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

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

[Out]

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

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Rubi [A]  time = 0.398931, 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 \[ \frac{(3 a B+A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{(3 a B+A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{(3 a B+A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} b^{7/4}}+\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} b^{7/4}}+\frac{x^{3/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 71.6248, size = 240, normalized size = 0.92 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{2 a b \left (a + b x^{2}\right )} + \frac{\sqrt{2} \left (A b + 3 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{5}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A b + 3 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{5}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A b + 3 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (A b + 3 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} b^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.307772, size = 228, normalized size = 0.87 \[ \frac{-\frac{8 \sqrt [4]{a} b^{3/4} x^{3/2} (a B-A b)}{a+b x^2}+\sqrt{2} (3 a B+A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-\sqrt{2} (3 a B+A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-2 \sqrt{2} (3 a B+A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+2 \sqrt{2} (3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{16 a^{5/4} b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 305, normalized size = 1.2 \[{\frac{Ab-Ba}{2\,ab \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{\sqrt{2}A}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}A}{16\,ab}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}A}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{16\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(A*b-B*a)*x^(3/2)/a/b/(b*x^2+a)+1/8/a/b/(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)*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)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+3/8/b^2/(a/b)^(1/4)*2^(1/
2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16/b^2/(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)))+3/8/b^2/(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)*sqrt(x)/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244753, size = 1072, normalized size = 4.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(4*(B*a - A*b)*x^(3/2) - 4*(a*b^2*x^2 + a^2*b)*(-(81*B^4*a^4 + 108*A*B^3*a^
3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(1/4)*arctan(a^4
*b^5*(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4
*b^4)/(a^5*b^7))^(3/4)/((27*B^3*a^3 + 27*A*B^2*a^2*b + 9*A^2*B*a*b^2 + A^3*b^3)*
sqrt(x) + sqrt((729*B^6*a^6 + 1458*A*B^5*a^5*b + 1215*A^2*B^4*a^4*b^2 + 540*A^3*
B^3*a^3*b^3 + 135*A^4*B^2*a^2*b^4 + 18*A^5*B*a*b^5 + A^6*b^6)*x - (81*B^4*a^7*b^
3 + 108*A*B^3*a^6*b^4 + 54*A^2*B^2*a^5*b^5 + 12*A^3*B*a^4*b^6 + A^4*a^3*b^7)*sqr
t(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4
)/(a^5*b^7))))) - (a*b^2*x^2 + a^2*b)*(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B
^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(1/4)*log(a^4*b^5*(-(81*B^4*a^
4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^
(3/4) + (27*B^3*a^3 + 27*A*B^2*a^2*b + 9*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) + (a*b^
2*x^2 + a^2*b)*(-(81*B^4*a^4 + 108*A*B^3*a^3*b + 54*A^2*B^2*a^2*b^2 + 12*A^3*B*a
*b^3 + A^4*b^4)/(a^5*b^7))^(1/4)*log(-a^4*b^5*(-(81*B^4*a^4 + 108*A*B^3*a^3*b +
54*A^2*B^2*a^2*b^2 + 12*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^7))^(3/4) + (27*B^3*a^3 +
27*A*B^2*a^2*b + 9*A^2*B*a*b^2 + A^3*b^3)*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)*x**(1/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.244316, size = 369, normalized size = 1.41 \[ -\frac{B a x^{\frac{3}{2}} - A b x^{\frac{3}{2}}}{2 \,{\left (b x^{2} + a\right )} a b} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b^{4}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b^{4}} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{4}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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