3.302 \(\int \frac{1}{x^{5/2} \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=230 \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4}}-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )} \]

[Out]

-7/(6*a^2*x^(3/2)) + 1/(2*a*x^(3/2)*(a + b*x^2)) + (7*b^(3/4)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)) - (7*b^(3/4)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)) + (7*b^(3/4)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)) - (7*b^(3/4)*Lo
g[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4))

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Rubi [A]  time = 0.421474, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4}}-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(a + b*x^2)^2),x]

[Out]

-7/(6*a^2*x^(3/2)) + 1/(2*a*x^(3/2)*(a + b*x^2)) + (7*b^(3/4)*ArcTan[1 - (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)) - (7*b^(3/4)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)) + (7*b^(3/4)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)) - (7*b^(3/4)*Lo
g[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4))

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Rubi in Sympy [A]  time = 68.1911, size = 218, normalized size = 0.95 \[ \frac{1}{2 a x^{\frac{3}{2}} \left (a + b x^{2}\right )} - \frac{7}{6 a^{2} x^{\frac{3}{2}}} + \frac{7 \sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{11}{4}}} - \frac{7 \sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{11}{4}}} + \frac{7 \sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}}} - \frac{7 \sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(b*x**2+a)**2,x)

[Out]

1/(2*a*x**(3/2)*(a + b*x**2)) - 7/(6*a**2*x**(3/2)) + 7*sqrt(2)*b**(3/4)*log(-sq
rt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(11/4)) - 7*sqrt(2
)*b**(3/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(
11/4)) + 7*sqrt(2)*b**(3/4)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(1
1/4)) - 7*sqrt(2)*b**(3/4)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(11
/4))

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Mathematica [A]  time = 0.337295, size = 212, normalized size = 0.92 \[ \frac{-\frac{24 a^{3/4} b \sqrt{x}}{a+b x^2}-\frac{32 a^{3/4}}{x^{3/2}}+21 \sqrt{2} b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-21 \sqrt{2} b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+42 \sqrt{2} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-42 \sqrt{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{48 a^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(a + b*x^2)^2),x]

[Out]

((-32*a^(3/4))/x^(3/2) - (24*a^(3/4)*b*Sqrt[x])/(a + b*x^2) + 42*Sqrt[2]*b^(3/4)
*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 42*Sqrt[2]*b^(3/4)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 21*Sqrt[2]*b^(3/4)*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 21*Sqrt[2]*b^(3/4)*Log[Sqrt[a] + Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(48*a^(11/4))

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Maple [A]  time = 0.02, size = 161, normalized size = 0.7 \[ -{\frac{2}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{b}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{7\,b\sqrt{2}}{16\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{7\,b\sqrt{2}}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{7\,b\sqrt{2}}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x^2+a)^2,x)

[Out]

-2/3/a^2/x^(3/2)-1/2*b/a^2*x^(1/2)/(b*x^2+a)-7/16*b/a^3*(a/b)^(1/4)*2^(1/2)*ln((
x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^
(1/2)))-7/8*b/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-7/8*
b/a^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*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(1/((b*x^2 + a)^2*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25439, size = 282, normalized size = 1.23 \[ -\frac{28 \, b x^{2} - 84 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{x} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}}}{b \sqrt{x} + \sqrt{a^{6} \sqrt{-\frac{b^{3}}{a^{11}}} + b^{2} x}}\right ) + 21 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{x} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} \log \left (7 \, a^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + 7 \, b \sqrt{x}\right ) - 21 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{x} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} \log \left (-7 \, a^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + 7 \, b \sqrt{x}\right ) + 16 \, a}{24 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*x^(5/2)),x, algorithm="fricas")

[Out]

-1/24*(28*b*x^2 - 84*(a^2*b*x^3 + a^3*x)*sqrt(x)*(-b^3/a^11)^(1/4)*arctan(a^3*(-
b^3/a^11)^(1/4)/(b*sqrt(x) + sqrt(a^6*sqrt(-b^3/a^11) + b^2*x))) + 21*(a^2*b*x^3
+ a^3*x)*sqrt(x)*(-b^3/a^11)^(1/4)*log(7*a^3*(-b^3/a^11)^(1/4) + 7*b*sqrt(x)) -
21*(a^2*b*x^3 + a^3*x)*sqrt(x)*(-b^3/a^11)^(1/4)*log(-7*a^3*(-b^3/a^11)^(1/4) +
7*b*sqrt(x)) + 16*a)/((a^2*b*x^3 + a^3*x)*sqrt(x))

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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(1/x**(5/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216723, size = 265, normalized size = 1.15 \[ -\frac{7 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3}} - \frac{7 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3}} - \frac{7 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3}} + \frac{7 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3}} - \frac{b \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a^{2}} - \frac{2}{3 \, a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^2*x^(5/2)),x, algorithm="giac")

[Out]

-7/8*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/
(a/b)^(1/4))/a^3 - 7/8*sqrt(2)*(a*b^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^
(1/4) - 2*sqrt(x))/(a/b)^(1/4))/a^3 - 7/16*sqrt(2)*(a*b^3)^(1/4)*ln(sqrt(2)*sqrt
(x)*(a/b)^(1/4) + x + sqrt(a/b))/a^3 + 7/16*sqrt(2)*(a*b^3)^(1/4)*ln(-sqrt(2)*sq
rt(x)*(a/b)^(1/4) + x + sqrt(a/b))/a^3 - 1/2*b*sqrt(x)/((b*x^2 + a)*a^2) - 2/3/(
a^2*x^(3/2))