3.762 \(\int \frac{(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=459 \[ \frac{3 d (d x)^{3/2}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{5/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(3*d*(d*x)^(3/2))/(16*a*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(3/2))/(4*
b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*d^(5/2)*(a + b*x^2)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(5/4)*b^(7/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3*d^(5/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(
1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(5/4)*b^(7/4)*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4]) + (3*d^(5/2)*(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^(5/4)*b^(7/4)*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4]) - (3*d^(5/2)*(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^(5/4)*b^(7/4)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.738773, antiderivative size = 459, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 d (d x)^{3/2}}{16 a b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{3/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{5/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{5/4} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(3*d*(d*x)^(3/2))/(16*a*b*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(3/2))/(4*
b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3*d^(5/2)*(a + b*x^2)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(5/4)*b^(7/4)*S
qrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3*d^(5/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(
1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(5/4)*b^(7/4)*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4]) + (3*d^(5/2)*(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^(5/4)*b^(7/4)*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4]) - (3*d^(5/2)*(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^(5/4)*b^(7/4)*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])

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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)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.782528, size = 436, normalized size = 0.95 \[ \frac{3 (d x)^{5/2} \left (a+b x^2\right )^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{5/4} b^{7/4} x^{5/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{3 (d x)^{5/2} \left (a+b x^2\right )^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{5/4} b^{7/4} x^{5/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{3 (d x)^{5/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{2 \sqrt [4]{b} \sqrt{x}-\sqrt{2} \sqrt [4]{a}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} a^{5/4} b^{7/4} x^{5/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{3 (d x)^{5/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt{x}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} a^{5/4} b^{7/4} x^{5/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{3 (d x)^{5/2} \left (a+b x^2\right )^2}{16 a b x \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{(d x)^{5/2} \left (a+b x^2\right )}{4 b x \left (\left (a+b x^2\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

-((d*x)^(5/2)*(a + b*x^2))/(4*b*x*((a + b*x^2)^2)^(3/2)) + (3*(d*x)^(5/2)*(a + b
*x^2)^2)/(16*a*b*x*((a + b*x^2)^2)^(3/2)) + (3*(d*x)^(5/2)*(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^(5/4)
*b^(7/4)*x^(5/2)*((a + b*x^2)^2)^(3/2)) + (3*(d*x)^(5/2)*(a + b*x^2)^3*ArcTan[(S
qrt[2]*a^(1/4) + 2*b^(1/4)*Sqrt[x])/(Sqrt[2]*a^(1/4))])/(32*Sqrt[2]*a^(5/4)*b^(7
/4)*x^(5/2)*((a + b*x^2)^2)^(3/2)) + (3*(d*x)^(5/2)*(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^(5/4)*b^(7/4)*x^(5/2
)*((a + b*x^2)^2)^(3/2)) - (3*(d*x)^(5/2)*(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^(5/4)*b^(7/4)*x^(5/2)*((a + b*
x^2)^2)^(3/2))

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Maple [B]  time = 0.024, size = 614, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

1/128*(3*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+6*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^4-6*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+2
4*(a*d^2/b)^(1/4)*(d*x)^(7/2)*b^2+6*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+12*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^4-12*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^4-8*(a*d^2/b)^(1/4)*(d*x)^(3/2)*a*b*d^2+3*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+6*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^4-6*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^2/a
/((b*x^2+a)^2)^(3/2)

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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)^(5/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298193, size = 416, normalized size = 0.91 \[ \frac{12 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{10}}{a^{5} b^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{27 \, a^{4} b^{5} \left (-\frac{d^{10}}{a^{5} b^{7}}\right )^{\frac{3}{4}}}{27 \, \sqrt{d x} d^{7} + \sqrt{-729 \, a^{3} b^{3} d^{10} \sqrt{-\frac{d^{10}}{a^{5} b^{7}}} + 729 \, d^{15} x}}\right ) + 3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{10}}{a^{5} b^{7}}\right )^{\frac{1}{4}} \log \left (27 \, a^{4} b^{5} \left (-\frac{d^{10}}{a^{5} b^{7}}\right )^{\frac{3}{4}} + 27 \, \sqrt{d x} d^{7}\right ) - 3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{d^{10}}{a^{5} b^{7}}\right )^{\frac{1}{4}} \log \left (-27 \, a^{4} b^{5} \left (-\frac{d^{10}}{a^{5} b^{7}}\right )^{\frac{3}{4}} + 27 \, \sqrt{d x} d^{7}\right ) + 4 \,{\left (3 \, b d^{2} x^{3} - a d^{2} x\right )} \sqrt{d x}}{64 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(5/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="fricas")

[Out]

1/64*(12*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-d^10/(a^5*b^7))^(1/4)*arctan(27*a
^4*b^5*(-d^10/(a^5*b^7))^(3/4)/(27*sqrt(d*x)*d^7 + sqrt(-729*a^3*b^3*d^10*sqrt(-
d^10/(a^5*b^7)) + 729*d^15*x))) + 3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-d^10/(
a^5*b^7))^(1/4)*log(27*a^4*b^5*(-d^10/(a^5*b^7))^(3/4) + 27*sqrt(d*x)*d^7) - 3*(
a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-d^10/(a^5*b^7))^(1/4)*log(-27*a^4*b^5*(-d^1
0/(a^5*b^7))^(3/4) + 27*sqrt(d*x)*d^7) + 4*(3*b*d^2*x^3 - a*d^2*x)*sqrt(d*x))/(a
*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{\frac{5}{2}}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral((d*x)**(5/2)/((a + b*x**2)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.290692, size = 498, normalized size = 1.08 \[ \frac{1}{128} \, d{\left (\frac{8 \,{\left (3 \, \sqrt{d x} b d^{5} x^{3} - \sqrt{d x} a d^{5} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a b{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{6 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{2} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{6 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{2} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{2} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{2} b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(5/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="giac")

[Out]

1/128*d*(8*(3*sqrt(d*x)*b*d^5*x^3 - sqrt(d*x)*a*d^5*x)/((b*d^2*x^2 + a*d^2)^2*a*
b*sign(b*d^4*x^2 + a*d^4)) + 6*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqr
t(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^2*b^4*sign(b*d^4*x^2 + a
*d^4)) + 6*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^4*sign(b*d^4*x^2 + a*d^4)) - 3*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^4*sign(b*d^4*x^2 + a*d^4)) + 3*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^4*sign(b*d^4*x^2 + a*d^4)))