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

Optimal. Leaf size=458 \[ -\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-7*d^3*(d*x)^(3/2))/(16*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(7/2))/
(4*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (21*d^(9/2)*(a + b*x^2)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(1/4)*b^(11
/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (21*d^(9/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(1/4)*b^(11/4)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) + (21*d^(9/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*
Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(1/4)*b^(11/4)*Sqr
t[a^2 + 2*a*b*x^2 + b^2*x^4]) - (21*d^(9/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sq
rt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(1/4)*b^(11/
4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.72581, antiderivative size = 458, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{7 d^3 (d x)^{3/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{7/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 d^{9/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-7*d^3*(d*x)^(3/2))/(16*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(7/2))/
(4*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (21*d^(9/2)*(a + b*x^2)*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(1/4)*b^(11
/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (21*d^(9/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(1/4)*b^(11/4)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4]) + (21*d^(9/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*
Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(1/4)*b^(11/4)*Sqr
t[a^2 + 2*a*b*x^2 + b^2*x^4]) - (21*d^(9/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sq
rt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(1/4)*b^(11/
4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

_______________________________________________________________________________________

Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

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)**(3/2),x)

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.408292, size = 434, normalized size = 0.95 \[ \frac{21 (d x)^{9/2} \left (a+b x^2\right )^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} x^{9/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{21 (d x)^{9/2} \left (a+b x^2\right )^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} \sqrt [4]{a} b^{11/4} x^{9/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{21 (d x)^{9/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{2 \sqrt [4]{b} \sqrt{x}-\sqrt{2} \sqrt [4]{a}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} x^{9/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{21 (d x)^{9/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt{x}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} \sqrt [4]{a} b^{11/4} x^{9/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{11 (d x)^{9/2} \left (a+b x^2\right )^2}{16 b^2 x^3 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{a (d x)^{9/2} \left (a+b x^2\right )}{4 b^2 x^3 \left (\left (a+b x^2\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*(d*x)^(9/2)*(a + b*x^2))/(4*b^2*x^3*((a + b*x^2)^2)^(3/2)) - (11*(d*x)^(9/2)*
(a + b*x^2)^2)/(16*b^2*x^3*((a + b*x^2)^2)^(3/2)) + (21*(d*x)^(9/2)*(a + b*x^2)^
3*ArcTan[(-(Sqrt[2]*a^(1/4)) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(32*Sqrt[2
]*a^(1/4)*b^(11/4)*x^(9/2)*((a + b*x^2)^2)^(3/2)) + (21*(d*x)^(9/2)*(a + b*x^2)^
3*ArcTan[(Sqrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(32*Sqrt[2]*a
^(1/4)*b^(11/4)*x^(9/2)*((a + b*x^2)^2)^(3/2)) + (21*(d*x)^(9/2)*(a + b*x^2)^3*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(1/4)*b
^(11/4)*x^(9/2)*((a + b*x^2)^2)^(3/2)) - (21*(d*x)^(9/2)*(a + b*x^2)^3*Log[Sqrt[
a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(1/4)*b^(11/4)*
x^(9/2)*((a + b*x^2)^2)^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.025, size = 609, normalized size = 1.3 \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

-1/128*(-21*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*x^4*b^2*d^4-42*2^(1
/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*b^2*d^4+42
*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*b^2*
d^4+88*(a*d^2/b)^(1/4)*(d*x)^(7/2)*b^2-42*2^(1/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/
2)*2^(1/2)-d*x-(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)))*x^2*a*b*d^4-84*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(
a*d^2/b)^(1/4))*x^2*a*b*d^4+84*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1
/4))/(a*d^2/b)^(1/4))*x^2*a*b*d^4+56*(a*d^2/b)^(1/4)*(d*x)^(3/2)*a*b*d^2-21*2^(1
/2)*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*a^2*d^4-42*2^(1/2)*arctan((2^(1/2)*
(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^2*d^4+42*2^(1/2)*arctan((-2^(1/2
)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^2*d^4)*d*(b*x^2+a)/(a*d^2/b)^(
1/4)/b^3/((b*x^2+a)^2)^(3/2)

_______________________________________________________________________________________

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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.298409, size = 396, normalized size = 0.86 \[ \frac{84 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{3}{4}} a b^{8}}{\sqrt{d x} d^{13} + \sqrt{d^{27} x - \sqrt{-\frac{d^{18}}{a b^{11}}} a b^{5} d^{18}}}\right ) + 21 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{1}{4}} \log \left (9261 \, \sqrt{d x} d^{13} + 9261 \, \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{3}{4}} a b^{8}\right ) - 21 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{1}{4}} \log \left (9261 \, \sqrt{d x} d^{13} - 9261 \, \left (-\frac{d^{18}}{a b^{11}}\right )^{\frac{3}{4}} a b^{8}\right ) - 4 \,{\left (11 \, b d^{4} x^{3} + 7 \, a d^{4} x\right )} \sqrt{d x}}{64 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

1/64*(84*(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)*(-d^18/(a*b^11))^(1/4)*arctan((-d^18/
(a*b^11))^(3/4)*a*b^8/(sqrt(d*x)*d^13 + sqrt(d^27*x - sqrt(-d^18/(a*b^11))*a*b^5
*d^18))) + 21*(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)*(-d^18/(a*b^11))^(1/4)*log(9261*
sqrt(d*x)*d^13 + 9261*(-d^18/(a*b^11))^(3/4)*a*b^8) - 21*(b^4*x^4 + 2*a*b^3*x^2
+ a^2*b^2)*(-d^18/(a*b^11))^(1/4)*log(9261*sqrt(d*x)*d^13 - 9261*(-d^18/(a*b^11)
)^(3/4)*a*b^8) - 4*(11*b*d^4*x^3 + 7*a*d^4*x)*sqrt(d*x))/(b^4*x^4 + 2*a*b^3*x^2
+ a^2*b^2)

_______________________________________________________________________________________

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)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.291303, size = 497, normalized size = 1.09 \[ -\frac{1}{128} \, d^{3}{\left (\frac{8 \,{\left (11 \, \sqrt{d x} b d^{5} x^{3} + 7 \, \sqrt{d x} a d^{5} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{42 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{42 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{21 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{21 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

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)^(3/2),x, algorithm="giac")

[Out]

-1/128*d^3*(8*(11*sqrt(d*x)*b*d^5*x^3 + 7*sqrt(d*x)*a*d^5*x)/((b*d^2*x^2 + a*d^2
)^2*b^2*sign(b*d^4*x^2 + a*d^4)) - 42*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^5*sign(b*d^4*x^
2 + a*d^4)) - 42*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^5*sign(b*d^4*x^2 + a*d^4)) + 21*sqr
t(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^5*sign(b*d^4*x^2 + a*d^4)) - 21*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^5*sign(b*d^4*x^2 + a*d^4)))