3.382 \(\int \frac{A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=310 \[ \frac{\sqrt [4]{b} (9 A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4}}-\frac{\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4}}-\frac{\sqrt [4]{b} (9 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4}}+\frac{\sqrt [4]{b} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{13/4}}+\frac{9 A b-5 a B}{2 a^3 \sqrt{x}}-\frac{9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac{A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )} \]

[Out]

-(9*A*b - 5*a*B)/(10*a^2*b*x^(5/2)) + (9*A*b - 5*a*B)/(2*a^3*Sqrt[x]) + (A*b - a
*B)/(2*a*b*x^(5/2)*(a + b*x^2)) - (b^(1/4)*(9*A*b - 5*a*B)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)) + (b^(1/4)*(9*A*b - 5*a*B)*ArcTan
[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)) + (b^(1/4)*(9*A*b
- 5*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*
a^(13/4)) - (b^(1/4)*(9*A*b - 5*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4))

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Rubi [A]  time = 0.522273, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{\sqrt [4]{b} (9 A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4}}-\frac{\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4}}-\frac{\sqrt [4]{b} (9 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4}}+\frac{\sqrt [4]{b} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{13/4}}+\frac{9 A b-5 a B}{2 a^3 \sqrt{x}}-\frac{9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac{A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(7/2)*(a + b*x^2)^2),x]

[Out]

-(9*A*b - 5*a*B)/(10*a^2*b*x^(5/2)) + (9*A*b - 5*a*B)/(2*a^3*Sqrt[x]) + (A*b - a
*B)/(2*a*b*x^(5/2)*(a + b*x^2)) - (b^(1/4)*(9*A*b - 5*a*B)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)) + (b^(1/4)*(9*A*b - 5*a*B)*ArcTan
[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)) + (b^(1/4)*(9*A*b
- 5*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*
a^(13/4)) - (b^(1/4)*(9*A*b - 5*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4))

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Rubi in Sympy [A]  time = 88.7177, size = 289, normalized size = 0.93 \[ \frac{A b - B a}{2 a b x^{\frac{5}{2}} \left (a + b x^{2}\right )} - \frac{9 A b - 5 B a}{10 a^{2} b x^{\frac{5}{2}}} + \frac{9 A b - 5 B a}{2 a^{3} \sqrt{x}} + \frac{\sqrt{2} \sqrt [4]{b} \left (9 A b - 5 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \left (9 A b - 5 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \left (9 A b - 5 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{13}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \left (9 A b - 5 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(7/2)/(b*x**2+a)**2,x)

[Out]

(A*b - B*a)/(2*a*b*x**(5/2)*(a + b*x**2)) - (9*A*b - 5*B*a)/(10*a**2*b*x**(5/2))
+ (9*A*b - 5*B*a)/(2*a**3*sqrt(x)) + sqrt(2)*b**(1/4)*(9*A*b - 5*B*a)*log(-sqrt
(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(13/4)) - sqrt(2)*b*
*(1/4)*(9*A*b - 5*B*a)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)
*x)/(16*a**(13/4)) - sqrt(2)*b**(1/4)*(9*A*b - 5*B*a)*atan(1 - sqrt(2)*b**(1/4)*
sqrt(x)/a**(1/4))/(8*a**(13/4)) + sqrt(2)*b**(1/4)*(9*A*b - 5*B*a)*atan(1 + sqrt
(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(13/4))

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Mathematica [A]  time = 0.463227, size = 277, normalized size = 0.89 \[ \frac{-\frac{32 a^{5/4} A}{x^{5/2}}-\frac{40 \sqrt [4]{a} b x^{3/2} (a B-A b)}{a+b x^2}-\frac{160 \sqrt [4]{a} (a B-2 A b)}{\sqrt{x}}+5 \sqrt{2} \sqrt [4]{b} (9 A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} \sqrt [4]{b} (5 a B-9 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-10 \sqrt{2} \sqrt [4]{b} (9 A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+10 \sqrt{2} \sqrt [4]{b} (9 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{80 a^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(7/2)*(a + b*x^2)^2),x]

[Out]

((-32*a^(5/4)*A)/x^(5/2) - (160*a^(1/4)*(-2*A*b + a*B))/Sqrt[x] - (40*a^(1/4)*b*
(-(A*b) + a*B)*x^(3/2))/(a + b*x^2) - 10*Sqrt[2]*b^(1/4)*(9*A*b - 5*a*B)*ArcTan[
1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 10*Sqrt[2]*b^(1/4)*(9*A*b - 5*a*B)*ArcT
an[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 5*Sqrt[2]*b^(1/4)*(9*A*b - 5*a*B)*Lo
g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 5*Sqrt[2]*b^(1/4)*(-9
*A*b + 5*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(80*a^
(13/4))

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Maple [A]  time = 0.026, size = 339, normalized size = 1.1 \[ -{\frac{2\,A}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}+4\,{\frac{Ab}{\sqrt{x}{a}^{3}}}-2\,{\frac{B}{\sqrt{x}{a}^{2}}}+{\frac{{b}^{2}A}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}-{\frac{Bb}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{9\,b\sqrt{2}A}{16\,{a}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{9\,b\sqrt{2}A}{8\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{9\,b\sqrt{2}A}{8\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,\sqrt{2}B}{16\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,\sqrt{2}B}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{5\,\sqrt{2}B}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(7/2)/(b*x^2+a)^2,x)

[Out]

-2/5*A/a^2/x^(5/2)+4/x^(1/2)/a^3*A*b-2/x^(1/2)/a^2*B+1/2/a^3*b^2*x^(3/2)/(b*x^2+
a)*A-1/2/a^2*b*x^(3/2)/(b*x^2+a)*B+9/16/a^3*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)))+
9/8/a^3*b/(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+9/8/a^3*b/
(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-5/16/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)))-5/8/a^2/(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*
x^(1/2)+1)-5/8/a^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)/((b*x^2 + a)^2*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26269, size = 1164, normalized size = 3.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^2*x^(7/2)),x, algorithm="fricas")

[Out]

-1/40*(20*(5*B*a*b - 9*A*b^2)*x^4 + 16*A*a^2 + 16*(5*B*a^2 - 9*A*a*b)*x^2 - 20*(
a^3*b*x^4 + a^4*x^2)*sqrt(x)*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B
^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(1/4)*arctan(-a^10*(-(625*B
^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561
*A^4*b^5)/a^13)^(3/4)/((125*B^3*a^3*b - 675*A*B^2*a^2*b^2 + 1215*A^2*B*a*b^3 - 7
29*A^3*b^4)*sqrt(x) - sqrt((15625*B^6*a^6*b^2 - 168750*A*B^5*a^5*b^3 + 759375*A^
2*B^4*a^4*b^4 - 1822500*A^3*B^3*a^3*b^5 + 2460375*A^4*B^2*a^2*b^6 - 1771470*A^5*
B*a*b^7 + 531441*A^6*b^8)*x - (625*B^4*a^11*b - 4500*A*B^3*a^10*b^2 + 12150*A^2*
B^2*a^9*b^3 - 14580*A^3*B*a^8*b^4 + 6561*A^4*a^7*b^5)*sqrt(-(625*B^4*a^4*b - 450
0*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13
)))) - 5*(a^3*b*x^4 + a^4*x^2)*sqrt(x)*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 1
2150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(1/4)*log(a^10*(-
(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4
+ 6561*A^4*b^5)/a^13)^(3/4) - (125*B^3*a^3*b - 675*A*B^2*a^2*b^2 + 1215*A^2*B*a*
b^3 - 729*A^3*b^4)*sqrt(x)) + 5*(a^3*b*x^4 + a^4*x^2)*sqrt(x)*(-(625*B^4*a^4*b -
4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/
a^13)^(1/4)*log(-a^10*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*
b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(3/4) - (125*B^3*a^3*b - 675*A*B^2
*a^2*b^2 + 1215*A^2*B*a*b^3 - 729*A^3*b^4)*sqrt(x)))/((a^3*b*x^4 + a^4*x^2)*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**(7/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270018, size = 409, normalized size = 1.32 \[ -\frac{B a b x^{\frac{3}{2}} - A b^{2} x^{\frac{3}{2}}}{2 \,{\left (b x^{2} + a\right )} a^{3}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{4} b^{2}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac{2 \,{\left (5 \, B a x^{2} - 10 \, A b x^{2} + A a\right )}}{5 \, a^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^2*x^(7/2)),x, algorithm="giac")

[Out]

-1/2*(B*a*b*x^(3/2) - A*b^2*x^(3/2))/((b*x^2 + a)*a^3) - 1/8*sqrt(2)*(5*(a*b^3)^
(3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqr
t(x))/(a/b)^(1/4))/(a^4*b^2) - 1/8*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 9*(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^4*b
^2) + 1/16*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)*ln(sqrt(2)*sqrt(x
)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 1/16*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 9
*(a*b^3)^(3/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) -
2/5*(5*B*a*x^2 - 10*A*b*x^2 + A*a)/(a^3*x^(5/2))