3.210 \(\int \frac{x^{17/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.487662, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{3 (A c+7 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{5/4} c^{11/4}}-\frac{3 (A c+7 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{5/4} c^{11/4}}-\frac{3 (A c+7 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{5/4} c^{11/4}}+\frac{3 (A c+7 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{5/4} c^{11/4}}-\frac{x^{3/2} (A c+7 b B)}{16 b c^2 \left (b+c x^2\right )}-\frac{x^{7/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(17/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 76.2193, size = 275, normalized size = 0.94 \[ \frac{x^{\frac{7}{2}} \left (A c - B b\right )}{4 b c \left (b + c x^{2}\right )^{2}} - \frac{x^{\frac{3}{2}} \left (A c + 7 B b\right )}{16 b c^{2} \left (b + c x^{2}\right )} + \frac{3 \sqrt{2} \left (A c + 7 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{5}{4}} c^{\frac{11}{4}}} - \frac{3 \sqrt{2} \left (A c + 7 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{5}{4}} c^{\frac{11}{4}}} - \frac{3 \sqrt{2} \left (A c + 7 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{5}{4}} c^{\frac{11}{4}}} + \frac{3 \sqrt{2} \left (A c + 7 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{5}{4}} c^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(17/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.676706, size = 272, normalized size = 0.93 \[ \frac{\frac{3 \sqrt{2} (A c+7 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{5/4}}-\frac{3 \sqrt{2} (A c+7 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{5/4}}-\frac{6 \sqrt{2} (A c+7 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{b^{5/4}}+\frac{6 \sqrt{2} (A c+7 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{b^{5/4}}+\frac{8 c^{3/4} x^{3/2} (3 A c-11 b B)}{b \left (b+c x^2\right )}-\frac{32 c^{3/4} x^{3/2} (A c-b B)}{\left (b+c x^2\right )^2}}{128 c^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(17/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

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

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Maple [A]  time = 0.025, size = 325, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( 3\,Ac-11\,Bb \right ){x}^{7/2}}{bc}}-1/32\,{\frac{ \left ( Ac+7\,Bb \right ){x}^{3/2}}{{c}^{2}}} \right ) }+{\frac{3\,\sqrt{2}A}{64\,b{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}A}{64\,b{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}A}{128\,b{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{21\,\sqrt{2}B}{64\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{21\,\sqrt{2}B}{64\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{21\,\sqrt{2}B}{128\,{c}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(17/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

2*(1/32*(3*A*c-11*B*b)/b/c*x^(7/2)-1/32*(A*c+7*B*b)/c^2*x^(3/2))/(c*x^2+b)^2+3/6
4/c^2/b/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+3/64/c^2/b/(
b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+3/128/c^2/b/(b/c)^(1/
4)*2^(1/2)*A*ln((x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x+(b/c)^(1/4)*x^(1/
2)*2^(1/2)+(b/c)^(1/2)))+21/64/c^3/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1
/4)*x^(1/2)+1)+21/64/c^3/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2
)-1)+21/128/c^3/(b/c)^(1/4)*2^(1/2)*B*ln((x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1
/2))/(x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(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^(17/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253421, 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^(17/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

1/64*(12*(b*c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2)*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*
c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(1/4)*arctan(b^4
*c^8*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 +
A^4*c^4)/(b^5*c^11))^(3/4)/((343*B^3*b^3 + 147*A*B^2*b^2*c + 21*A^2*B*b*c^2 + A
^3*c^3)*sqrt(x) + sqrt((117649*B^6*b^6 + 100842*A*B^5*b^5*c + 36015*A^2*B^4*b^4*
c^2 + 6860*A^3*B^3*b^3*c^3 + 735*A^4*B^2*b^2*c^4 + 42*A^5*B*b*c^5 + A^6*c^6)*x -
(2401*B^4*b^7*c^5 + 1372*A*B^3*b^6*c^6 + 294*A^2*B^2*b^5*c^7 + 28*A^3*B*b^4*c^8
+ A^4*b^3*c^9)*sqrt(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 2
8*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))))) + 3*(b*c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2
)*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^
4*c^4)/(b^5*c^11))^(1/4)*log(27*b^4*c^8*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294
*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(3/4) + 27*(343*B^3*b^3
+ 147*A*B^2*b^2*c + 21*A^2*B*b*c^2 + A^3*c^3)*sqrt(x)) - 3*(b*c^4*x^4 + 2*b^2*c
^3*x^2 + b^3*c^2)*(-(2401*B^4*b^4 + 1372*A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*
A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(1/4)*log(-27*b^4*c^8*(-(2401*B^4*b^4 + 1372*
A*B^3*b^3*c + 294*A^2*B^2*b^2*c^2 + 28*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^11))^(3/4)
+ 27*(343*B^3*b^3 + 147*A*B^2*b^2*c + 21*A^2*B*b*c^2 + A^3*c^3)*sqrt(x)) - 4*((1
1*B*b*c - 3*A*c^2)*x^3 + (7*B*b^2 + A*b*c)*x)*sqrt(x))/(b*c^4*x^4 + 2*b^2*c^3*x^
2 + b^3*c^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**(17/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226861, size = 396, normalized size = 1.35 \[ -\frac{11 \, B b c x^{\frac{7}{2}} - 3 \, A c^{2} x^{\frac{7}{2}} + 7 \, B b^{2} x^{\frac{3}{2}} + A b c x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b c^{2}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{5}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{5}} - \frac{3 \, \sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{5}} + \frac{3 \, \sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(17/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

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