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

Optimal. Leaf size=255 \[ \frac{\sqrt [4]{a} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}+\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{9/4}}+\frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{2 B x^{5/2}}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.443393, antiderivative size = 255, 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 \[ \frac{\sqrt [4]{a} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{9/4}}+\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{9/4}}-\frac{\sqrt [4]{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{9/4}}+\frac{2 \sqrt{x} (A b-a B)}{b^2}+\frac{2 B x^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 78.6494, size = 238, normalized size = 0.93 \[ \frac{2 B x^{\frac{5}{2}}}{5 b} + \frac{\sqrt{2} \sqrt [4]{a} \left (A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{9}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \left (A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{9}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \left (A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{9}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \left (A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{9}{4}}} + \frac{2 \sqrt{x} \left (A b - B a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x**(5/2)/(5*b) + sqrt(2)*a**(1/4)*(A*b - B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)
*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*b**(9/4)) - sqrt(2)*a**(1/4)*(A*b - B*a)*log(
sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*b**(9/4)) + sqrt(2)*
a**(1/4)*(A*b - B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b**(9/4)) -
sqrt(2)*a**(1/4)*(A*b - B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b**(
9/4)) + 2*sqrt(x)*(A*b - B*a)/b**2

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Mathematica [A]  time = 0.221948, size = 243, normalized size = 0.95 \[ \frac{40 \sqrt [4]{b} \sqrt{x} (A b-a B)-5 \sqrt{2} \sqrt [4]{a} (a B-A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} \sqrt [4]{a} (a B-A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-10 \sqrt{2} \sqrt [4]{a} (a B-A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+10 \sqrt{2} \sqrt [4]{a} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+8 b^{5/4} B x^{5/2}}{20 b^{9/4}} \]

Antiderivative was successfully verified.

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

[Out]

(40*b^(1/4)*(A*b - a*B)*Sqrt[x] + 8*b^(5/4)*B*x^(5/2) - 10*Sqrt[2]*a^(1/4)*(-(A*
b) + a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 10*Sqrt[2]*a^(1/4)*(-(
A*b) + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 5*Sqrt[2]*a^(1/4)*(-
(A*b) + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 5*Sqrt
[2]*a^(1/4)*(-(A*b) + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[
b]*x])/(20*b^(9/4))

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Maple [A]  time = 0.011, size = 299, normalized size = 1.2 \[{\frac{2\,B}{5\,b}{x}^{{\frac{5}{2}}}}+2\,{\frac{A\sqrt{x}}{b}}-2\,{\frac{B\sqrt{x}a}{{b}^{2}}}-{\frac{\sqrt{2}A}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{\sqrt{2}A}{2\,b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{\sqrt{2}A}{4\,b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}Ba}{2\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}Ba}{2\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{\sqrt{2}Ba}{4\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*B*x^(5/2)/b+2/b*A*x^(1/2)-2/b^2*B*x^(1/2)*a-1/2/b*(a/b)^(1/4)*2^(1/2)*A*arct
an(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-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^2*(a/b)^(1/4)*
2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*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+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

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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)*x^(3/2)/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25064, size = 737, normalized size = 2.89 \[ \frac{20 \, b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}}}{{\left (B a - A b\right )} \sqrt{x} - \sqrt{b^{4} \sqrt{-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}} +{\left (B^{2} a^{2} - 2 \, A B a b + A^{2} b^{2}\right )} x}}\right ) - 5 \, b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} \log \left (b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} -{\left (B a - A b\right )} \sqrt{x}\right ) + 5 \, b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-b^{2} \left (-\frac{B^{4} a^{5} - 4 \, A B^{3} a^{4} b + 6 \, A^{2} B^{2} a^{3} b^{2} - 4 \, A^{3} B a^{2} b^{3} + A^{4} a b^{4}}{b^{9}}\right )^{\frac{1}{4}} -{\left (B a - A b\right )} \sqrt{x}\right ) + 4 \,{\left (B b x^{2} - 5 \, B a + 5 \, A b\right )} \sqrt{x}}{10 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(20*b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4*A^3*B*a^2*b^3 +
A^4*a*b^4)/b^9)^(1/4)*arctan(-b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2
- 4*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4)/((B*a - A*b)*sqrt(x) - sqrt(b^4*sqrt(
-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9
) + (B^2*a^2 - 2*A*B*a*b + A^2*b^2)*x))) - 5*b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*
A^2*B^2*a^3*b^2 - 4*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4)*log(b^2*(-(B^4*a^5 - 4
*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4) - (B*
a - A*b)*sqrt(x)) + 5*b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B^2*a^3*b^2 - 4*A^3
*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4)*log(-b^2*(-(B^4*a^5 - 4*A*B^3*a^4*b + 6*A^2*B
^2*a^3*b^2 - 4*A^3*B*a^2*b^3 + A^4*a*b^4)/b^9)^(1/4) - (B*a - A*b)*sqrt(x)) + 4*
(B*b*x^2 - 5*B*a + 5*A*b)*sqrt(x))/b^2

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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(x**(3/2)*(B*x**2+A)/(b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.259624, size = 355, normalized size = 1.39 \[ \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{3}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{3}} + \frac{2 \,{\left (B b^{4} x^{\frac{5}{2}} - 5 \, B a b^{3} \sqrt{x} + 5 \, A b^{4} \sqrt{x}\right )}}{5 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*((a*b^3)^(1/4)*B*a - (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))/b^3 + 1/2*sqrt(2)*((a*b^3)^(1/4)*B*a - (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
))/b^3 + 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4)*A*b)*ln(sqrt(2)*sqrt(x)*
(a/b)^(1/4) + x + sqrt(a/b))/b^3 - 1/4*sqrt(2)*((a*b^3)^(1/4)*B*a - (a*b^3)^(1/4
)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^3 + 2/5*(B*b^4*x^(5/2)
- 5*B*a*b^3*sqrt(x) + 5*A*b^4*sqrt(x))/b^5