3.384 \(\int \frac{x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=293 \[ \frac{3 (7 a B+A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{5/4} b^{11/4}}-\frac{3 (7 a B+A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{5/4} b^{11/4}}-\frac{3 (7 a B+A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{5/4} b^{11/4}}+\frac{3 (7 a B+A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{5/4} b^{11/4}}-\frac{x^{3/2} (7 a B+A b)}{16 a b^2 \left (a+b x^2\right )}+\frac{x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

[Out]

((A*b - a*B)*x^(7/2))/(4*a*b*(a + b*x^2)^2) - ((A*b + 7*a*B)*x^(3/2))/(16*a*b^2*
(a + b*x^2)) - (3*(A*b + 7*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
32*Sqrt[2]*a^(5/4)*b^(11/4)) + (3*(A*b + 7*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/(32*Sqrt[2]*a^(5/4)*b^(11/4)) + (3*(A*b + 7*a*B)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(5/4)*b^(11/4)) - (3*(
A*b + 7*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqr
t[2]*a^(5/4)*b^(11/4))

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Rubi [A]  time = 0.471919, antiderivative size = 293, 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 \[ \frac{3 (7 a B+A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{5/4} b^{11/4}}-\frac{3 (7 a B+A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{5/4} b^{11/4}}-\frac{3 (7 a B+A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{5/4} b^{11/4}}+\frac{3 (7 a B+A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{5/4} b^{11/4}}-\frac{x^{3/2} (7 a B+A b)}{16 a b^2 \left (a+b x^2\right )}+\frac{x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

((A*b - a*B)*x^(7/2))/(4*a*b*(a + b*x^2)^2) - ((A*b + 7*a*B)*x^(3/2))/(16*a*b^2*
(a + b*x^2)) - (3*(A*b + 7*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(
32*Sqrt[2]*a^(5/4)*b^(11/4)) + (3*(A*b + 7*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/(32*Sqrt[2]*a^(5/4)*b^(11/4)) + (3*(A*b + 7*a*B)*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(5/4)*b^(11/4)) - (3*(
A*b + 7*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqr
t[2]*a^(5/4)*b^(11/4))

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Rubi in Sympy [A]  time = 80.7179, size = 275, normalized size = 0.94 \[ \frac{x^{\frac{7}{2}} \left (A b - B a\right )}{4 a b \left (a + b x^{2}\right )^{2}} - \frac{x^{\frac{3}{2}} \left (A b + 7 B a\right )}{16 a b^{2} \left (a + b x^{2}\right )} + \frac{3 \sqrt{2} \left (A b + 7 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{5}{4}} b^{\frac{11}{4}}} - \frac{3 \sqrt{2} \left (A b + 7 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{5}{4}} b^{\frac{11}{4}}} - \frac{3 \sqrt{2} \left (A b + 7 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{5}{4}} b^{\frac{11}{4}}} + \frac{3 \sqrt{2} \left (A b + 7 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{5}{4}} b^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

x**(7/2)*(A*b - B*a)/(4*a*b*(a + b*x**2)**2) - x**(3/2)*(A*b + 7*B*a)/(16*a*b**2
*(a + b*x**2)) + 3*sqrt(2)*(A*b + 7*B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x)
+ sqrt(a) + sqrt(b)*x)/(128*a**(5/4)*b**(11/4)) - 3*sqrt(2)*(A*b + 7*B*a)*log(sq
rt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a**(5/4)*b**(11/4))
- 3*sqrt(2)*(A*b + 7*B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(5/
4)*b**(11/4)) + 3*sqrt(2)*(A*b + 7*B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/
4))/(64*a**(5/4)*b**(11/4))

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Mathematica [A]  time = 0.634128, size = 272, normalized size = 0.93 \[ \frac{\frac{3 \sqrt{2} (7 a B+A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4}}-\frac{3 \sqrt{2} (7 a B+A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4}}-\frac{6 \sqrt{2} (7 a B+A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac{6 \sqrt{2} (7 a B+A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+\frac{8 b^{3/4} x^{3/2} (3 A b-11 a B)}{a \left (a+b x^2\right )}-\frac{32 b^{3/4} x^{3/2} (A b-a B)}{\left (a+b x^2\right )^2}}{128 b^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

((-32*b^(3/4)*(A*b - a*B)*x^(3/2))/(a + b*x^2)^2 + (8*b^(3/4)*(3*A*b - 11*a*B)*x
^(3/2))/(a*(a + b*x^2)) - (6*Sqrt[2]*(A*b + 7*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)])/a^(5/4) + (6*Sqrt[2]*(A*b + 7*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/a^(5/4) + (3*Sqrt[2]*(A*b + 7*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4) - (3*Sqrt[2]*(A*b + 7*a*B)*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4))/(128*b^(11/4))

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Maple [A]  time = 0.023, size = 325, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( 3\,Ab-11\,Ba \right ){x}^{7/2}}{ab}}-1/32\,{\frac{ \left ( Ab+7\,Ba \right ){x}^{3/2}}{{b}^{2}}} \right ) }+{\frac{3\,\sqrt{2}A}{64\,{b}^{2}a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}A}{64\,{b}^{2}a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}A}{128\,{b}^{2}a}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{21\,\sqrt{2}B}{64\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{21\,\sqrt{2}B}{64\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{21\,\sqrt{2}B}{128\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

2*(1/32*(3*A*b-11*B*a)/a/b*x^(7/2)-1/32*(A*b+7*B*a)/b^2*x^(3/2))/(b*x^2+a)^2+3/6
4/b^2/a/(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+3/64/b^2/a/(
a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/128/b^2/a/(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)))+21/64/b^3/(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1
/4)*x^(1/2)+1)+21/64/b^3/(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2
)-1)+21/128/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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259484, size = 1177, normalized size = 4.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(5/2)/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/64*(12*(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*
b + 294*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(1/4)*arctan(a^4
*b^8*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 +
A^4*b^4)/(a^5*b^11))^(3/4)/((343*B^3*a^3 + 147*A*B^2*a^2*b + 21*A^2*B*a*b^2 + A
^3*b^3)*sqrt(x) + sqrt((117649*B^6*a^6 + 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*
b^2 + 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 + 42*A^5*B*a*b^5 + A^6*b^6)*x -
(2401*B^4*a^7*b^5 + 1372*A*B^3*a^6*b^6 + 294*A^2*B^2*a^5*b^7 + 28*A^3*B*a^4*b^8
+ A^4*a^3*b^9)*sqrt(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 2
8*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))))) + 3*(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2
)*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^
4*b^4)/(a^5*b^11))^(1/4)*log(27*a^4*b^8*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 294
*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(3/4) + 27*(343*B^3*a^3
+ 147*A*B^2*a^2*b + 21*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) - 3*(a*b^4*x^4 + 2*a^2*b
^3*x^2 + a^3*b^2)*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 28*
A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(1/4)*log(-27*a^4*b^8*(-(2401*B^4*a^4 + 1372*
A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(3/4)
+ 27*(343*B^3*a^3 + 147*A*B^2*a^2*b + 21*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) - 4*((1
1*B*a*b - 3*A*b^2)*x^3 + (7*B*a^2 + A*a*b)*x)*sqrt(x))/(a*b^4*x^4 + 2*a^2*b^3*x^
2 + a^3*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.248179, size = 396, normalized size = 1.35 \[ -\frac{11 \, B a b x^{\frac{7}{2}} - 3 \, A b^{2} x^{\frac{7}{2}} + 7 \, B a^{2} x^{\frac{3}{2}} + A a b x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a b^{2}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} b^{5}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} b^{5}} - \frac{3 \, \sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{2} b^{5}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{2} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(5/2)/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

-1/16*(11*B*a*b*x^(7/2) - 3*A*b^2*x^(7/2) + 7*B*a^2*x^(3/2) + A*a*b*x^(3/2))/((b
*x^2 + a)^2*a*b^2) + 3/64*sqrt(2)*(7*(a*b^3)^(3/4)*B*a + (a*b^3)^(3/4)*A*b)*arct
an(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^5) + 3/64*s
qrt(2)*(7*(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^2*b^5) - 3/128*sqrt(2)*(7*(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^2*b^5
) + 3/128*sqrt(2)*(7*(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^2*b^5)