3.709 \(\int \frac{1}{(d x)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\)

Optimal. Leaf size=370 \[ \frac{663 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3} \]

[Out]

-663/(320*a^4*d*(d*x)^(5/2)) + (663*b)/(64*a^5*d^3*Sqrt[d*x]) + 1/(6*a*d*(d*x)^(
5/2)*(a + b*x^2)^3) + 17/(48*a^2*d*(d*x)^(5/2)*(a + b*x^2)^2) + 221/(192*a^3*d*(
d*x)^(5/2)*(a + b*x^2)) - (663*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a
^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(21/4)*d^(7/2)) + (663*b^(5/4)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(21/4)*d^(7/2)) + (663
*b^(5/4)*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^(21/4)*d^(7/2)) - (663*b^(5/4)*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^(21/4)*d^(7/2)
)

_______________________________________________________________________________________

Rubi [A]  time = 0.917915, antiderivative size = 370, 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 \[ \frac{663 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{21/4} d^{7/2}}-\frac{663 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{21/4} d^{7/2}}+\frac{663 b}{64 a^5 d^3 \sqrt{d x}}-\frac{663}{320 a^4 d (d x)^{5/2}}+\frac{221}{192 a^3 d (d x)^{5/2} \left (a+b x^2\right )}+\frac{17}{48 a^2 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac{1}{6 a d (d x)^{5/2} \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]

[Out]

-663/(320*a^4*d*(d*x)^(5/2)) + (663*b)/(64*a^5*d^3*Sqrt[d*x]) + 1/(6*a*d*(d*x)^(
5/2)*(a + b*x^2)^3) + 17/(48*a^2*d*(d*x)^(5/2)*(a + b*x^2)^2) + 221/(192*a^3*d*(
d*x)^(5/2)*(a + b*x^2)) - (663*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a
^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(21/4)*d^(7/2)) + (663*b^(5/4)*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(21/4)*d^(7/2)) + (663
*b^(5/4)*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^(21/4)*d^(7/2)) - (663*b^(5/4)*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^(21/4)*d^(7/2)
)

_______________________________________________________________________________________

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(1/(d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.298428, size = 295, normalized size = 0.8 \[ \frac{\sqrt{d x} \left (\frac{5280 a^{5/4} b^2 x^4}{\left (a+b x^2\right )^2}+\frac{1280 a^{9/4} b^2 x^4}{\left (a+b x^2\right )^3}-3072 a^{5/4}+9945 \sqrt{2} b^{5/4} x^{5/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-9945 \sqrt{2} b^{5/4} x^{5/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-19890 \sqrt{2} b^{5/4} x^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+19890 \sqrt{2} b^{5/4} x^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{18120 \sqrt [4]{a} b^2 x^4}{a+b x^2}+61440 \sqrt [4]{a} b x^2\right )}{7680 a^{21/4} d^4 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^2),x]

[Out]

(Sqrt[d*x]*(-3072*a^(5/4) + 61440*a^(1/4)*b*x^2 + (1280*a^(9/4)*b^2*x^4)/(a + b*
x^2)^3 + (5280*a^(5/4)*b^2*x^4)/(a + b*x^2)^2 + (18120*a^(1/4)*b^2*x^4)/(a + b*x
^2) - 19890*Sqrt[2]*b^(5/4)*x^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)
] + 19890*Sqrt[2]*b^(5/4)*x^(5/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
+ 9945*Sqrt[2]*b^(5/4)*x^(5/2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x] - 9945*Sqrt[2]*b^(5/4)*x^(5/2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[x] + Sqrt[b]*x]))/(7680*a^(21/4)*d^4*x^3)

_______________________________________________________________________________________

Maple [A]  time = 0.036, size = 304, normalized size = 0.8 \[ -{\frac{2}{5\,{a}^{4}d} \left ( dx \right ) ^{-{\frac{5}{2}}}}+8\,{\frac{b}{{a}^{5}{d}^{3}\sqrt{dx}}}+{\frac{151\,{b}^{4}}{64\,{a}^{5}{d}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{173\,{b}^{3}}{32\,{a}^{4}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{617\,{b}^{2}d}{192\,{a}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{663\,b\sqrt{2}}{512\,{a}^{5}{d}^{3}}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{663\,b\sqrt{2}}{256\,{a}^{5}{d}^{3}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{663\,b\sqrt{2}}{256\,{a}^{5}{d}^{3}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

-2/5/a^4/d/(d*x)^(5/2)+8*b/a^5/d^3/(d*x)^(1/2)+151/64/d^3*b^4/a^5/(b*d^2*x^2+a*d
^2)^3*(d*x)^(11/2)+173/32/d*b^3/a^4/(b*d^2*x^2+a*d^2)^3*(d*x)^(7/2)+617/192*d*b^
2/a^3/(b*d^2*x^2+a*d^2)^3*(d*x)^(3/2)+663/512/d^3*b/a^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)))+663/256/d^3*b/a^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)+663/256/d^3*b/a^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)

_______________________________________________________________________________________

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(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^2*(d*x)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.302338, size = 603, normalized size = 1.63 \[ \frac{39780 \, b^{4} x^{8} + 111384 \, a b^{3} x^{6} + 99892 \, a^{2} b^{2} x^{4} + 26112 \, a^{3} b x^{2} - 1536 \, a^{4} + 39780 \,{\left (a^{5} b^{3} d^{3} x^{8} + 3 \, a^{6} b^{2} d^{3} x^{6} + 3 \, a^{7} b d^{3} x^{4} + a^{8} d^{3} x^{2}\right )} \sqrt{d x} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{1}{4}} \arctan \left (\frac{291434247 \, a^{16} d^{11} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{3}{4}}}{291434247 \, \sqrt{d x} b^{4} + \sqrt{-84933920324457009 \, a^{11} b^{5} d^{8} \sqrt{-\frac{b^{5}}{a^{21} d^{14}}} + 84933920324457009 \, b^{8} d x}}\right ) + 9945 \,{\left (a^{5} b^{3} d^{3} x^{8} + 3 \, a^{6} b^{2} d^{3} x^{6} + 3 \, a^{7} b d^{3} x^{4} + a^{8} d^{3} x^{2}\right )} \sqrt{d x} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{1}{4}} \log \left (291434247 \, a^{16} d^{11} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{3}{4}} + 291434247 \, \sqrt{d x} b^{4}\right ) - 9945 \,{\left (a^{5} b^{3} d^{3} x^{8} + 3 \, a^{6} b^{2} d^{3} x^{6} + 3 \, a^{7} b d^{3} x^{4} + a^{8} d^{3} x^{2}\right )} \sqrt{d x} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{1}{4}} \log \left (-291434247 \, a^{16} d^{11} \left (-\frac{b^{5}}{a^{21} d^{14}}\right )^{\frac{3}{4}} + 291434247 \, \sqrt{d x} b^{4}\right )}{3840 \,{\left (a^{5} b^{3} d^{3} x^{8} + 3 \, a^{6} b^{2} d^{3} x^{6} + 3 \, a^{7} b d^{3} x^{4} + a^{8} d^{3} x^{2}\right )} \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^2*(d*x)^(7/2)),x, algorithm="fricas")

[Out]

1/3840*(39780*b^4*x^8 + 111384*a*b^3*x^6 + 99892*a^2*b^2*x^4 + 26112*a^3*b*x^2 -
1536*a^4 + 39780*(a^5*b^3*d^3*x^8 + 3*a^6*b^2*d^3*x^6 + 3*a^7*b*d^3*x^4 + a^8*d
^3*x^2)*sqrt(d*x)*(-b^5/(a^21*d^14))^(1/4)*arctan(291434247*a^16*d^11*(-b^5/(a^2
1*d^14))^(3/4)/(291434247*sqrt(d*x)*b^4 + sqrt(-84933920324457009*a^11*b^5*d^8*s
qrt(-b^5/(a^21*d^14)) + 84933920324457009*b^8*d*x))) + 9945*(a^5*b^3*d^3*x^8 + 3
*a^6*b^2*d^3*x^6 + 3*a^7*b*d^3*x^4 + a^8*d^3*x^2)*sqrt(d*x)*(-b^5/(a^21*d^14))^(
1/4)*log(291434247*a^16*d^11*(-b^5/(a^21*d^14))^(3/4) + 291434247*sqrt(d*x)*b^4)
- 9945*(a^5*b^3*d^3*x^8 + 3*a^6*b^2*d^3*x^6 + 3*a^7*b*d^3*x^4 + a^8*d^3*x^2)*sq
rt(d*x)*(-b^5/(a^21*d^14))^(1/4)*log(-291434247*a^16*d^11*(-b^5/(a^21*d^14))^(3/
4) + 291434247*sqrt(d*x)*b^4))/((a^5*b^3*d^3*x^8 + 3*a^6*b^2*d^3*x^6 + 3*a^7*b*d
^3*x^4 + a^8*d^3*x^2)*sqrt(d*x))

_______________________________________________________________________________________

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(1/(d*x)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.274729, size = 471, normalized size = 1.27 \[ \frac{663 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{6} b d^{5}} + \frac{663 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{6} b d^{5}} - \frac{663 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{6} b d^{5}} + \frac{663 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{6} b d^{5}} + \frac{453 \, \sqrt{d x} b^{4} d^{5} x^{5} + 1038 \, \sqrt{d x} a b^{3} d^{5} x^{3} + 617 \, \sqrt{d x} a^{2} b^{2} d^{5} x}{192 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{5} d^{3}} + \frac{2 \,{\left (20 \, b d^{2} x^{2} - a d^{2}\right )}}{5 \, \sqrt{d x} a^{5} d^{5} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^2*(d*x)^(7/2)),x, algorithm="giac")

[Out]

663/256*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^6*b*d^5) + 663/256*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^6*
b*d^5) - 663/512*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^6*b*d^5) + 663/512*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^6*b*d^5) + 1/192*(453*sqrt
(d*x)*b^4*d^5*x^5 + 1038*sqrt(d*x)*a*b^3*d^5*x^3 + 617*sqrt(d*x)*a^2*b^2*d^5*x)/
((b*d^2*x^2 + a*d^2)^3*a^5*d^3) + 2/5*(20*b*d^2*x^2 - a*d^2)/(sqrt(d*x)*a^5*d^5*
x^2)