3.380 \(\int \frac{A+B x^2}{x^{3/2} \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=289 \[ -\frac{(5 A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} b^{3/4}}+\frac{(5 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} b^{3/4}}+\frac{(5 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} b^{3/4}}-\frac{(5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 A b-a B}{2 a^2 b \sqrt{x}}+\frac{A b-a B}{2 a b \sqrt{x} \left (a+b x^2\right )} \]

[Out]

-(5*A*b - a*B)/(2*a^2*b*Sqrt[x]) + (A*b - a*B)/(2*a*b*Sqrt[x]*(a + b*x^2)) + ((5
*A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*b^
(3/4)) - ((5*A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2
]*a^(9/4)*b^(3/4)) - ((5*A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(3/4)) + ((5*A*b - a*B)*Log[Sqrt[a] + Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(3/4))

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Rubi [A]  time = 0.463966, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{(5 A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} b^{3/4}}+\frac{(5 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} b^{3/4}}+\frac{(5 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} b^{3/4}}-\frac{(5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} b^{3/4}}-\frac{5 A b-a B}{2 a^2 b \sqrt{x}}+\frac{A b-a B}{2 a b \sqrt{x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^2),x]

[Out]

-(5*A*b - a*B)/(2*a^2*b*Sqrt[x]) + (A*b - a*B)/(2*a*b*Sqrt[x]*(a + b*x^2)) + ((5
*A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*b^
(3/4)) - ((5*A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2
]*a^(9/4)*b^(3/4)) - ((5*A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(3/4)) + ((5*A*b - a*B)*Log[Sqrt[a] + Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(3/4))

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Rubi in Sympy [A]  time = 80.1584, size = 260, normalized size = 0.9 \[ \frac{A b - B a}{2 a b \sqrt{x} \left (a + b x^{2}\right )} - \frac{5 A b - B a}{2 a^{2} b \sqrt{x}} - \frac{\sqrt{2} \left (5 A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{9}{4}} b^{\frac{3}{4}}} + \frac{\sqrt{2} \left (5 A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{9}{4}} b^{\frac{3}{4}}} + \frac{\sqrt{2} \left (5 A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}} b^{\frac{3}{4}}} - \frac{\sqrt{2} \left (5 A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(3/2)/(b*x**2+a)**2,x)

[Out]

(A*b - B*a)/(2*a*b*sqrt(x)*(a + b*x**2)) - (5*A*b - B*a)/(2*a**2*b*sqrt(x)) - sq
rt(2)*(5*A*b - B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x
)/(16*a**(9/4)*b**(3/4)) + sqrt(2)*(5*A*b - B*a)*log(sqrt(2)*a**(1/4)*b**(1/4)*s
qrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(9/4)*b**(3/4)) + sqrt(2)*(5*A*b - B*a)*ata
n(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(9/4)*b**(3/4)) - sqrt(2)*(5*A*b
- B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(9/4)*b**(3/4))

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Mathematica [A]  time = 0.458713, size = 253, normalized size = 0.88 \[ \frac{\frac{\sqrt{2} (a B-5 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}+\frac{\sqrt{2} (5 A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}+\frac{2 \sqrt{2} (5 A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{3/4}}-\frac{2 \sqrt{2} (5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{3/4}}+\frac{8 \sqrt [4]{a} x^{3/2} (a B-A b)}{a+b x^2}-\frac{32 \sqrt [4]{a} A}{\sqrt{x}}}{16 a^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(3/2)*(a + b*x^2)^2),x]

[Out]

((-32*a^(1/4)*A)/Sqrt[x] + (8*a^(1/4)*(-(A*b) + a*B)*x^(3/2))/(a + b*x^2) + (2*S
qrt[2]*(5*A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(3/4) - (2
*Sqrt[2]*(5*A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(3/4) +
(Sqrt[2]*(-5*A*b + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*
x])/b^(3/4) + (Sqrt[2]*(5*A*b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x])/b^(3/4))/(16*a^(9/4))

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Maple [A]  time = 0.023, size = 323, normalized size = 1.1 \[ -2\,{\frac{A}{{a}^{2}\sqrt{x}}}-{\frac{Ab}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{B}{2\,a \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}-{\frac{5\,\sqrt{2}A}{16\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,\sqrt{2}A}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,\sqrt{2}A}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}B}{16\,ab}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}B}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}B}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(3/2)/(b*x^2+a)^2,x)

[Out]

-2*A/a^2/x^(1/2)-1/2/a^2*x^(3/2)/(b*x^2+a)*A*b+1/2/a*x^(3/2)/(b*x^2+a)*B-5/16/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)))-5/8/a^2/(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)
/(a/b)^(1/4)*x^(1/2)+1)-5/8/a^2/(a/b)^(1/4)*2^(1/2)*A*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)))+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)

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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*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259386, size = 1087, normalized size = 3.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(4*(B*a - 5*A*b)*x^2 - 4*(a^2*b*x^2 + a^3)*sqrt(x)*(-(B^4*a^4 - 20*A*B^3*a^3
*b + 150*A^2*B^2*a^2*b^2 - 500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^3))^(1/4)*arcta
n(-a^7*b^2*(-(B^4*a^4 - 20*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 500*A^3*B*a*b^3 +
625*A^4*b^4)/(a^9*b^3))^(3/4)/((B^3*a^3 - 15*A*B^2*a^2*b + 75*A^2*B*a*b^2 - 125
*A^3*b^3)*sqrt(x) - sqrt((B^6*a^6 - 30*A*B^5*a^5*b + 375*A^2*B^4*a^4*b^2 - 2500*
A^3*B^3*a^3*b^3 + 9375*A^4*B^2*a^2*b^4 - 18750*A^5*B*a*b^5 + 15625*A^6*b^6)*x -
(B^4*a^9*b - 20*A*B^3*a^8*b^2 + 150*A^2*B^2*a^7*b^3 - 500*A^3*B*a^6*b^4 + 625*A^
4*a^5*b^5)*sqrt(-(B^4*a^4 - 20*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 500*A^3*B*a*b
^3 + 625*A^4*b^4)/(a^9*b^3))))) - (a^2*b*x^2 + a^3)*sqrt(x)*(-(B^4*a^4 - 20*A*B^
3*a^3*b + 150*A^2*B^2*a^2*b^2 - 500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^3))^(1/4)*
log(a^7*b^2*(-(B^4*a^4 - 20*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 500*A^3*B*a*b^3
+ 625*A^4*b^4)/(a^9*b^3))^(3/4) - (B^3*a^3 - 15*A*B^2*a^2*b + 75*A^2*B*a*b^2 - 1
25*A^3*b^3)*sqrt(x)) + (a^2*b*x^2 + a^3)*sqrt(x)*(-(B^4*a^4 - 20*A*B^3*a^3*b + 1
50*A^2*B^2*a^2*b^2 - 500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^3))^(1/4)*log(-a^7*b^
2*(-(B^4*a^4 - 20*A*B^3*a^3*b + 150*A^2*B^2*a^2*b^2 - 500*A^3*B*a*b^3 + 625*A^4*
b^4)/(a^9*b^3))^(3/4) - (B^3*a^3 - 15*A*B^2*a^2*b + 75*A^2*B*a*b^2 - 125*A^3*b^3
)*sqrt(x)) - 16*A*a)/((a^2*b*x^2 + a^3)*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((B*x**2+A)/x**(3/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.255788, size = 375, normalized size = 1.3 \[ \frac{B a x^{2} - 5 \, A b x^{2} - 4 \, A a}{2 \,{\left (b x^{\frac{5}{2}} + a \sqrt{x}\right )} a^{2}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b^{3}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*a*x^2 - 5*A*b*x^2 - 4*A*a)/((b*x^(5/2) + a*sqrt(x))*a^2) + 1/8*sqrt(2)*((
a*b^3)^(3/4)*B*a - 5*(a*b^3)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4)
+ 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^3) + 1/8*sqrt(2)*((a*b^3)^(3/4)*B*a - 5*(a*b^3)
^(3/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(
a^3*b^3) - 1/16*sqrt(2)*((a*b^3)^(3/4)*B*a - 5*(a*b^3)^(3/4)*A*b)*ln(sqrt(2)*sqr
t(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3) + 1/16*sqrt(2)*((a*b^3)^(3/4)*B*a -
5*(a*b^3)^(3/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^3)