3.341 \(\int \frac{x^{19/2}}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=239 \[ -\frac{5 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{3/4} c^{9/4}}+\frac{5 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{3/4} c^{9/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{3/4} c^{9/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{3/4} c^{9/4}}-\frac{5 \sqrt{x}}{16 c^2 \left (b+c x^2\right )}-\frac{x^{5/2}}{4 c \left (b+c x^2\right )^2} \]

[Out]

-x^(5/2)/(4*c*(b + c*x^2)^2) - (5*Sqrt[x])/(16*c^2*(b + c*x^2)) - (5*ArcTan[1 -
(Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(3/4)*c^(9/4)) + (5*ArcTan[1 +
(Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(3/4)*c^(9/4)) - (5*Log[Sqrt[
b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(3/4)*c^(9/4))
+ (5*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(
3/4)*c^(9/4))

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Rubi [A]  time = 0.391573, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{5 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{3/4} c^{9/4}}+\frac{5 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{3/4} c^{9/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{3/4} c^{9/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{3/4} c^{9/4}}-\frac{5 \sqrt{x}}{16 c^2 \left (b+c x^2\right )}-\frac{x^{5/2}}{4 c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(19/2)/(b*x^2 + c*x^4)^3,x]

[Out]

-x^(5/2)/(4*c*(b + c*x^2)^2) - (5*Sqrt[x])/(16*c^2*(b + c*x^2)) - (5*ArcTan[1 -
(Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(3/4)*c^(9/4)) + (5*ArcTan[1 +
(Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(3/4)*c^(9/4)) - (5*Log[Sqrt[
b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(3/4)*c^(9/4))
+ (5*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(
3/4)*c^(9/4))

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Rubi in Sympy [A]  time = 72.7818, size = 224, normalized size = 0.94 \[ - \frac{x^{\frac{5}{2}}}{4 c \left (b + c x^{2}\right )^{2}} - \frac{5 \sqrt{x}}{16 c^{2} \left (b + c x^{2}\right )} - \frac{5 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{3}{4}} c^{\frac{9}{4}}} + \frac{5 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{3}{4}} c^{\frac{9}{4}}} - \frac{5 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{3}{4}} c^{\frac{9}{4}}} + \frac{5 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{3}{4}} c^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(19/2)/(c*x**4+b*x**2)**3,x)

[Out]

-x**(5/2)/(4*c*(b + c*x**2)**2) - 5*sqrt(x)/(16*c**2*(b + c*x**2)) - 5*sqrt(2)*l
og(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(3/4)*c**(9
/4)) + 5*sqrt(2)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(1
28*b**(3/4)*c**(9/4)) - 5*sqrt(2)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(6
4*b**(3/4)*c**(9/4)) + 5*sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64
*b**(3/4)*c**(9/4))

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Mathematica [A]  time = 0.201094, size = 221, normalized size = 0.92 \[ \frac{-\frac{5 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{3/4}}+\frac{5 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{3/4}}-\frac{10 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{b^{3/4}}+\frac{10 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{b^{3/4}}+\frac{32 b \sqrt [4]{c} \sqrt{x}}{\left (b+c x^2\right )^2}-\frac{72 \sqrt [4]{c} \sqrt{x}}{b+c x^2}}{128 c^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(19/2)/(b*x^2 + c*x^4)^3,x]

[Out]

((32*b*c^(1/4)*Sqrt[x])/(b + c*x^2)^2 - (72*c^(1/4)*Sqrt[x])/(b + c*x^2) - (10*S
qrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) + (10*Sqrt[2]*ArcT
an[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(3/4) - (5*Sqrt[2]*Log[Sqrt[b] - Sq
rt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4) + (5*Sqrt[2]*Log[Sqrt[b] + S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(3/4))/(128*c^(9/4))

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Maple [A]  time = 0.022, size = 170, normalized size = 0.7 \[ 2\,{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ( -{\frac{9\,{x}^{5/2}}{32\,c}}-{\frac{5\,b\sqrt{x}}{32\,{c}^{2}}} \right ) }+{\frac{5\,\sqrt{2}}{128\,b{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{5\,\sqrt{2}}{64\,b{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}}{64\,b{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(19/2)/(c*x^4+b*x^2)^3,x)

[Out]

2*(-9/32*x^(5/2)/c-5/32*b*x^(1/2)/c^2)/(c*x^2+b)^2+5/128/c^2*(b/c)^(1/4)/b*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)))+5/64/c^2*(b/c)^(1/4)/b*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)
+1)+5/64/c^2*(b/c)^(1/4)/b*2^(1/2)*arctan(2^(1/2)/(b/c)^(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(x^(19/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285217, size = 319, normalized size = 1.33 \[ -\frac{20 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )} \left (-\frac{1}{b^{3} c^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{b c^{2} \left (-\frac{1}{b^{3} c^{9}}\right )^{\frac{1}{4}}}{\sqrt{b^{2} c^{4} \sqrt{-\frac{1}{b^{3} c^{9}}} + x} + \sqrt{x}}\right ) - 5 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )} \left (-\frac{1}{b^{3} c^{9}}\right )^{\frac{1}{4}} \log \left (b c^{2} \left (-\frac{1}{b^{3} c^{9}}\right )^{\frac{1}{4}} + \sqrt{x}\right ) + 5 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )} \left (-\frac{1}{b^{3} c^{9}}\right )^{\frac{1}{4}} \log \left (-b c^{2} \left (-\frac{1}{b^{3} c^{9}}\right )^{\frac{1}{4}} + \sqrt{x}\right ) + 4 \,{\left (9 \, c x^{2} + 5 \, b\right )} \sqrt{x}}{64 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(19/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

-1/64*(20*(c^4*x^4 + 2*b*c^3*x^2 + b^2*c^2)*(-1/(b^3*c^9))^(1/4)*arctan(b*c^2*(-
1/(b^3*c^9))^(1/4)/(sqrt(b^2*c^4*sqrt(-1/(b^3*c^9)) + x) + sqrt(x))) - 5*(c^4*x^
4 + 2*b*c^3*x^2 + b^2*c^2)*(-1/(b^3*c^9))^(1/4)*log(b*c^2*(-1/(b^3*c^9))^(1/4) +
sqrt(x)) + 5*(c^4*x^4 + 2*b*c^3*x^2 + b^2*c^2)*(-1/(b^3*c^9))^(1/4)*log(-b*c^2*
(-1/(b^3*c^9))^(1/4) + sqrt(x)) + 4*(9*c*x^2 + 5*b)*sqrt(x))/(c^4*x^4 + 2*b*c^3*
x^2 + b^2*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**(19/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278622, size = 282, normalized size = 1.18 \[ \frac{5 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b c^{3}} + \frac{5 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b c^{3}} + \frac{5 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b c^{3}} - \frac{5 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b c^{3}} - \frac{9 \, c x^{\frac{5}{2}} + 5 \, b \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(19/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

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