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

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

[Out]

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

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Rubi [A]  time = 0.391432, 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 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{5/4} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{5/4} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{5/4} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{5/4} c^{7/4}}+\frac{3 x^{3/2}}{16 b c \left (b+c x^2\right )}-\frac{x^{3/2}}{4 c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 72.7242, size = 224, normalized size = 0.93 \[ - \frac{x^{\frac{3}{2}}}{4 c \left (b + c x^{2}\right )^{2}} + \frac{3 x^{\frac{3}{2}}}{16 b c \left (b + c x^{2}\right )} + \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{5}{4}} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{5}{4}} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{5}{4}} c^{\frac{7}{4}}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{5}{4}} c^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**(3/2)/(4*c*(b + c*x**2)**2) + 3*x**(3/2)/(16*b*c*(b + c*x**2)) + 3*sqrt(2)*l
og(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(5/4)*c**(7
/4)) - 3*sqrt(2)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(1
28*b**(5/4)*c**(7/4)) - 3*sqrt(2)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(6
4*b**(5/4)*c**(7/4)) + 3*sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64
*b**(5/4)*c**(7/4))

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Mathematica [A]  time = 0.21586, size = 223, normalized size = 0.92 \[ \frac{\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{5/4}}-\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{b^{5/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{b^{5/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{b^{5/4}}+\frac{24 c^{3/4} x^{3/2}}{b^2+b c x^2}-\frac{32 c^{3/4} x^{3/2}}{\left (b+c x^2\right )^2}}{128 c^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*c^(3/4)*x^(3/2))/(b + c*x^2)^2 + (24*c^(3/4)*x^(3/2))/(b^2 + b*c*x^2) - (6
*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(5/4) + (6*Sqrt[2]*Arc
Tan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(5/4) + (3*Sqrt[2]*Log[Sqrt[b] - S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(5/4) - (3*Sqrt[2]*Log[Sqrt[b] +
Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(5/4))/(128*c^(7/4))

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Maple [A]  time = 0.02, size = 169, normalized size = 0.7 \[ 2\,{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ({\frac{3\,{x}^{7/2}}{32\,b}}-1/32\,{\frac{{x}^{3/2}}{c}} \right ) }+{\frac{3\,\sqrt{2}}{128\,b{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}}{64\,b{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}}{64\,b{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(17/2)/(c*x^4+b*x^2)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282984, size = 335, normalized size = 1.38 \[ \frac{12 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{5} c^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} c^{5} \left (-\frac{1}{b^{5} c^{7}}\right )^{\frac{3}{4}}}{\sqrt{-b^{3} c^{3} \sqrt{-\frac{1}{b^{5} c^{7}}} + x} + \sqrt{x}}\right ) + 3 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{5} c^{7}}\right )^{\frac{1}{4}} \log \left (b^{4} c^{5} \left (-\frac{1}{b^{5} c^{7}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) - 3 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )} \left (-\frac{1}{b^{5} c^{7}}\right )^{\frac{1}{4}} \log \left (-b^{4} c^{5} \left (-\frac{1}{b^{5} c^{7}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) + 4 \,{\left (3 \, c x^{3} - b x\right )} \sqrt{x}}{64 \,{\left (b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(17/2)/(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^5*c^7))^(1/4)*arctan(b^4*c^5
*(-1/(b^5*c^7))^(3/4)/(sqrt(-b^3*c^3*sqrt(-1/(b^5*c^7)) + x) + sqrt(x))) + 3*(b*
c^3*x^4 + 2*b^2*c^2*x^2 + b^3*c)*(-1/(b^5*c^7))^(1/4)*log(b^4*c^5*(-1/(b^5*c^7))
^(3/4) + sqrt(x)) - 3*(b*c^3*x^4 + 2*b^2*c^2*x^2 + b^3*c)*(-1/(b^5*c^7))^(1/4)*l
og(-b^4*c^5*(-1/(b^5*c^7))^(3/4) + sqrt(x)) + 4*(3*c*x^3 - b*x)*sqrt(x))/(b*c^3*
x^4 + 2*b^2*c^2*x^2 + b^3*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**(17/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280892, size = 286, normalized size = 1.18 \[ \frac{3 \, c x^{\frac{7}{2}} - b x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b c} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{4}} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{2} c^{4}} - \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{4}} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{2} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(3*c*x^(7/2) - b*x^(3/2))/((c*x^2 + b)^2*b*c) + 3/64*sqrt(2)*(b*c^3)^(3/4)*
arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^2*c^4) + 3/
64*sqrt(2)*(b*c^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(
b/c)^(1/4))/(b^2*c^4) - 3/128*sqrt(2)*(b*c^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/
4) + x + sqrt(b/c))/(b^2*c^4) + 3/128*sqrt(2)*(b*c^3)^(3/4)*ln(-sqrt(2)*sqrt(x)*
(b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^4)