3.379 \(\int \frac{A+B x^2}{\sqrt{x} \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=261 \[ -\frac{(a B+3 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a B+3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{5/4}}-\frac{(a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\sqrt{x} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[Out]

((A*b - a*B)*Sqrt[x])/(2*a*b*(a + b*x^2)) - ((3*A*b + a*B)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*b^(5/4)) + ((3*A*b + a*B)*ArcTan[1
+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*b^(5/4)) - ((3*A*b + a*
B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4
)*b^(5/4)) + ((3*A*b + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt
[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(5/4))

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Rubi [A]  time = 0.396892, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{(a B+3 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a B+3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} b^{5/4}}-\frac{(a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} b^{5/4}}+\frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\sqrt{x} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^2),x]

[Out]

((A*b - a*B)*Sqrt[x])/(2*a*b*(a + b*x^2)) - ((3*A*b + a*B)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*b^(5/4)) + ((3*A*b + a*B)*ArcTan[1
+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*b^(5/4)) - ((3*A*b + a*
B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4
)*b^(5/4)) + ((3*A*b + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt
[b]*x])/(8*Sqrt[2]*a^(7/4)*b^(5/4))

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Rubi in Sympy [A]  time = 69.8223, size = 240, normalized size = 0.92 \[ \frac{\sqrt{x} \left (A b - B a\right )}{2 a b \left (a + b x^{2}\right )} - \frac{\sqrt{2} \left (3 A b + B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{7}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (3 A b + B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{7}{4}} b^{\frac{5}{4}}} - \frac{\sqrt{2} \left (3 A b + B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (3 A b + B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)*(A*b - B*a)/(2*a*b*(a + b*x**2)) - sqrt(2)*(3*A*b + B*a)*log(-sqrt(2)*a*
*(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(7/4)*b**(5/4)) + sqrt(2)*
(3*A*b + B*a)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a
**(7/4)*b**(5/4)) - sqrt(2)*(3*A*b + B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(
1/4))/(8*a**(7/4)*b**(5/4)) + sqrt(2)*(3*A*b + B*a)*atan(1 + sqrt(2)*b**(1/4)*sq
rt(x)/a**(1/4))/(8*a**(7/4)*b**(5/4))

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Mathematica [A]  time = 0.299087, size = 228, normalized size = 0.87 \[ \frac{-\frac{8 a^{3/4} \sqrt [4]{b} \sqrt{x} (a B-A b)}{a+b x^2}-\sqrt{2} (a B+3 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+\sqrt{2} (a B+3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-2 \sqrt{2} (a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+2 \sqrt{2} (a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{16 a^{7/4} b^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^2),x]

[Out]

((-8*a^(3/4)*b^(1/4)*(-(A*b) + a*B)*Sqrt[x])/(a + b*x^2) - 2*Sqrt[2]*(3*A*b + a*
B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*Sqrt[2]*(3*A*b + a*B)*ArcTa
n[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - Sqrt[2]*(3*A*b + a*B)*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + Sqrt[2]*(3*A*b + a*B)*Log[Sqrt[a]
+ Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(16*a^(7/4)*b^(5/4))

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Maple [A]  time = 0.017, size = 305, normalized size = 1.2 \[{\frac{Ab-Ba}{2\,ab \left ( b{x}^{2}+a \right ) }\sqrt{x}}+{\frac{3\,\sqrt{2}A}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}A}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}A}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}B}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}B}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{\sqrt{2}B}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/(b*x^2+a)^2/x^(1/2),x)

[Out]

1/2*(A*b-B*a)*x^(1/2)/a/b/(b*x^2+a)+3/8/a^2*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)
/(a/b)^(1/4)*x^(1/2)+1)+3/8/a^2*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)
*x^(1/2)-1)+3/16/a^2*(a/b)^(1/4)*2^(1/2)*A*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)))+1/8/a/b*(a/b)^(1/4)*2^(1/
2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/8/a/b*(a/b)^(1/4)*2^(1/2)*B*arctan(
2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+1/16/a/b*(a/b)^(1/4)*2^(1/2)*B*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)))

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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((B*x^2 + A)/((b*x^2 + a)^2*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24986, size = 801, normalized size = 3.07 \[ -\frac{4 \,{\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac{B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} b \left (-\frac{B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}}}{{\left (B a + 3 \, A b\right )} \sqrt{x} + \sqrt{a^{4} b^{2} \sqrt{-\frac{B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}} +{\left (B^{2} a^{2} + 6 \, A B a b + 9 \, A^{2} b^{2}\right )} x}}\right ) -{\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac{B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (a^{2} b \left (-\frac{B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} +{\left (B a + 3 \, A b\right )} \sqrt{x}\right ) +{\left (a b^{2} x^{2} + a^{2} b\right )} \left (-\frac{B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (-a^{2} b \left (-\frac{B^{4} a^{4} + 12 \, A B^{3} a^{3} b + 54 \, A^{2} B^{2} a^{2} b^{2} + 108 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{5}}\right )^{\frac{1}{4}} +{\left (B a + 3 \, A b\right )} \sqrt{x}\right ) + 4 \,{\left (B a - A b\right )} \sqrt{x}}{8 \,{\left (a b^{2} x^{2} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^2*sqrt(x)),x, algorithm="fricas")

[Out]

-1/8*(4*(a*b^2*x^2 + a^2*b)*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 1
08*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)*arctan(a^2*b*(-(B^4*a^4 + 12*A*B^3
*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)/((B
*a + 3*A*b)*sqrt(x) + sqrt(a^4*b^2*sqrt(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*
a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5)) + (B^2*a^2 + 6*A*B*a*b + 9*A^
2*b^2)*x))) - (a*b^2*x^2 + a^2*b)*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b
^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)*log(a^2*b*(-(B^4*a^4 + 12*A*
B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)
+ (B*a + 3*A*b)*sqrt(x)) + (a*b^2*x^2 + a^2*b)*(-(B^4*a^4 + 12*A*B^3*a^3*b + 54*
A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^7*b^5))^(1/4)*log(-a^2*b*(-(B
^4*a^4 + 12*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 108*A^3*B*a*b^3 + 81*A^4*b^4)/(a^
7*b^5))^(1/4) + (B*a + 3*A*b)*sqrt(x)) + 4*(B*a - A*b)*sqrt(x))/(a*b^2*x^2 + a^2
*b)

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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((B*x**2+A)/(b*x**2+a)**2/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.244352, size = 369, normalized size = 1.41 \[ \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b^{2}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b^{2}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{2}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{2}} - \frac{B a \sqrt{x} - A b \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^2*sqrt(x)),x, algorithm="giac")

[Out]

1/8*sqrt(2)*((a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2
)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^2) + 1/8*sqrt(2)*((a*b^3)^(1/4)*B
*a + 3*(a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/
(a/b)^(1/4))/(a^2*b^2) + 1/16*sqrt(2)*((a*b^3)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*
ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^2) - 1/16*sqrt(2)*((a*b^3
)^(1/4)*B*a + 3*(a*b^3)^(1/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/
b))/(a^2*b^2) - 1/2*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^2 + a)*a*b)