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

Optimal. Leaf size=289 \[ \frac{(7 A b-3 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{(7 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )} \]

[Out]

-(7*A*b - 3*a*B)/(6*a^2*b*x^(3/2)) + (A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x^2)) + (
(7*A*b - 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/
4)*b^(1/4)) - ((7*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(11/4)*b^(1/4)) + ((7*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^(11/4)*b^(1/4)) - ((7*A*b - 3*a*B)*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(1/4
))

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Rubi [A]  time = 0.456884, 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 \[ \frac{(7 A b-3 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4} \sqrt [4]{b}}+\frac{(7 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{(7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4} \sqrt [4]{b}}-\frac{7 A b-3 a B}{6 a^2 b x^{3/2}}+\frac{A b-a B}{2 a b x^{3/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(7*A*b - 3*a*B)/(6*a^2*b*x^(3/2)) + (A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x^2)) + (
(7*A*b - 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/
4)*b^(1/4)) - ((7*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4
*Sqrt[2]*a^(11/4)*b^(1/4)) + ((7*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^(11/4)*b^(1/4)) - ((7*A*b - 3*a*B)*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*b^(1/4
))

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Rubi in Sympy [A]  time = 78.7587, size = 269, normalized size = 0.93 \[ \frac{A b - B a}{2 a b x^{\frac{3}{2}} \left (a + b x^{2}\right )} - \frac{7 A b - 3 B a}{6 a^{2} b x^{\frac{3}{2}}} + \frac{\sqrt{2} \left (7 A b - 3 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{11}{4}} \sqrt [4]{b}} - \frac{\sqrt{2} \left (7 A b - 3 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{11}{4}} \sqrt [4]{b}} + \frac{\sqrt{2} \left (7 A b - 3 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}} \sqrt [4]{b}} - \frac{\sqrt{2} \left (7 A b - 3 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}} \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*b - B*a)/(2*a*b*x**(3/2)*(a + b*x**2)) - (7*A*b - 3*B*a)/(6*a**2*b*x**(3/2))
+ sqrt(2)*(7*A*b - 3*B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqr
t(b)*x)/(16*a**(11/4)*b**(1/4)) - sqrt(2)*(7*A*b - 3*B*a)*log(sqrt(2)*a**(1/4)*b
**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(11/4)*b**(1/4)) + sqrt(2)*(7*A*b
- 3*B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(11/4)*b**(1/4)) - sq
rt(2)*(7*A*b - 3*B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(11/4)*b
**(1/4))

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Mathematica [A]  time = 0.599642, size = 256, normalized size = 0.89 \[ \frac{\frac{24 a^{3/4} \sqrt{x} (a B-A b)}{a+b x^2}-\frac{32 a^{3/4} A}{x^{3/2}}+\frac{3 \sqrt{2} (7 A b-3 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{3 \sqrt{2} (3 a B-7 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{6 \sqrt{2} (7 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}-\frac{6 \sqrt{2} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}}{48 a^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a^(3/4)*A)/x^(3/2) + (24*a^(3/4)*(-(A*b) + a*B)*Sqrt[x])/(a + b*x^2) + (6*
Sqrt[2]*(7*A*b - 3*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) -
(6*Sqrt[2]*(7*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/
4) + (3*Sqrt[2]*(7*A*b - 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/b^(1/4) + (3*Sqrt[2]*(-7*A*b + 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4))/(48*a^(11/4))

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Maple [A]  time = 0.023, size = 317, normalized size = 1.1 \[ -{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{Ab}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}+{\frac{B}{2\,a \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{7\,\sqrt{2}Ab}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{7\,\sqrt{2}Ab}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{7\,\sqrt{2}Ab}{16\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}B}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}B}{8\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}B}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*A/a^2/x^(3/2)-1/2/a^2*x^(1/2)/(b*x^2+a)*A*b+1/2/a*x^(1/2)/(b*x^2+a)*B-7/8/a
^3*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*b-7/8/a^3*(a/b)^(
1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b-7/16/a^3*(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)))*b+3/8/a^2*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(
1/2)+1)+3/8/a^2*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16
/a^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)))

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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^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258555, size = 838, normalized size = 2.9 \[ \frac{4 \,{\left (3 \, B a - 7 \, A b\right )} x^{2} + 12 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{3} \left (-\frac{81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac{1}{4}}}{{\left (3 \, B a - 7 \, A b\right )} \sqrt{x} - \sqrt{a^{6} \sqrt{-\frac{81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}} +{\left (9 \, B^{2} a^{2} - 42 \, A B a b + 49 \, A^{2} b^{2}\right )} x}}\right ) - 3 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac{1}{4}} \log \left (a^{3} \left (-\frac{81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac{1}{4}} -{\left (3 \, B a - 7 \, A b\right )} \sqrt{x}\right ) + 3 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac{1}{4}} \log \left (-a^{3} \left (-\frac{81 \, B^{4} a^{4} - 756 \, A B^{3} a^{3} b + 2646 \, A^{2} B^{2} a^{2} b^{2} - 4116 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b}\right )^{\frac{1}{4}} -{\left (3 \, B a - 7 \, A b\right )} \sqrt{x}\right ) - 16 \, A a}{24 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(4*(3*B*a - 7*A*b)*x^2 + 12*(a^2*b*x^3 + a^3*x)*sqrt(x)*(-(81*B^4*a^4 - 756
*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))
^(1/4)*arctan(-a^3*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116
*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4)/((3*B*a - 7*A*b)*sqrt(x) - sqrt(a^6
*sqrt(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 +
2401*A^4*b^4)/(a^11*b)) + (9*B^2*a^2 - 42*A*B*a*b + 49*A^2*b^2)*x))) - 3*(a^2*b
*x^3 + a^3*x)*sqrt(x)*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4
116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4)*log(a^3*(-(81*B^4*a^4 - 756*A*B^
3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4
) - (3*B*a - 7*A*b)*sqrt(x)) + 3*(a^2*b*x^3 + a^3*x)*sqrt(x)*(-(81*B^4*a^4 - 756
*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))
^(1/4)*log(-a^3*(-(81*B^4*a^4 - 756*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 4116*A^
3*B*a*b^3 + 2401*A^4*b^4)/(a^11*b))^(1/4) - (3*B*a - 7*A*b)*sqrt(x)) - 16*A*a)/(
(a^2*b*x^3 + a^3*x)*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**(5/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239638, size = 382, normalized size = 1.32 \[ \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b} + \frac{B a \sqrt{x} - A b \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a^{2}} - \frac{2 \, A}{3 \, a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 7*(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))/(a^3*b) + 1/8*sqrt(2)*(3*(a*b^3)^(1/4)
*B*a - 7*(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))/(a^3*b) + 1/16*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 7*(a*b^3)^(1/4)*A*b
)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b) - 1/16*sqrt(2)*(3*(a*b
^3)^(1/4)*B*a - 7*(a*b^3)^(1/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(
a/b))/(a^3*b) + 1/2*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^2 + a)*a^2) - 2/3*A/(a^2*x
^(3/2))