3.761 \(\int \frac{(d x)^{7/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=458 \[ -\frac{5 d^3 \sqrt{d x}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-5*d^3*Sqrt[d*x])/(16*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(5/2))/(4
*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*d^(7/2)*(a + b*x^2)*ArcTan[
1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(3/4)*b^(9/4)*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (5*d^(7/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^
(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(3/4)*b^(9/4)*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4]) - (5*d^(7/2)*(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^(3/4)*b^(9/4)*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) + (5*d^(7/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(3/4)*b^(9/4)*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.727506, 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 \[ -\frac{5 d^3 \sqrt{d x}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(-5*d^3*Sqrt[d*x])/(16*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(5/2))/(4
*b*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (5*d^(7/2)*(a + b*x^2)*ArcTan[
1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(3/4)*b^(9/4)*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (5*d^(7/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^
(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(3/4)*b^(9/4)*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4]) - (5*d^(7/2)*(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^(3/4)*b^(9/4)*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) + (5*d^(7/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(3/4)*b^(9/4)*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)**(7/2)/(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 = 0.407838, size = 272, normalized size = 0.59 \[ \frac{(d x)^{7/2} \left (a+b x^2\right ) \left (-72 a^{3/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )+32 a^{7/4} \sqrt [4]{b} \sqrt{x}-5 \sqrt{2} \left (a+b x^2\right )^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} \left (a+b x^2\right )^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-10 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+10 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{128 a^{3/4} b^{9/4} x^{7/2} \left (\left (a+b x^2\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

((d*x)^(7/2)*(a + b*x^2)*(32*a^(7/4)*b^(1/4)*Sqrt[x] - 72*a^(3/4)*b^(1/4)*Sqrt[x
]*(a + b*x^2) - 10*Sqrt[2]*(a + b*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^
(1/4)] + 10*Sqrt[2]*(a + b*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
- 5*Sqrt[2]*(a + b*x^2)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b
]*x] + 5*Sqrt[2]*(a + b*x^2)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x]))/(128*a^(3/4)*b^(9/4)*x^(7/2)*((a + b*x^2)^2)^(3/2))

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Maple [B]  time = 0.024, size = 672, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

1/128*(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))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*x^4*b^
2*d^2+10*(a*d^2/b)^(1/4)*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^2-10*(a*d^2/b)^(1/4)*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^2+10*(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))/((a*d^2/b)^(1/4)*(d*x)
^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*x^2*a*b*d^2+20*(a*d^2/b)^(1/4)*2^(1/2)*arct
an((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2*a*b*d^2-20*(a*d^2/
b)^(1/4)*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^2+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))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*a
^2*d^2+10*(a*d^2/b)^(1/4)*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^2-10*(a*d^2/b)^(1/4)*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^2-72*(d*x)^(5/2)*a*b-40*(d*x)^(1/2)*a^2*
d^2)*d*(b*x^2+a)/a/b^2/((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)^(7/2)/(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 = 0.300424, size = 392, normalized size = 0.86 \[ -\frac{20 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} a b^{2}}{\sqrt{d x} d^{3} + \sqrt{d^{7} x + \sqrt{-\frac{d^{14}}{a^{3} b^{9}}} a^{2} b^{4}}}\right ) - 5 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \log \left (5 \, \sqrt{d x} d^{3} + 5 \, \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} a b^{2}\right ) + 5 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \log \left (5 \, \sqrt{d x} d^{3} - 5 \, \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} a b^{2}\right ) + 4 \,{\left (9 \, b d^{3} x^{2} + 5 \, a d^{3}\right )} \sqrt{d x}}{64 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="fricas")

[Out]

-1/64*(20*(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)*(-d^14/(a^3*b^9))^(1/4)*arctan((-d^1
4/(a^3*b^9))^(1/4)*a*b^2/(sqrt(d*x)*d^3 + sqrt(d^7*x + sqrt(-d^14/(a^3*b^9))*a^2
*b^4))) - 5*(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)*(-d^14/(a^3*b^9))^(1/4)*log(5*sqrt
(d*x)*d^3 + 5*(-d^14/(a^3*b^9))^(1/4)*a*b^2) + 5*(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^
2)*(-d^14/(a^3*b^9))^(1/4)*log(5*sqrt(d*x)*d^3 - 5*(-d^14/(a^3*b^9))^(1/4)*a*b^2
) + 4*(9*b*d^3*x^2 + 5*a*d^3)*sqrt(d*x))/(b^4*x^4 + 2*a*b^3*x^2 + a^2*b^2)

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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)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.288672, size = 501, normalized size = 1.09 \[ \frac{1}{128} \, d^{2}{\left (\frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (9 \, \sqrt{d x} b d^{5} x^{2} + 5 \, \sqrt{d x} a d^{5}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="giac")

[Out]

1/128*d^2*(10*sqrt(2)*(a*b^3*d^2)^(1/4)*d*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^3*sign(b*d^4*x^2 + a*d^4)) + 10*sqrt(
2)*(a*b^3*d^2)^(1/4)*d*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^3*sign(b*d^4*x^2 + a*d^4)) + 5*sqrt(2)*(a*b^3*d^2)^(1/4
)*d*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^3*sign(b*d^
4*x^2 + a*d^4)) - 5*sqrt(2)*(a*b^3*d^2)^(1/4)*d*ln(d*x - sqrt(2)*(a*d^2/b)^(1/4)
*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^3*sign(b*d^4*x^2 + a*d^4)) - 8*(9*sqrt(d*x)*b*d
^5*x^2 + 5*sqrt(d*x)*a*d^5)/((b*d^2*x^2 + a*d^2)^2*b^2*sign(b*d^4*x^2 + a*d^4)))