3.343 \(\int \frac{x^{15/2}}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=242 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.397999, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{7/4} c^{5/4}}+\frac{\sqrt{x}}{16 b c \left (b+c x^2\right )}-\frac{\sqrt{x}}{4 c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(15/2)/(b*x^2 + c*x^4)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 71.7558, size = 223, normalized size = 0.92 \[ - \frac{\sqrt{x}}{4 c \left (b + c x^{2}\right )^{2}} + \frac{\sqrt{x}}{16 b c \left (b + c x^{2}\right )} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{7}{4}} c^{\frac{5}{4}}} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{7}{4}} c^{\frac{5}{4}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{7}{4}} c^{\frac{5}{4}}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{7}{4}} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(15/2)/(c*x**4+b*x**2)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.223664, size = 223, normalized size = 0.92 \[ \frac{-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{7/4}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{7/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{b^{7/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{b^{7/4}}+\frac{8 \sqrt [4]{c} \sqrt{x}}{b^2+b c x^2}-\frac{32 \sqrt [4]{c} \sqrt{x}}{\left (b+c x^2\right )^2}}{128 c^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(15/2)/(b*x^2 + c*x^4)^3,x]

[Out]

((-32*c^(1/4)*Sqrt[x])/(b + c*x^2)^2 + (8*c^(1/4)*Sqrt[x])/(b^2 + b*c*x^2) - (6*
Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(7/4) + (6*Sqrt[2]*ArcT
an[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(7/4) - (3*Sqrt[2]*Log[Sqrt[b] - Sq
rt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(7/4) + (3*Sqrt[2]*Log[Sqrt[b] + S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(7/4))/(128*c^(5/4))

_______________________________________________________________________________________

Maple [A]  time = 0.022, size = 169, normalized size = 0.7 \[ 2\,{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ( 1/32\,{\frac{{x}^{5/2}}{b}}-{\frac{3\,\sqrt{x}}{32\,c}} \right ) }+{\frac{3\,\sqrt{2}}{128\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{64\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}}{64\,{b}^{2}c}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(15/2)/(c*x^4+b*x^2)^3,x)

[Out]

2*(1/32/b*x^(5/2)-3/32*x^(1/2)/c)/(c*x^2+b)^2+3/128/c/b^2*(b/c)^(1/4)*2^(1/2)*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/c/b^2*(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+3
/64/c/b^2*(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)

_______________________________________________________________________________________

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(x^(15/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.284425, size = 323, normalized size = 1.33 \[ -\frac{12 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} c \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}}}{\sqrt{b^{4} c^{2} \sqrt{-\frac{1}{b^{7} c^{5}}} + x} + \sqrt{x}}\right ) - 3 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (b^{2} c \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} + \sqrt{x}\right ) + 3 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (-b^{2} c \left (-\frac{1}{b^{7} c^{5}}\right )^{\frac{1}{4}} + \sqrt{x}\right ) - 4 \,{\left (c x^{2} - 3 \, b\right )} \sqrt{x}}{64 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

-1/64*(12*(b*c^3*x^4 + 2*b^2*c^2*x^2 + b^3*c)*(-1/(b^7*c^5))^(1/4)*arctan(b^2*c*
(-1/(b^7*c^5))^(1/4)/(sqrt(b^4*c^2*sqrt(-1/(b^7*c^5)) + x) + sqrt(x))) - 3*(b*c^
3*x^4 + 2*b^2*c^2*x^2 + b^3*c)*(-1/(b^7*c^5))^(1/4)*log(b^2*c*(-1/(b^7*c^5))^(1/
4) + sqrt(x)) + 3*(b*c^3*x^4 + 2*b^2*c^2*x^2 + b^3*c)*(-1/(b^7*c^5))^(1/4)*log(-
b^2*c*(-1/(b^7*c^5))^(1/4) + sqrt(x)) - 4*(c*x^2 - 3*b)*sqrt(x))/(b*c^3*x^4 + 2*
b^2*c^2*x^2 + b^3*c)

_______________________________________________________________________________________

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**(15/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278388, size = 285, normalized size = 1.18 \[ \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{2}} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{2}} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{2}} - \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{2}} + \frac{c x^{\frac{5}{2}} - 3 \, b \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(15/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

3/64*sqrt(2)*(b*c^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/
(b/c)^(1/4))/(b^2*c^2) + 3/64*sqrt(2)*(b*c^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)
*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/(b^2*c^2) + 3/128*sqrt(2)*(b*c^3)^(1/4)*l
n(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^2) - 3/128*sqrt(2)*(b*c^3)
^(1/4)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^2) + 1/16*(c*x^(5
/2) - 3*b*sqrt(x))/((c*x^2 + b)^2*b*c)