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

Optimal. Leaf size=237 \[ \frac{(A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{(A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{(A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{2 B x^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.405857, antiderivative size = 237, 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 \[ \frac{(A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{(A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{(A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{2 B x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 70.0701, size = 221, normalized size = 0.93 \[ \frac{2 B x^{\frac{3}{2}}}{3 b} + \frac{\sqrt{2} \left (A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 \sqrt [4]{a} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 \sqrt [4]{a} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 \sqrt [4]{a} b^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x**(3/2)/(3*b) + sqrt(2)*(A*b - B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x)
+ sqrt(a) + sqrt(b)*x)/(4*a**(1/4)*b**(7/4)) - sqrt(2)*(A*b - B*a)*log(sqrt(2)*a
**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(1/4)*b**(7/4)) - sqrt(2)*
(A*b - B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(1/4)*b**(7/4)) +
sqrt(2)*(A*b - B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*a**(1/4)*b**(
7/4))

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Mathematica [A]  time = 0.1969, size = 213, normalized size = 0.9 \[ \frac{3 \sqrt{2} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-3 \sqrt{2} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-6 \sqrt{2} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+6 \sqrt{2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+8 \sqrt [4]{a} b^{3/4} B x^{3/2}}{12 \sqrt [4]{a} b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

(8*a^(1/4)*b^(3/4)*B*x^(3/2) - 6*Sqrt[2]*(A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)] + 6*Sqrt[2]*(A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/
a^(1/4)] + 3*Sqrt[2]*(A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x] - 3*Sqrt[2]*(A*b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/(12*a^(1/4)*b^(7/4))

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Maple [A]  time = 0.011, size = 280, normalized size = 1.2 \[{\frac{2\,B}{3\,b}{x}^{{\frac{3}{2}}}}+{\frac{\sqrt{2}A}{2\,b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}A}{4\,b}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}A}{2\,b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}Ba}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}Ba}{4\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}Ba}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251612, size = 977, normalized size = 4.12 \[ \frac{4 \, B x^{\frac{3}{2}} + 12 \, b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a b^{7}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a b^{5} \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a b^{7}}\right )^{\frac{3}{4}}}{{\left (B^{3} a^{3} - 3 \, A B^{2} a^{2} b + 3 \, A^{2} B a b^{2} - A^{3} b^{3}\right )} \sqrt{x} - \sqrt{{\left (B^{6} a^{6} - 6 \, A B^{5} a^{5} b + 15 \, A^{2} B^{4} a^{4} b^{2} - 20 \, A^{3} B^{3} a^{3} b^{3} + 15 \, A^{4} B^{2} a^{2} b^{4} - 6 \, A^{5} B a b^{5} + A^{6} b^{6}\right )} x -{\left (B^{4} a^{5} b^{3} - 4 \, A B^{3} a^{4} b^{4} + 6 \, A^{2} B^{2} a^{3} b^{5} - 4 \, A^{3} B a^{2} b^{6} + A^{4} a b^{7}\right )} \sqrt{-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a b^{7}}}}}\right ) + 3 \, b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a b^{7}}\right )^{\frac{1}{4}} \log \left (a b^{5} \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a b^{7}}\right )^{\frac{3}{4}} -{\left (B^{3} a^{3} - 3 \, A B^{2} a^{2} b + 3 \, A^{2} B a b^{2} - A^{3} b^{3}\right )} \sqrt{x}\right ) - 3 \, b \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a b^{7}}\right )^{\frac{1}{4}} \log \left (-a b^{5} \left (-\frac{B^{4} a^{4} - 4 \, A B^{3} a^{3} b + 6 \, A^{2} B^{2} a^{2} b^{2} - 4 \, A^{3} B a b^{3} + A^{4} b^{4}}{a b^{7}}\right )^{\frac{3}{4}} -{\left (B^{3} a^{3} - 3 \, A B^{2} a^{2} b + 3 \, A^{2} B a b^{2} - A^{3} b^{3}\right )} \sqrt{x}\right )}{6 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(4*B*x^(3/2) + 12*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B
*a*b^3 + A^4*b^4)/(a*b^7))^(1/4)*arctan(-a*b^5*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^
2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a*b^7))^(3/4)/((B^3*a^3 - 3*A*B^2*a^2*
b + 3*A^2*B*a*b^2 - A^3*b^3)*sqrt(x) - sqrt((B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^
4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)*x
- (B^4*a^5*b^3 - 4*A*B^3*a^4*b^4 + 6*A^2*B^2*a^3*b^5 - 4*A^3*B*a^2*b^6 + A^4*a*
b^7)*sqrt(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^
4)/(a*b^7))))) + 3*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*
b^3 + A^4*b^4)/(a*b^7))^(1/4)*log(a*b^5*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a
^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a*b^7))^(3/4) - (B^3*a^3 - 3*A*B^2*a^2*b + 3*
A^2*B*a*b^2 - A^3*b^3)*sqrt(x)) - 3*b*(-(B^4*a^4 - 4*A*B^3*a^3*b + 6*A^2*B^2*a^2
*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a*b^7))^(1/4)*log(-a*b^5*(-(B^4*a^4 - 4*A*B^3*a
^3*b + 6*A^2*B^2*a^2*b^2 - 4*A^3*B*a*b^3 + A^4*b^4)/(a*b^7))^(3/4) - (B^3*a^3 -
3*A*B^2*a^2*b + 3*A^2*B*a*b^2 - A^3*b^3)*sqrt(x)))/b

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.324584, size = 339, normalized size = 1.43 \[ \frac{2 \, B x^{\frac{3}{2}}}{3 \, b} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*B*x^(3/2)/b - 1/2*sqrt(2)*((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*b^4) - 1/2*sqrt(2)*((
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*b^4) + 1/4*sqrt(2)*((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*b^4) - 1/4*sqrt(2)*((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*b^4)