3.368 \(\int \frac{x^{5/2} \left (A+B x^2\right )}{a+b x^2} \, dx\)

Optimal. Leaf size=257 \[ -\frac{a^{3/4} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{a^{3/4} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{a^{3/4} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{11/4}}-\frac{a^{3/4} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{11/4}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{7/2}}{7 b} \]

[Out]

(2*(A*b - a*B)*x^(3/2))/(3*b^2) + (2*B*x^(7/2))/(7*b) + (a^(3/4)*(A*b - a*B)*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(11/4)) - (a^(3/4)*(A*b -
a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(11/4)) - (a^(3/
4)*(A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sq
rt[2]*b^(11/4)) + (a^(3/4)*(A*b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(11/4))

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Rubi [A]  time = 0.477297, antiderivative size = 257, 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 \[ -\frac{a^{3/4} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{a^{3/4} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{a^{3/4} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{11/4}}-\frac{a^{3/4} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{11/4}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x^2))/(a + b*x^2),x]

[Out]

(2*(A*b - a*B)*x^(3/2))/(3*b^2) + (2*B*x^(7/2))/(7*b) + (a^(3/4)*(A*b - a*B)*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(11/4)) - (a^(3/4)*(A*b -
a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(11/4)) - (a^(3/
4)*(A*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sq
rt[2]*b^(11/4)) + (a^(3/4)*(A*b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(11/4))

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Rubi in Sympy [A]  time = 78.2852, size = 240, normalized size = 0.93 \[ \frac{2 B x^{\frac{7}{2}}}{7 b} - \frac{\sqrt{2} a^{\frac{3}{4}} \left (A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{11}{4}}} + \frac{\sqrt{2} a^{\frac{3}{4}} \left (A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{11}{4}}} + \frac{\sqrt{2} a^{\frac{3}{4}} \left (A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{11}{4}}} - \frac{\sqrt{2} a^{\frac{3}{4}} \left (A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{11}{4}}} + \frac{2 x^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x**2+A)/(b*x**2+a),x)

[Out]

2*B*x**(7/2)/(7*b) - sqrt(2)*a**(3/4)*(A*b - B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)
*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*b**(11/4)) + sqrt(2)*a**(3/4)*(A*b - B*a)*log
(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*b**(11/4)) + sqrt(2
)*a**(3/4)*(A*b - B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b**(11/4))
- sqrt(2)*a**(3/4)*(A*b - B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b
**(11/4)) + 2*x**(3/2)*(A*b - B*a)/(3*b**2)

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Mathematica [A]  time = 0.234142, size = 243, normalized size = 0.95 \[ \frac{21 \sqrt{2} a^{3/4} (a B-A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-21 \sqrt{2} a^{3/4} (a B-A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-42 \sqrt{2} a^{3/4} (a B-A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+42 \sqrt{2} a^{3/4} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+56 b^{3/4} x^{3/2} (A b-a B)+24 b^{7/4} B x^{7/2}}{84 b^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x^2))/(a + b*x^2),x]

[Out]

(56*b^(3/4)*(A*b - a*B)*x^(3/2) + 24*b^(7/4)*B*x^(7/2) - 42*Sqrt[2]*a^(3/4)*(-(A
*b) + a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 42*Sqrt[2]*a^(3/4)*(-
(A*b) + a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 21*Sqrt[2]*a^(3/4)*
(-(A*b) + a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 21*S
qrt[2]*a^(3/4)*(-(A*b) + a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sq
rt[b]*x])/(84*b^(11/4))

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Maple [A]  time = 0.011, size = 308, normalized size = 1.2 \[{\frac{2\,B}{7\,b}{x}^{{\frac{7}{2}}}}+{\frac{2\,A}{3\,b}{x}^{{\frac{3}{2}}}}-{\frac{2\,Ba}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{a\sqrt{2}A}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{2}A}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{2}A}{4\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{a}^{2}\sqrt{2}B}{2\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{a}^{2}\sqrt{2}B}{2\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{{a}^{2}\sqrt{2}B}{4\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x^2+A)/(b*x^2+a),x)

[Out]

2/7*B*x^(7/2)/b+2/3/b*x^(3/2)*A-2/3/b^2*x^(3/2)*B*a-1/2*a/b^2/(a/b)^(1/4)*2^(1/2
)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-1/2*a/b^2/(a/b)^(1/4)*2^(1/2)*A*arctan
(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-1/4*a/b^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)))+1/2*
a^2/b^3/(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/2*a^2/b^3/
(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+1/4*a^2/b^3/(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)*x^(5/2)/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254527, size = 1054, normalized size = 4.1 \[ -\frac{84 \, b^{2} \left (-\frac{B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{8} \left (-\frac{B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac{3}{4}}}{{\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt{x} - \sqrt{{\left (B^{6} a^{10} - 6 \, A B^{5} a^{9} b + 15 \, A^{2} B^{4} a^{8} b^{2} - 20 \, A^{3} B^{3} a^{7} b^{3} + 15 \, A^{4} B^{2} a^{6} b^{4} - 6 \, A^{5} B a^{5} b^{5} + A^{6} a^{4} b^{6}\right )} x -{\left (B^{4} a^{7} b^{5} - 4 \, A B^{3} a^{6} b^{6} + 6 \, A^{2} B^{2} a^{5} b^{7} - 4 \, A^{3} B a^{4} b^{8} + A^{4} a^{3} b^{9}\right )} \sqrt{-\frac{B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}}}}\right ) + 21 \, b^{2} \left (-\frac{B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac{1}{4}} \log \left (b^{8} \left (-\frac{B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac{3}{4}} -{\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt{x}\right ) - 21 \, b^{2} \left (-\frac{B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac{1}{4}} \log \left (-b^{8} \left (-\frac{B^{4} a^{7} - 4 \, A B^{3} a^{6} b + 6 \, A^{2} B^{2} a^{5} b^{2} - 4 \, A^{3} B a^{4} b^{3} + A^{4} a^{3} b^{4}}{b^{11}}\right )^{\frac{3}{4}} -{\left (B^{3} a^{5} - 3 \, A B^{2} a^{4} b + 3 \, A^{2} B a^{3} b^{2} - A^{3} a^{2} b^{3}\right )} \sqrt{x}\right ) - 4 \,{\left (3 \, B b x^{3} - 7 \,{\left (B a - A b\right )} x\right )} \sqrt{x}}{42 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(5/2)/(b*x^2 + a),x, algorithm="fricas")

[Out]

-1/42*(84*b^2*(-(B^4*a^7 - 4*A*B^3*a^6*b + 6*A^2*B^2*a^5*b^2 - 4*A^3*B*a^4*b^3 +
A^4*a^3*b^4)/b^11)^(1/4)*arctan(-b^8*(-(B^4*a^7 - 4*A*B^3*a^6*b + 6*A^2*B^2*a^5
*b^2 - 4*A^3*B*a^4*b^3 + A^4*a^3*b^4)/b^11)^(3/4)/((B^3*a^5 - 3*A*B^2*a^4*b + 3*
A^2*B*a^3*b^2 - A^3*a^2*b^3)*sqrt(x) - sqrt((B^6*a^10 - 6*A*B^5*a^9*b + 15*A^2*B
^4*a^8*b^2 - 20*A^3*B^3*a^7*b^3 + 15*A^4*B^2*a^6*b^4 - 6*A^5*B*a^5*b^5 + A^6*a^4
*b^6)*x - (B^4*a^7*b^5 - 4*A*B^3*a^6*b^6 + 6*A^2*B^2*a^5*b^7 - 4*A^3*B*a^4*b^8 +
A^4*a^3*b^9)*sqrt(-(B^4*a^7 - 4*A*B^3*a^6*b + 6*A^2*B^2*a^5*b^2 - 4*A^3*B*a^4*b
^3 + A^4*a^3*b^4)/b^11)))) + 21*b^2*(-(B^4*a^7 - 4*A*B^3*a^6*b + 6*A^2*B^2*a^5*b
^2 - 4*A^3*B*a^4*b^3 + A^4*a^3*b^4)/b^11)^(1/4)*log(b^8*(-(B^4*a^7 - 4*A*B^3*a^6
*b + 6*A^2*B^2*a^5*b^2 - 4*A^3*B*a^4*b^3 + A^4*a^3*b^4)/b^11)^(3/4) - (B^3*a^5 -
3*A*B^2*a^4*b + 3*A^2*B*a^3*b^2 - A^3*a^2*b^3)*sqrt(x)) - 21*b^2*(-(B^4*a^7 - 4
*A*B^3*a^6*b + 6*A^2*B^2*a^5*b^2 - 4*A^3*B*a^4*b^3 + A^4*a^3*b^4)/b^11)^(1/4)*lo
g(-b^8*(-(B^4*a^7 - 4*A*B^3*a^6*b + 6*A^2*B^2*a^5*b^2 - 4*A^3*B*a^4*b^3 + A^4*a^
3*b^4)/b^11)^(3/4) - (B^3*a^5 - 3*A*B^2*a^4*b + 3*A^2*B*a^3*b^2 - A^3*a^2*b^3)*s
qrt(x)) - 4*(3*B*b*x^3 - 7*(B*a - A*b)*x)*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**(5/2)*(B*x**2+A)/(b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.256861, size = 356, normalized size = 1.39 \[ \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{5}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, b^{5}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{5}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, b^{5}} + \frac{2 \,{\left (3 \, B b^{6} x^{\frac{7}{2}} - 7 \, B a b^{5} x^{\frac{3}{2}} + 7 \, A b^{6} x^{\frac{3}{2}}\right )}}{21 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(5/2)/(b*x^2 + a),x, algorithm="giac")

[Out]

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))/b^5 + 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
))/b^5 - 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))/b^5 + 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))/b^5 + 2/21*(3*B*b^6*x^(7
/2) - 7*B*a*b^5*x^(3/2) + 7*A*b^6*x^(3/2))/b^7