3.326 \(\int \frac{1}{x^{7/2} \left (b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=230 \[ -\frac{c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{13/4}}+\frac{c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{13/4}}+\frac{c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{13/4}}-\frac{c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{13/4}}-\frac{2 c^2}{b^3 \sqrt{x}}+\frac{2 c}{5 b^2 x^{5/2}}-\frac{2}{9 b x^{9/2}} \]

[Out]

-2/(9*b*x^(9/2)) + (2*c)/(5*b^2*x^(5/2)) - (2*c^2)/(b^3*Sqrt[x]) + (c^(9/4)*ArcT
an[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(13/4)) - (c^(9/4)*ArcTan[
1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(13/4)) - (c^(9/4)*Log[Sqrt[b
] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(13/4)) + (c^(9/4
)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(13/4
))

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Rubi [A]  time = 0.430817, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{13/4}}+\frac{c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{13/4}}+\frac{c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{13/4}}-\frac{c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{13/4}}-\frac{2 c^2}{b^3 \sqrt{x}}+\frac{2 c}{5 b^2 x^{5/2}}-\frac{2}{9 b x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(b*x^2 + c*x^4)),x]

[Out]

-2/(9*b*x^(9/2)) + (2*c)/(5*b^2*x^(5/2)) - (2*c^2)/(b^3*Sqrt[x]) + (c^(9/4)*ArcT
an[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(13/4)) - (c^(9/4)*ArcTan[
1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*b^(13/4)) - (c^(9/4)*Log[Sqrt[b
] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(13/4)) + (c^(9/4
)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(13/4
))

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Rubi in Sympy [A]  time = 80.3726, size = 219, normalized size = 0.95 \[ - \frac{2}{9 b x^{\frac{9}{2}}} + \frac{2 c}{5 b^{2} x^{\frac{5}{2}}} - \frac{2 c^{2}}{b^{3} \sqrt{x}} - \frac{\sqrt{2} c^{\frac{9}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{13}{4}}} + \frac{\sqrt{2} c^{\frac{9}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{13}{4}}} + \frac{\sqrt{2} c^{\frac{9}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{13}{4}}} - \frac{\sqrt{2} c^{\frac{9}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(7/2)/(c*x**4+b*x**2),x)

[Out]

-2/(9*b*x**(9/2)) + 2*c/(5*b**2*x**(5/2)) - 2*c**2/(b**3*sqrt(x)) - sqrt(2)*c**(
9/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(4*b**(13/4))
+ sqrt(2)*c**(9/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)
/(4*b**(13/4)) + sqrt(2)*c**(9/4)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2
*b**(13/4)) - sqrt(2)*c**(9/4)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b*
*(13/4))

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Mathematica [A]  time = 0.104406, size = 234, normalized size = 1.02 \[ \frac{72 b^{5/4} c x^2-40 b^{9/4}-45 \sqrt{2} c^{9/4} x^{9/2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} c^{9/4} x^{9/2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+90 \sqrt{2} c^{9/4} x^{9/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-90 \sqrt{2} c^{9/4} x^{9/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-360 \sqrt [4]{b} c^2 x^4}{180 b^{13/4} x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(b*x^2 + c*x^4)),x]

[Out]

(-40*b^(9/4) + 72*b^(5/4)*c*x^2 - 360*b^(1/4)*c^2*x^4 + 90*Sqrt[2]*c^(9/4)*x^(9/
2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 90*Sqrt[2]*c^(9/4)*x^(9/2)*Ar
cTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 45*Sqrt[2]*c^(9/4)*x^(9/2)*Log[Sqr
t[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 45*Sqrt[2]*c^(9/4)*x^(9/2)
*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(180*b^(13/4)*x^(9/
2))

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Maple [A]  time = 0.016, size = 169, normalized size = 0.7 \[ -{\frac{{c}^{2}\sqrt{2}}{4\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{{c}^{2}\sqrt{2}}{2\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{{c}^{2}\sqrt{2}}{2\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{2}{9\,b}{x}^{-{\frac{9}{2}}}}-2\,{\frac{{c}^{2}}{{b}^{3}\sqrt{x}}}+{\frac{2\,c}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(c*x^4+b*x^2),x)

[Out]

-1/4*c^2/b^3/(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)))-1/2*c^2/b^3/(b/c)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-1/2*c^2/b^3/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2
)/(b/c)^(1/4)*x^(1/2)-1)-2/9/b/x^(9/2)-2*c^2/b^3/x^(1/2)+2/5*c/b^2/x^(5/2)

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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(1/((c*x^4 + b*x^2)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28539, size = 244, normalized size = 1.06 \[ -\frac{180 \, b^{3} x^{\frac{9}{2}} \left (-\frac{c^{9}}{b^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{10} \left (-\frac{c^{9}}{b^{13}}\right )^{\frac{3}{4}}}{c^{7} \sqrt{x} + \sqrt{-b^{7} c^{9} \sqrt{-\frac{c^{9}}{b^{13}}} + c^{14} x}}\right ) + 45 \, b^{3} x^{\frac{9}{2}} \left (-\frac{c^{9}}{b^{13}}\right )^{\frac{1}{4}} \log \left (b^{10} \left (-\frac{c^{9}}{b^{13}}\right )^{\frac{3}{4}} + c^{7} \sqrt{x}\right ) - 45 \, b^{3} x^{\frac{9}{2}} \left (-\frac{c^{9}}{b^{13}}\right )^{\frac{1}{4}} \log \left (-b^{10} \left (-\frac{c^{9}}{b^{13}}\right )^{\frac{3}{4}} + c^{7} \sqrt{x}\right ) + 180 \, c^{2} x^{4} - 36 \, b c x^{2} + 20 \, b^{2}}{90 \, b^{3} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^(7/2)),x, algorithm="fricas")

[Out]

-1/90*(180*b^3*x^(9/2)*(-c^9/b^13)^(1/4)*arctan(b^10*(-c^9/b^13)^(3/4)/(c^7*sqrt
(x) + sqrt(-b^7*c^9*sqrt(-c^9/b^13) + c^14*x))) + 45*b^3*x^(9/2)*(-c^9/b^13)^(1/
4)*log(b^10*(-c^9/b^13)^(3/4) + c^7*sqrt(x)) - 45*b^3*x^(9/2)*(-c^9/b^13)^(1/4)*
log(-b^10*(-c^9/b^13)^(3/4) + c^7*sqrt(x)) + 180*c^2*x^4 - 36*b*c*x^2 + 20*b^2)/
(b^3*x^(9/2))

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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(1/x**(7/2)/(c*x**4+b*x**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276245, size = 269, normalized size = 1.17 \[ -\frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{4}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{4}} - \frac{2 \,{\left (45 \, c^{2} x^{4} - 9 \, b c x^{2} + 5 \, b^{2}\right )}}{45 \, b^{3} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^(7/2)),x, algorithm="giac")

[Out]

-1/2*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^4 - 1/2*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^4 + 1/4*sqrt(2)*(b*c^3)^(3/4)*ln(sqrt(2)*sqrt(
x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 - 1/4*sqrt(2)*(b*c^3)^(3/4)*ln(-sqrt(2)*sqrt
(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 - 2/45*(45*c^2*x^4 - 9*b*c*x^2 + 5*b^2)/(b^
3*x^(9/2))