1038 \, \sqrt{d x} a^{3} b d^{7} x^{2} + 453 \, \sqrt{d x} a^{4} d^{7}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{5}} + \frac{3072 \,{\left (\sqrt{d x} b^{16} d^{6} x^{2} - 20 \, \sqrt{d x} a b^{15} d^{6}\right )}}{b^{20} d^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(19/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="giac")

[Out]

1/7680*d^8*(19890*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d
^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/b^6 + 19890*sqrt(2)*(a*b^3*d^2)^(1/4
)*a*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4
))/b^6 + 9945*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqr
t(d*x) + sqrt(a*d^2/b))/b^6 - 9945*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*ln(d*x - sqrt(2
)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/b^6 - 40*(617*sqrt(d*x)*a^2*b^2*d^7
*x^4 + 1038*sqrt(d*x)*a^3*b*d^7*x^2 + 453*sqrt(d*x)*a^4*d^7)/((b*d^2*x^2 + a*d^2
)^3*b^5) + 3072*(sqrt(d*x)*b^16*d^6*x^2 - 20*sqrt(d*x)*a*b^15*d^6)/(b^20*d^5))