3.212 \(\int \frac{x^{13/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=298 \[ \frac{(5 A c+3 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{9/4} c^{7/4}}-\frac{(5 A c+3 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{9/4} c^{7/4}}-\frac{(5 A c+3 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{9/4} c^{7/4}}+\frac{(5 A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{9/4} c^{7/4}}+\frac{x^{3/2} (5 A c+3 b B)}{16 b^2 c \left (b+c x^2\right )}-\frac{x^{3/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

[Out]

-((b*B - A*c)*x^(3/2))/(4*b*c*(b + c*x^2)^2) + ((3*b*B + 5*A*c)*x^(3/2))/(16*b^2
*c*(b + c*x^2)) - ((3*b*B + 5*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]
)/(32*Sqrt[2]*b^(9/4)*c^(7/4)) + ((3*b*B + 5*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(32*Sqrt[2]*b^(9/4)*c^(7/4)) + ((3*b*B + 5*A*c)*Log[Sqrt[b] - S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(9/4)*c^(7/4)) - ((3*
b*B + 5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqr
t[2]*b^(9/4)*c^(7/4))

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Rubi [A]  time = 0.492634, antiderivative size = 298, 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 \[ \frac{(5 A c+3 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{9/4} c^{7/4}}-\frac{(5 A c+3 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{9/4} c^{7/4}}-\frac{(5 A c+3 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{9/4} c^{7/4}}+\frac{(5 A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{9/4} c^{7/4}}+\frac{x^{3/2} (5 A c+3 b B)}{16 b^2 c \left (b+c x^2\right )}-\frac{x^{3/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

-((b*B - A*c)*x^(3/2))/(4*b*c*(b + c*x^2)^2) + ((3*b*B + 5*A*c)*x^(3/2))/(16*b^2
*c*(b + c*x^2)) - ((3*b*B + 5*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]
)/(32*Sqrt[2]*b^(9/4)*c^(7/4)) + ((3*b*B + 5*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(32*Sqrt[2]*b^(9/4)*c^(7/4)) + ((3*b*B + 5*A*c)*Log[Sqrt[b] - S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(9/4)*c^(7/4)) - ((3*
b*B + 5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqr
t[2]*b^(9/4)*c^(7/4))

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Rubi in Sympy [A]  time = 76.036, size = 277, normalized size = 0.93 \[ \frac{x^{\frac{3}{2}} \left (A c - B b\right )}{4 b c \left (b + c x^{2}\right )^{2}} + \frac{x^{\frac{3}{2}} \left (5 A c + 3 B b\right )}{16 b^{2} c \left (b + c x^{2}\right )} + \frac{\sqrt{2} \left (5 A c + 3 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{9}{4}} c^{\frac{7}{4}}} - \frac{\sqrt{2} \left (5 A c + 3 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{9}{4}} c^{\frac{7}{4}}} - \frac{\sqrt{2} \left (5 A c + 3 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{9}{4}} c^{\frac{7}{4}}} + \frac{\sqrt{2} \left (5 A c + 3 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{9}{4}} c^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(13/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

x**(3/2)*(A*c - B*b)/(4*b*c*(b + c*x**2)**2) + x**(3/2)*(5*A*c + 3*B*b)/(16*b**2
*c*(b + c*x**2)) + sqrt(2)*(5*A*c + 3*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x
) + sqrt(b) + sqrt(c)*x)/(128*b**(9/4)*c**(7/4)) - sqrt(2)*(5*A*c + 3*B*b)*log(s
qrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(9/4)*c**(7/4))
- sqrt(2)*(5*A*c + 3*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(9/
4)*c**(7/4)) + sqrt(2)*(5*A*c + 3*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4
))/(64*b**(9/4)*c**(7/4))

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Mathematica [A]  time = 0.389132, size = 267, normalized size = 0.9 \[ \frac{-\frac{32 b^{5/4} c^{3/4} x^{3/2} (b B-A c)}{\left (b+c x^2\right )^2}+\frac{8 \sqrt [4]{b} c^{3/4} x^{3/2} (5 A c+3 b B)}{b+c x^2}+\sqrt{2} (5 A c+3 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-\sqrt{2} (5 A c+3 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-2 \sqrt{2} (5 A c+3 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+2 \sqrt{2} (5 A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{128 b^{9/4} c^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

((-32*b^(5/4)*c^(3/4)*(b*B - A*c)*x^(3/2))/(b + c*x^2)^2 + (8*b^(1/4)*c^(3/4)*(3
*b*B + 5*A*c)*x^(3/2))/(b + c*x^2) - 2*Sqrt[2]*(3*b*B + 5*A*c)*ArcTan[1 - (Sqrt[
2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 2*Sqrt[2]*(3*b*B + 5*A*c)*ArcTan[1 + (Sqrt[2]*c^(
1/4)*Sqrt[x])/b^(1/4)] + Sqrt[2]*(3*b*B + 5*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c
^(1/4)*Sqrt[x] + Sqrt[c]*x] - Sqrt[2]*(3*b*B + 5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1
/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(128*b^(9/4)*c^(7/4))

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Maple [A]  time = 0.025, size = 335, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( 5\,Ac+3\,Bb \right ){x}^{7/2}}{{b}^{2}}}+1/32\,{\frac{ \left ( 9\,Ac-Bb \right ){x}^{3/2}}{bc}} \right ) }+{\frac{5\,\sqrt{2}A}{64\,{b}^{2}c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{5\,\sqrt{2}A}{64\,{b}^{2}c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{5\,\sqrt{2}A}{128\,{b}^{2}c}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}B}{64\,b{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}B}{64\,b{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}B}{128\,b{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(13/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

2*(1/32*(5*A*c+3*B*b)/b^2*x^(7/2)+1/32*(9*A*c-B*b)/b/c*x^(3/2))/(c*x^2+b)^2+5/64
/b^2/c/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+5/64/b^2/c/(b
/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+5/128/b^2/c/(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)))+3/64/b/c^2/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1
/4)*x^(1/2)+1)+3/64/b/c^2/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/
2)-1)+3/128/b/c^2/(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^(13/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254269, size = 1189, normalized size = 3.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(13/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

1/64*(4*(b^2*c^3*x^4 + 2*b^3*c^2*x^2 + b^4*c)*(-(81*B^4*b^4 + 540*A*B^3*b^3*c +
1350*A^2*B^2*b^2*c^2 + 1500*A^3*B*b*c^3 + 625*A^4*c^4)/(b^9*c^7))^(1/4)*arctan(b
^7*c^5*(-(81*B^4*b^4 + 540*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2 + 1500*A^3*B*b*c^3
+ 625*A^4*c^4)/(b^9*c^7))^(3/4)/((27*B^3*b^3 + 135*A*B^2*b^2*c + 225*A^2*B*b*c^
2 + 125*A^3*c^3)*sqrt(x) + sqrt((729*B^6*b^6 + 7290*A*B^5*b^5*c + 30375*A^2*B^4*
b^4*c^2 + 67500*A^3*B^3*b^3*c^3 + 84375*A^4*B^2*b^2*c^4 + 56250*A^5*B*b*c^5 + 15
625*A^6*c^6)*x - (81*B^4*b^9*c^3 + 540*A*B^3*b^8*c^4 + 1350*A^2*B^2*b^7*c^5 + 15
00*A^3*B*b^6*c^6 + 625*A^4*b^5*c^7)*sqrt(-(81*B^4*b^4 + 540*A*B^3*b^3*c + 1350*A
^2*B^2*b^2*c^2 + 1500*A^3*B*b*c^3 + 625*A^4*c^4)/(b^9*c^7))))) + (b^2*c^3*x^4 +
2*b^3*c^2*x^2 + b^4*c)*(-(81*B^4*b^4 + 540*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2 +
1500*A^3*B*b*c^3 + 625*A^4*c^4)/(b^9*c^7))^(1/4)*log(b^7*c^5*(-(81*B^4*b^4 + 540
*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2 + 1500*A^3*B*b*c^3 + 625*A^4*c^4)/(b^9*c^7))
^(3/4) + (27*B^3*b^3 + 135*A*B^2*b^2*c + 225*A^2*B*b*c^2 + 125*A^3*c^3)*sqrt(x))
- (b^2*c^3*x^4 + 2*b^3*c^2*x^2 + b^4*c)*(-(81*B^4*b^4 + 540*A*B^3*b^3*c + 1350*
A^2*B^2*b^2*c^2 + 1500*A^3*B*b*c^3 + 625*A^4*c^4)/(b^9*c^7))^(1/4)*log(-b^7*c^5*
(-(81*B^4*b^4 + 540*A*B^3*b^3*c + 1350*A^2*B^2*b^2*c^2 + 1500*A^3*B*b*c^3 + 625*
A^4*c^4)/(b^9*c^7))^(3/4) + (27*B^3*b^3 + 135*A*B^2*b^2*c + 225*A^2*B*b*c^2 + 12
5*A^3*c^3)*sqrt(x)) + 4*((3*B*b*c + 5*A*c^2)*x^3 - (B*b^2 - 9*A*b*c)*x)*sqrt(x))
/(b^2*c^3*x^4 + 2*b^3*c^2*x^2 + b^4*c)

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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**(13/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226797, size = 402, normalized size = 1.35 \[ \frac{3 \, B b c x^{\frac{7}{2}} + 5 \, A c^{2} x^{\frac{7}{2}} - B b^{2} x^{\frac{3}{2}} + 9 \, A b c x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{2} c} + \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + 5 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{3} c^{4}} + \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + 5 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{3} c^{4}} - \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + 5 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{3} c^{4}} + \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + 5 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(13/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

1/16*(3*B*b*c*x^(7/2) + 5*A*c^2*x^(7/2) - B*b^2*x^(3/2) + 9*A*b*c*x^(3/2))/((c*x
^2 + b)^2*b^2*c) + 1/64*sqrt(2)*(3*(b*c^3)^(3/4)*B*b + 5*(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^3*c^4) + 1/64*s
qrt(2)*(3*(b*c^3)^(3/4)*B*b + 5*(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^3*c^4) - 1/128*sqrt(2)*(3*(b*c^3)^(3/4)
*B*b + 5*(b*c^3)^(3/4)*A*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^3
*c^4) + 1/128*sqrt(2)*(3*(b*c^3)^(3/4)*B*b + 5*(b*c^3)^(3/4)*A*c)*ln(-sqrt(2)*sq
rt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^3*c^4)